Getting the lowest score possible
Colin Fahey

1. Introduction

This document describes my effort to get the lowest score possible on the SAT exam.

2. SAT

The SAT is a test designed to predict how well a high school student would perform as a freshman at a college or university in the United States of America (USA).
More than 1000 (80%) of colleges and universities in the United States of America (USA) accept or require SAT scores from people applying for admission.
The SAT was first administered on 1926.6.23 to 8040 people.
In the 2003-2004 SAT testing year, 1419007 high school seniors took the SAT.
On 2005.3.12, a new version of the SAT was administered for the first time to approximately 300000 people.

3. Trust only the College Board for information about the SAT

The College Board creates all versions of the SAT, and is the only authority on all aspects of the SAT.
The College Board Internet site describes the content of the SAT and various testing administration conditions and policies.

4. SAT question types

This section describes all question types on the SAT, for each of the three divisions of the SAT: mathematics (M), critical reading (CR), and writing (W).
All sample questions shown here appeared on the SAT version with form code BWBA that was administered on 2005.3.12.
Most of the sample questions shown here have a difficulty rating of "5", the highest difficulty rating on a scale from 1 through 5.
Answers to the sample questions appear at the end of this section.

4.1 Math (M)

This section describes questions types appearing in the math (M) division of the SAT.

4.1.1 General instructions

This section shows the general math instructions, and the instructions for "student-produced response" grids.

4.1.2 Number and operations

The following question is an example of the "number and operations" type of math question.

4.1.3 Algebra and functions

The following question is an example of the "algebra and functions" type of math question.

4.1.4 Geometry and measurement

The following question is an example of the "geometry and measurement" type of math question.

4.1.5 Data analysis, statistics, and probability

The following question is an example of the "data analysis, statistics, and probability" type of math question.

4.2 Critical reading (CR)

This section describes questions types appearing in the critical reading (CR) division of the SAT.

4.2.1 Passage-based reading

The following question is an example of the "passage-based reading" type of critical reading question.

4.2.2 Sentence completion

The following question is an example of the "sentence completion" type of critical reading question.

4.3 Writing (W)

This section describes questions types appearing in the writing (W) division of the SAT.

4.3.1 Identifying sentence errors

The following question is an example of the "identifying sentence errors" type of writing question.

4.3.2 Improving sentences

The following question is an example of the "improving sentences" type of writing question.

4.3.3 Improving paragraphs

The following question is an example of the "improving paragraphs" type of writing question.

4.3.4 Essay

The following question is an example of the "essay" type of writing question.

4.4 Answers to the sample questions

math (M)

  number and operations                        answer: C

  algebra and functions                        answer: 5/2  or  2.5

  geometry and measurement                     answer: A

  data analysis, statistics, and probability   answer: C

critical reading (CR)

  passage-based reading                        answer: B

  sentence completion                          answer: E

writing (W)

  identifying sentence errors                  answer: A

  improving sentences                          answer: D

  improving paragraphs                         answer: C

  essay                                        answer: Yes

5. SAT structure

5.1 SAT question raw score points by format

question format raw points
if wrong
raw points
if omitted
raw points
if correct
5-choice (-1/4) 0 +1
12700-choice 0 0 +1
essay 0 0 +2 ... +12

5.2 SAT score structure by division

division question
5-choice 44 -11 +44
12700-choice 10 0 +10
5-choice 67 -17 +67
5-choice 49 -12 +49
essay 1 0 +12

5.3 SAT question totals by format

question format total
5-choice 160
12700-choice 10
essay 1

5.4 SAT question subjects by division

division question
questions division
number and operations 11 ... 13 54
algebra and functions 19 ... 21
geometry and measurement 14 ... 16
data analysis, statistics, and probability 6 ... 7
passage-based reading extended reasoning 36 ... 40 48 67
literal comprehension 4 ... 6
vocabulary in context 4 ... 6
sentence completion 19
improving sentence errors 18 49
improving sentences 25
improving paragraphs 6
essay 1 1

5.5 SAT chronological structure

(1) There are ten, independently-timed sections, with the following sequence of durations in minutes: {25,25,25,25,25,25,25,20,20,10}, for a total testing duration of 225 minutes (3 hours, 45 minutes).
(2) There is a five-minute break (leaving the room to go to the bathroom is allowed) after section #2, and a one-minute "stretching break" (leaving the room is not allowed) after section #4, and another five-minute break (leaving the room to go to the bathroom is allowed) after section #6.
(3) Section #1 is always the essay section of the Writing (W) division.
(4) Section #10 is always a 14-question section of the Writing (W) division.
(5) Sections #8 and #9 always include a 16-question section of the Math (M) division, and a 19-question section of the Critical Reading (CR) division, but in either of the two possible orderings.
(6) Sections {2,3,4,5,6,7} always include: two 24-question sections from the Critical Reading (CR) division, one 20-question section from the Math (M) division, one 18-question section from the Math (M) division, one 35-question section from the Writing (W) division, and one "variable" section that has the same format as one of the other sections in the set of these six sections.
The order of the section kinds is "random", and the identity of the "variable" section is intended to not be discovered while taking the test.
I took the SAT on 2005.3.12.
The following was the chronology of my particular test day experience:
section duration
division total
1 25 min Writing (W) 1 (essay) essay is always first
2 25 min Math (M) 18 8(5-choice);10(12700-choice)
(BREAK) 5 min ---- ---- long/bathroom break
3 25 min Writing (W) 35 sent. errors, imp. paragraphs
4 25 min Critical Reading (CR) 23 passages and sentence comp.
(BREAK) 1 min ---- ---- short/stretch break
5 25 min Math (M) 20 ----
6 25 min Critical Reading (CR) 25 long reading passage!
(BREAK) 5 min ---- ---- long/bathroom break
7 25 min ***VARIABLE*** ???? ----
8 20 min Math (M) 16 geometry; number and op.
9 20 min Critical Reading (CR) 19 (1-6;7-19)
10 10 min Writing (W) 14 always last; improve sentences
The book entitled "The Official SAT STUDY GUIDE: For the New SAT", published by the College Board, copyright 2004, has eight practice SATs.
Here are the chronologies of those eight practice SATs:
section number
#1 #2 #3 #4 #5 #6 #7 #8 #9 #10
#1 WE CR24 M20 VAR CR24 M18 W35 CR19 M16 W14
#2 WE CR24 M20 VAR CR24 M18 W35 CR19 M16 W14
#3 WE M20 CR24 M18 VAR W35 CR24 M16 CR19 W14
#4 WE M20 CR24 M18 VAR W35 CR24 M16 CR19 W14
#5 WE CR24 M18 W35 CR24 VAR M20 CR19 M16 W14
#6 WE CR24 M18 W35 CR24 VAR M20 CR19 M16 W14
#7 WE M18 W35 CR24 M20 CR24 VAR M16 CR19 W14
#8 WE M18 W35 CR24 M20 CR24 VAR M16 CR19 W14
The chronologies of these eight practice SATs only illustrate possible orders of the sections, given the constraints.
One should not try to form other conclusions based on these chronologies of the practice tests; this sample size is very small relative to the large number of possible chronologies, and the College Board has no incentive to describe any additional constraints they might use to form an acceptable chronology.
For example, although the variable section appears in sections {4,5,6,7} in the practice tests listed above, there is no basis for concluding that it is not just as likely that the variable section can appear in section #2 or section #3.
Also, there is no basis for concluding that sections within the same division won't ever appear in consecutive sections in the chronology.
For example, there might be a version of the SAT with two consecutive sections in the Math (M) division.

5.6 Determining the "variable" section while taking the SAT

One of the sections {2,3,4,5,6,7} will be for research purposes only and will not be given a score.
The section used for research purposes is named the "variable" section.
Consider the non-variable sections that must appear in the set of sections {2,3,4,5,6,7}:
division              questions       total by division
Math (M)                 20              38
Math (M)                 18
Critical Reading (CR)    24 (+\-1)       48
Critical Reading (CR)    24 (+\-1)
Writing (W)              35              35
Therefore, the division with the variable section well be known as soon as one encounters:
(1) a third section in the Math (M) division;
(2) a third section in the Critical Reading (CR) division;
or, (3) a second section in the Writing (W) division.
This will happen while taking the SAT.
Therefore, one will know, before starting work on a section that matches one of the three cases, that there is a (1/3) chance (cases 1 and 2) or a (1/2) chance (case 3) that the current section is the variable section and will not be given a score.
Regardless of the sequence of sections, when a test taker encounters a section that proves which division has the variable section, the test taker has a (1/3) chance (cases 1 and 2) or a (1/2) chance (case 3) that the current section is the variable section.
Before gaining this information, the probability was only (1/6).
Also, the probability for all subsequent sections becomes zero.
Furthermore, if a math section with 20 questions has been encountered, and then another math section with 20 questions is encountered, then one knows, before starting work on the second math section with 20 questions, that the variable section is in the math division, and also that there is a (1/2) chance that the current section is the variable section and will not be given a score.
Similarly, encountering a math section with 18 questions, and later encountering another math section with 18 questions, leads to the same conclusions.
The probability of any subsequent section being the variable section becomes zero.
Okay, now consider this information leak from the perspective of the authors of the SAT.
The goal of the authors of the SAT is to have test takers work on the variable section with the same concern and effort given to all other sections of the SAT, giving the authors of the SAT a method of linking performance on the particular version of the SAT to performances on versions administered throughout the long history of the SAT.
Therefore, the authors of the SAT want to minimize the chance that a person taking the test will determine that a particular section is the variable section.
For example, the authors of the SAT probably would avoid having sections #2 and #3 be math sections with exactly 20 questions each, because the test taker would know, at the very beginning of section #3, that either section #2 or section #3 must be the variable section.
Also, the test taker would know that all subsequent sections will be scored.
In this hypothetical situation, the test taker gets information about the variable section at the earliest possible time in the sequence of sections in the SAT.
The test taker can use this information in a few ways.
If the test taker is consciously or tacitly "taking advantage" of the general (1/6) chance that each section, of the sections {2,3,4,5,6,7}, is the variable section, putting in only (5/6) of maximum personal effort to avoid wasting a full effort on the variable section, then the test taker would change this conservative strategy after learning that the sections {4,5,6,7} will be graded, investing full effort in to those sections.
The test taker might also take advantage of the (1/2) chance that section #3 is the variable section, putting in less effort, or skipping the section entirely and instead resting or doing work on another section (against SAT rules; don't cheat!).
It is my guess that the authors of the SAT delay conclusive evidence of the division containing the variable section until section #7.
This minimizes any advantage to the test taker.
Section #7 itself need not be the variable section, but I believe that delaying conclusive evidence of the division having the variable section until section #7 is best for the test authors.
( Note: On the 2005.3.12 administration of the SAT, and form code BWBA, section #7 itself happened to be variable section. )
In conclusion, there is a way to be certain which division has the variable section while taking the SAT -- and this information might offer, at the very least, some psychological relief (closure, or SATisfaction of morbid curiosity) for a person taking the SAT.

6. Calculating SAT scores

This section describes how to convert total numbers of correct and incorrect responses to scaled scores, for each of the three divisions of the SAT: mathematics (M), critical reading (CR), and writing (W).

6.1 Math (M)

This section describes the procedure to compute the raw and scaled scores for the math (M) division of the SAT.

6.1.1 Calculations


// [Answers left blank are neither counted as correct nor counted as wrong.]

// Number of math multiple-choice questions answered correctly
// [an integer from 0 through 44]:
int mathMCCorrect;

// Number of math multiple-choice questions answered incorrectly
// [an integer from 0 through 44]:
int mathMCWrong;

// Number of correct math "student-produced responses"
// [an integer from 0 through 10]:
int mathSPRCorrect;


// Overall number of correct answers
// [an integer from 0 through 54]:
int mathCorrect   =  ( mathMCCorrect + mathSPRCorrect );

// Overall number of incorrect answers (such that blank answers are ignored)
// [an integer from 0 through 44]:
int mathWrong     =  ( mathMCWrong );

// Raw score with fractional part
// [a decimal number from -11.0 through +54.0]
decimal mathRawFractional =
             (decimal) mathCorrect - ( (decimal) mathWrong / (decimal4 );

// Raw score rounded to the nearest integer
// [an integer from -11 through +54]:
int mathRawScore = Nearest( mathRawFractional );

// Scaled score
// [an integer, multiple of 10, from 200 through 800]:
// (The MathRawToScaledScore() function is shown below as a graph.)
int mathScaledScore =  MathRawToScaledScore( mathRawScore );

6.1.2 Graph

The following graph shows the conversion from a multiple-choice raw score (-11 through +54) to a scaled score (200 through 800, in multiples of 10) for the Math (M) division of the SAT.

6.2 Critical reading (CR)

This section describes the procedure to compute the raw and scaled scores for the critical reading (CR) division of the SAT.

6.2.1 Calculations


// [Answers left blank are neither counted as correct nor counted as wrong.]

// Number of critical reading multiple-choice questions answered correctly
// [an integer from 0 through 67]:
int criticalReadingCorrect;

// Number of critical reading multiple-choice questions answered
// incorrectly [an integer from 0 through 67]:
int criticalReadingWrong;


// Raw score with fractional part
// [a decimal number from -(67/4) = -16.75 through +67.0]:
decimal criticalReadingRawFractional =
                 (decimal) criticalReadingCorrect
              - ((decimal) criticalReadingWrong / (decimal4);

// Raw score rounded to the nearest integer
// [an integer from -17 through +67]:
int criticalReadingRawScore = Nearest( criticalReadingRawFractional );

// Scaled score
// [an integer, multiple of 10, from 200 through 800]:
// (The CriticalReadingRawToScaledScore() function is shown below
// as a graph.)
criticalReadingScaledScore =
              CriticalReadingRawToScaledScore( criticalReadingRawScore );

6.2.2 Graph

The following graph shows the conversion from a multiple-choice raw score (-17 through +67) to a scaled score (200 through 800, in multiples of 10) for the critical reading (CR) division of the SAT.

6.3 Writing (W)

This section describes the procedure to compute the raw and scaled scores for the writing (W) division of the SAT.

6.3.1 Calculations


// [Answers left blank are neither counted as correct nor counted as wrong.]

// Number of writing multiple-choice questions answered correctly
// [an integer from 0 through 49]:
int writingMCCorrect;

// Number of writing multiple-choice questions answered incorrectly
// [an integer from 0 through 49]:
int writingMCWrong;

// Essay score
// [an integer; zero, or, 2 through 12; { 0, 2..12 }]:
int writingEssayScore;


// Raw score with fractional part
// [a decimal number from -(49/4) = -12.25 through +49]:
decimal writingMCRawFractional =
               (decimal) writingMCCorrect
            - ((decimal) writingMCWrong / (decimal4);

// Raw score rounded to the nearest integer
// [an integer from -12 through +49]:
int writingMCRawScore = Nearest( writingMCRawFractional );

// Scaled score
// [an integer, from 20 through 80]:
// (The writingMCRawToScaledScore() function is shown below, as a graph.)
int writingMCScaledScore = writingMCRawToScaledScore( writingMCRawScore );

// Combined score
// [an integer, multiple of 10, from 200 through 800]:
// (The writingCSRawToScaledScore() function is shown below as a graph.)
int writingCSScaledScore =
        writingCSRawToScaledScore( writingMCRawScore, writingEssayScore );

6.3.2 Graphs

The following graph shows the conversion from a multiple-choice raw score (-12 through +49) to a scaled score (20 through 80) for the writing (W) division of the SAT.
The following graph shows the conversion from a multiple-choice raw score (-12 through +49), and the essay raw score (0, +2 ... +12), to a combined scaled score (200 through 800, in multiples of 10) for the Writing (W) division of the SAT.
Notice that the graph is a family of curves.
Thus, one finds the proper horizontal coordinate using the multiple-choice raw score, and then selects the proper curve using the essay score.
The point on that curve at the proper horizontal coordinate is the composite scaled score.
The graph is missing data for a small region of score combinations, marked by the question mark ("?") on the graph.
The College Board did not provide data for this region in the table in the Question and Answer Service (QAS) report.
I suppose the College Board doesn't think there will be many people who write competent essays (with a pair of scores adding to "6" or higher), and, at the same time, get a multiple-choice raw score less than "-2".
But, hey, it could happen.

7. SAT scaled score distributions from 2004

Although the following graphs pertain to an old SAT format, the distributions of scaled scores are likely to be maintained for the new SAT format.
The following graphs show the percentages of graduating seniors in 2004 whose scaled scores were within particular ranges, for the math and verbal divisions of the old SAT format (prior to the introduction of the new SAT on 2005.3.12).
These distributions were designed by the College Board, and achieved by computing, and using, appropriate "raw score to scaled score" conversion curves.
For the 2004 SAT testing year, the average math score was 518, with a standard deviation of 114.
For the 2004 SAT testing year, the average verbal score was 508, with a standard deviation of 112.

8. Analysis of "student-produced response" grid encodings

8.1 Introduction

The following image shows the format of the "student-produced response" grid as it appears on the SAT answer sheet.
Various consequences of this response grid format are also shown.

8.2 Fractions may always be avoided

It is always possible to encode a correct response value using a decimal format on the SAT.
Fractions may always be avoided.
There are encodable fractions whose exact decimal format encodings cannot fit in the space provided, but it is always acceptable to encode the values in a decimal format, as long as the decimal encoding is as precise as possible, given the limited space.
It is acceptable to "truncate" the decimal encoding, which involves simply stopping the writing of digits beyond the most-significant digits that fit in the space provided.
It is also acceptable to "round" the value, choosing the decimal encoding, among all decimal encodings that will fit in the space provided, that has a value that is closest to the exact value of the correct result.
The following image shows how a fractional value without a corresponding exact decimal format encoding may nonetheless be correctly encoded on the SAT in a decimal format after a process of either truncation or rounding.

8.3 All 22308 encodings

8.3.1 Introduction

The symbols available in the four columns of the "student-produced response" grid imply a total of ((11)*(13)*(13)*(12)) = 22308 encodings.
This section describes various classifications of the encodings.

8.3.2 Hierarchy of classifications for all 22308 encodings

The following chart shows a hierarchy of classifications of subsets of all possible encodings.
Blank (No Response; Skipped; Omitted)
No Fraction
Value is
No Decimal (implies Integer Value)
With Decimal
Integer Value
Non-integer Value
With Fraction
Integer Value
Non-integer Value
With Exact Encoding
Without Exact Encoding
Truncated is Closest
Truncated is not Closest
Syntax Error
Undefined Value

8.3.3 Blank

If the entire response is blank, the response is considered "omitted"; the test-taker decided not to respond to the corresponding question.
The response is given a score of zero points.
( An incorrect "student-produced response" is also given a score of zero points. )
Blank response:

     ( blank, blank, blank, blank )     "    "

8.3.4 Invalid syntax

Encoding the values of numbers requires a grammar, or syntax, so that a reader can unambiguously interpret the encoding as a specific numeric value.
The existence of a grammar implies the existence of encodings that are inconsistent with that grammar.
[ This generalization is not true for the all-inclusive, non-constraining, trivial "non-grammar grammar", often exploited in advertising, online chat, and spam messages. ]
The following encodings are examples of encodings that violate the implicit grammar of the "student-produced responses grid".
Examples of invalid syntax:

    Involving only punctuation and blanks:

        ( blank, blank, blank, point }     "   ."
        ( point, point, point, point )     "...."
        ( blank, slash, blank, blank )     " /  "
        ( blank, slash, slash, blank )     " // "
        ( point, slash, slash, point )     ".//."

        ( blank, slash, point, blank )     " /. "

          -->  News for nerds: slash-dot will never be
               a correct response on the SAT!

    Involving only digits and blanks:

        ( blank,     0, blank,     0 )     " 1 2"
        (     1, blank, blank,     2 )     "1  2"
        (     1,     2, blank,     3 )     "12 3"

    Involving only digits and points:

        (     1, point, point,     2 )     "1..2"
        (     1, point,     2, point )     "1.2."
        ( point,     1,     2, point )     ".12."

    Involving only digits, slashes, and blanks:

        ( blank, slash,     2, blank )     " /2 "
        ( blank, slash, blank,     2 )     " / 2"
        ( blank,     1, slash, blank )     " 1/ "

    Involving only digits, slashes, points, and blanks:

        ( point, slash, blank,     2 )     "./ 2"
        ( blank, slash,     2, point )     " /2."

8.3.5 Division-by-zero error

Some encodings express the idea of dividing a number by zero.
The result of dividing a number by zero cannot be defined without contradicting one or more axioms of mathematics.
Therefore, encodings which express the idea of dividing a number by zero cannot be interpreted as having any particular numerical value.
I wondered what would happen if I wrote " 1/0" as a "student-produced response" on the SAT.
Would I sabotage the SAT-grading computer software?
I wanted to try this out, but I didn't want to jeopardize getting my score report.
Example of the division-by-zero error:

     ( blank,     1, slash,     0 )     " 1/0"

8.3.6 Zero-over-zero error

Zero over zero is even more of a mathematical horror than division by zero.
Zero over zero is at the impossible crossroads of zero, the finite, and infinity!
Example of the zero-over-zero error:

     ( blank,     0, slash,     0 )     " 0/0"

8.3.7 Problem cases

Some encodings unambiguously specify numeric values, but might nonetheless be rejected by the computer software grading the SAT due to unconventional features.
In particular, the SAT grading software might not expect or allow for redundant or superfluous characteristics in an encoding.
The following encodings are unconventional, but unambiguous. It seems reasonable to expect that the SAT grading software would interpret such encodings in a way that yields the numeric value intended by the human who did the encoding, but perhaps these encodings would be rejected.
Examples of unconventional encodings that are probably accepted:

     ( blank,     0,     0,     0 )     " 000"
     ( blank,     0,     0, blank )     " 00 "
     ( point,     0,     0,     0 )     ".000"
     ( blank, point,     0,     0 )     " .00"
     ( blank,     0, point,     0 )     " 0.0"
     ( blank,     0,     0, point )     " 00."
     ( blank,     0, slash,     1 )     " 0/1"
     ( point,     0, slash,     1 )     ".0/1"
     ( blank,     0, slash,     9 )     " 0/9"

     ( blank,     0,     0,     1 )     " 001"
     ( blank,     0,     1, blank )     " 01 "
     ( blank,     1, point,     0 )     " 1.0"
     ( blank,     0,     1, point )     " 01."
     (     1, slash,     0,     1 )     "1/01"
     (     1, slash,     1, point )     "1/1."
     (     1, slash, blank,     1 )     "1/ 1"

8.3.8 Various statistics

The following table contains various statistics relating to the 22308 encodings.
Various subsets of the 22308 encodings:

  Total integer encodings      = (11229 + 1243 + 474 ) = 12946

        (Note that there are only 10000 distinct integer values.)

  Total non-integer encodings  = (3096         + 1893) =  4989

        (Note that only there are only 2700 non-integer values in
         a non-fraction encoding format.)

  (Note that total integer encodings, plus total non-integer encodings,
   plus the one blank encoding, adds to 17936, the total number of
   valid encodings.)


  Total distinct values that can be encoded (where values of fractions
    are exact) = 13526

        (Note that it is only necessary to use one of 12700 values
         to encode any response correctly (in a non-fraction format).
         Therefore, there are (13526 - 12700) = 826 distinct
         encodable values (via fractions) that are not exactly
         encodable in non-fraction format.)


  Total number of encodings with fractions whose values cannot be encoded
    exactly in a non-fraction format in the available space = 1347

        (Note that these particular 1347 encodings with fractions only
         include 826 distinct values.  For example, "1/11" and "2/22"
         are have equal values.)


  Summary of the 12700 distinct values sufficient to respond correctly
  to any question requiring a "student-produced response":

    Sub-Range       Total Values  Total Integers  Total Non-Integers
       0                   1           1                 0
    .001 --> .999        999           0               999
    1.00 --> 9.99        900           9               891
    10.0 --> 99.9        900          90               810
    100. --> 999.        900         900                 0
    1000 --> 9999       9000        9000                 0
       0 --> 9999      12700       10000              2700


8.3.9 File with all 22308 encodings

The following file contains a complete list of all 22308 encodings, along with the truncated values, truncation error values, closest decimal values, closest decimal value errors, and full classifications.

299727 bytes
MD5: 2ac0842c3bc726f446908cecc589c45e

8.3.10 Equivalent encodings

There are 22308 encodings, of which 17936 are acceptable.
The test-taker need only consider 12700 distinct numerical values, because it is always acceptable to avoid the use of encodings with fractions (digits separated by exactly one slash).
Clearly, some of the 12700 encodable values must have more than one equivalent encoding.
The average, (17936 encodings/12700 values) = 1.412 encodings per value, is informative, but the actual distribution of the number of encodings per value is very uneven.
For example, there are 77 ways to encode the value "1" exactly:
(Note: "_" indicates a blank space in the response.)

___1   __1_   _1__   1___
       __1.   _1._   1.__
              _1.0   1.0_

__01   _01_
_001   _01.

_1/1   1_/1   1/_1   1/01   1/1_   1/1.   1./1
_2/2   2_/2   2/_2   2/02   2/2_   2/2.   2./2
_3/3   3_/3   3/_3   3/03   3/3_   3/3.   3./3
_4/4   4_/4   4/_4   4/04   4/4_   4/4.   4./4
_5/5   5_/5   5/_5   5/05   5/5_   5/5.   5./5
_6/6   6_/6   6/_6   6/06   6/6_   6/6.   6./6
_7/7   7_/7   7/_7   7/07   7/7_   7/7.   7./7
_8/8   8_/8   8/_8   8/08   8/8_   8/8.   8./8
_9/9   9_/9   9/_9   9/09   9/9_   9/9.   9./9
The challenge of finding all possible encodings for a specified value reminds me of the obscure game "EQUATIONS: The Game of Creative Mathematics", part of the "WFF 'N PROOF"(*) family of "well-formed formula" (wff) logic games developed by Dr. Layman E. Allen (currently a professor at Yale Law School).
In the EQUATIONS game, the challenge is to use numbers (examples: "1", "2", "3", "4") and operators (examples: "-" (subtract), "-" (subtract), "*" (multiply), "/" (divide), "/" (divide)), to form as many expressions as possible to yield a goal value (example: "6").
The high-intelligence group named the "Mega Society", which requires each member to have an IQ of at least 176, has a journal entitled "Noesis" (Greek: "understanding"; "to perceive"), which featured, in "Issue 169", the following text (in a section entitled "Part 2 - "EQUATIONS" - continued"):
Dr. Allen mentioned in a telephone conversation that NASA rocket scientists were unable to get all twelve solutions to Elementary problem E1. Test yourself! The youngsters who have mastered Equations, however, plow through problems like these quickly and decisively. The problem with our normal mathematics education, Dr. Allen explained, is we are taught to “calculate” going forward but not to reason abstractly. The Equations game teaches the user to “think” and to be able to reason in reverse – given a conclusion, how are we able to get to this conclusion – or GOAL in the Math-Quest exercises? See how many solutions you can produce in, say, 15 minutes. It’s r-e-a-l-l-y easy. Right? In Elementary problem E1 we are only using two “-“ signs, one “x” sign, two “/” operators, and the numbers 1 through 4 (one each). If you get all 12 correct, the online GUI-based system will let you know with a blinking “CONGRATULATIONS” indicator. For those readers that send me their 12 correct, “unique” solutions, the editor will send them an additional wallet-size Mega Society card with their next issue of Noesis.
Puzzle "E1"
I think it would be funny to have standardized admission tests for religions, such as Christianity, so that the next time I am accosted by someone proselytizing their faith, I can ask to see credentials (such as a religious ID card).
Take a look at the quizzes at the "Landover Baptist Church" Internet site -- such as "The Bible Poop Quiz", "The Bible Logic Quiz", and the "God vs. Allah Quiz".
"Landover Baptist Church" quiz list
The Weekly World News, a (defunct, later resurrected) tabloid, had a cover on 2005 June 27 (issue Vol. 26, No. 42) that had a drawing of Moses on Sinai mountain holding two stone tablets, and the associated headline was: "10 MORE COMMANDMENTS FOUND! YOU WON'T BELIEVE WHAT THEY SAY!" Weekly World News (June 27, 2005; Vol. 26, No. 42)
There are { 34, 77, 48, 41, 36, 33, 30, 29, 30, 29, 25 } ways to exactly encode the values { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }, respectively.
The following graph shows the number of equivalent encodings as a function of the value to be encoded.
Notice the data point on the graph for the value "1", indicating 77 encodings, just as explicitly enumerated above.
It is a beautiful pattern, indicating the greatest degeneracy of encodings centered about the value "1", and a decrease of both degeneracy and resolution for values requiring more precision (digits).
For each integer value from 1000 through 9999, there is obviously only one encoding that will fit in the space provided.
For each integer value from 100 through 999, there are three equivalent encodings:
"_###", "###_", "###."; where "_" represents a blank, and "#" represents a digit.
For each of the smallest, non-zero values, ".001" through ".009", there is only one encoding.
Of the 17935 valid, non-blank encodings, only "1347" lack an exact decimal encoding.
These 1347 encodings (which have only 826 distinct exact values) are the only encodings that lead to non-zero truncation (or rounding) errors when encoded instead using one of the 12700 non-fractional values (particularly, the subset of 2700 non-integer values).
Thus, if one insists on encoding one of the 1347 encodings mentioned above in a non-fraction encoding instead, then one must choose to truncate or round (to the nearest encodable value).
Truncation means chopping off less-significant digits.
Therefore, the error that results is always negative.
(Observed) = ((Exact) + (Error)).
Or, (Error) = ((Observed) - (Exact)).

8.3.11 Truncation and closest errors

The following table contains various facts relating to the truncation and closest error values.
About Error Values

  Observed = (Actual + Error);   ==>   Error = (Observed - Actual);

  In this context, "Observed" corresponds to the value one interprets from
the encoding.

  In this context, "Actual" corresponds to the exact value one desires to
represent by an encoding.

  The only circumstance which leads to non-zero error is when one desires to
encode a value corresponding to an exactly encodable fraction but intentionally
avoids the use of fractions (which is always acceptable) and instead encodes
the value in a decimal format.  In this circumstance, one is forced to choose
either a TRUNCATED decimal value or CLOSEST decimal value to encode.  In this
circumstance, the truncated value (Observed) is always less than the desired
exact value (Actual), and thus results in an error that is always negative.
Meanwhile, the closest value is either the same as the truncated value, or is
equal to the truncated value plus one more unit in the least-significant digit
available in the encoding.  The absolute error of the closest value is always
less than or equal to the absolute value of the truncation error.

Interesting error values


  Maximum truncation error = 0.0 (for all exact integer and decimal encodings)

  Least-negative truncation error = ( (   .043   ) - (   4/   93) )
                                  = ( (  43/ 1000) - (   4/   93) )
                                  = ( (3999/93000) - (4000/93000) )
                                  = (           -1 / 93000        )
                                  = -0.00001075268...
     (occurs when "4/93" is encoded as the truncated decimal number ".043")

  Minimum truncation error        = ( ( 10.8     ) - (  98/    9) )
                                  = ( ( 108/   10) - (  98/    9) )
                                  = ( ( 972/   90) - ( 980/   90) )
                                  = (           -8 / 90           )
                                  = (           -4 / 45           )
                                  = -0.08888888888...
     (occurs when "98/9" is encoded as the truncated decimal number "10.8")



  Maximum closest error           = ( ( 10.6     ) - (  95/    9) )
                                  = ( ( 106/   10) - (  95/    9) )
                                  = ( ( 954/   90) - ( 950/   90) )
                                  = (           +4 / 90           )
                                  = (           +2 / 45           )
                                  = +0.04444444444...
     (occurs when "95/9" is encoded as the closest decimal number "10.6")
     *CAUTION: See the minimum closest error comments below.

  Least-positive closest error    = ( (   .011   ) - (   1/   91) )
                                  = ( (  11/ 1000) - (   1/   91) )
                                  = ( (1001/91000) - (1000/91000) )
                                  = (           +1 / 91000        )
                                  = +0.00001098901...
     (occurs when "1/91" is encoded as the closest decimal number ".011")

  Least-negative closest error    = ( (   .043   ) - (   4/   93) )
                                  = ( (  43/ 1000) - (   4/   93) )
                                  = ( (3999/93000) - (4000/93000) )
                                  = (           -1 / 93000        )
                                  = -0.00001075268...
     (occurs when "4/93" is encoded as the closest decimal number ".043")

  Minimum closest error           = ( ( 10.2     ) - (  41/    4) )
                                  = ( ( 102/   10) - (  41/    4) )
                                  = ( ( 408/   40) - ( 410/   40) )
                                  = (           -2 / 40           )
                                  = (           -1 / 20           )
                                  = -0.05
     (occurs when "41/4" is encoded as the closest decimal number "10.2")
     *CAUTION: If "10.3" was selected as closest in this boundary case, then
     the error would be +0.05, and would become the maximum closest error.



  The greatest absolute error that can be achieved is the absolute
  value of (-4/45) (or (-0.088888...)), resulting when the encoding
  "98/9" is encoded instead as the truncated decimal number "10.8".

  The smallest, non-zero, absolute error that can be achieved is
  the absolute value of (-1/93000) (or (-0.00001075268...)), resulting
  when the encoding "4/93" is encoded instead as the truncated and closest
  decimal number ".043".


8.4 Fun encodings : Morse code

One can use the "student-produced response" grids to encode private jokes, using Morse" code.

9. SAT preparation resources

The new SAT is very different from the previous version of the SAT.
I bought all of the books listed in this section to help me learn about the new SAT format.

9.1 "The Official SAT Study Guide: For the New SAT"

"The Official SAT Study Guide: For the New SAT"
I highly-recommend the book entitled "The Official SAT Study Guide: For the New SAT", created by the College Board.
If you buy any book on the SAT, this should be your first choice.
The College Board creates the actual SAT, so this book provides reliable information about the SAT.
The book also includes 8 actual SATs, thus offering the most realistic examples possible of what a person will see during an actual SAT test administration.
All SAT question types are described in detail in this book.
However, this book does not contain a substantial amount of teaching and practice material, so I recommend getting a second book, such as the Gruber book mentioned in the next section, to complement this book.
Information: "The Official SAT Study Guide: For the New SAT" (for the March 2005 test and beyond) (c)2004; College Entrance Examination Board, New York; Distributed by Henry Holt & Co., New York. (International Standard Book Number (ISBN): 0-87447-718-2); USA $19.95 (2005); Canadian $29.95 (2005).

9.2 "Gruber's Complete Preparation for the New SAT : 10th edition"

"Gruber's Complete Preparation for the New SAT : 10th edition"
I highly-recommend the book entitled "Gruber's Complete Preparation for the New SAT : 10th edition", as a second book to complement the book created by the College Board (mentioned in the previous section).
Gruber's book is filled with numerous lessons and practice questions for all question types on the SAT.
Information: "Gruber's Complete Preparation for the New SAT : 10th edition"; Gary R. Gruber, Ph.D. (; (c) 2005; ISBN: 0-06-058170-0, ISBN 1068-7262; USA $18.95 (2005); Canadian $26.95 (2005).

9.3 "Kaplan New SAT 2400 : Advanced Prep for Advanced Students : 2005 Edition"

"Kaplan New SAT 2400 : Advanced Prep for Advanced Students : 2005 Edition"
The book entitled "Kaplan New SAT 2400 : Advanced Prep for Advanced Students : 2005 Edition" is an interesting book.
This book is filled with challenging questions similar to those on the new SAT.
But the most interesting feature of this book is its advice on analyzing SAT questions and determining correct answers.
Many books that have example problems merely show the key steps to their solutions.
In this Kaplan book, the correct answers to example problems are described by indicating the succession of thoughts that a person "should have" when thinking about the problem.
This book describes "The Kaplan Method" for solving the various types of SAT questions, including the essay question.
I only glanced at a few of the methods, and I didn't have enough time before my test date to really try to learn any of these methods, but I can easily imagine that having such methods and procedures to structure one's use of time while solving problems on the SAT could be a big advance for people who haven't already internalized good, proven methods.
Information: "Kaplan New SAT 2400 : Advanced Prep for Advanced Students : 2005 Edition"; (c) 2004, Kaplan; (; ISBN: 0-7432-6035-X; USA $20 (2005); Canadian $29 (2005).

9.4 Practice by taking the actual SAT

One good form of preparation for the SAT is to register for, and take, the actual SAT.
The SAT is offered seven times each year.
It costs more to take the SAT than to buy various books about the SAT, but taking the SAT is the most realistic experience of the SAT imaginable.
In fact, the experience remains real unless a person explicitly asks to cancel the scoring of the answer sheet.
Optionally-real is pretty damned realistic!
If a person does not want an SAT attempt to be scored -- you coward! -- then, during any break or after the entire test, a person can ask the testing supervisor to cancel the scoring of the answer sheet.
I believe a person also has several days after a test date to contact the College Board to cancel the scoring of an answer sheet.
Heck, with some ingenuity you could probably cancel the scoring of someone else's answer sheet!
Given my lack of trust in the system, I would rip my answer sheet to shreds, burn it, and eat its ashes, all in an attempt to prevent the answer sheet from being optically scanned, maybe scored, maybe reported, maybe acquired by legal subpoena, or possibly leaked in an "accidental" disclosure of data.
The College Board can honor a promise to not report scores for an answer sheet, and can, at the same time, optically scan the answer sheet and use its contents for research purposes.
So, in my opinion, anything less than immediate disposal of the answer sheet introduces unnecessary risk.
Risk of what?
Neuroses don't tolerate such questions!

10. Mathematics used in this document

10.1 Introduction

This section of the document introduces mathematical terminology as it is used elsewhere in this document.
Unfortunately, establishing the formula for the "standard deviation of a sub-population of a multinomial distribution", for example, to allow the reader to interpret some of the brief comments I make in this document, requires intermediate terms and procedures to be defined.
I expressed the information in a way that can be typed using ordinary characters, and I used a syntax similar to the various syntaxes used in common computer languages, making the information more accessible to people who are familiar with the syntax and semantics of popular computer languages ( C, C++, Java, C#, ... ).

10.2 Basic mathematical definitions

(See appendix for mathematical definitions.)

10.3 Converting "real numbers" to integers

The following graphs show the various ways "real numbers" can be converted to integer numbers.
The College Board uses the "Nearest" integer method, with the additional convention of rounding the otherwise-ambiguous, halfway cases upward.
For this variety of "Nearest": Nearest( -0.5 ) = 0; Nearest( 0.5 ) = 1.

11. Guessing on the SAT

11.1 Introduction

Suppose a person decides to respond to all 171 scored questions on the SAT without opening the test book to actually read any of the questions.
This section describes calculations that predict raw scores based on responding to questions without actually knowing the content of the questions.

11.2 5-choice questions

Each 5-choice question is independent of all others.
If a response is selected at random, there is a (1/5) chance that the response is correct, and there is a (4/5) chance that the response is incorrect.
For a specific total number of questions, the probability of answering some specific number of questions correctly can be computed using the binomial distribution (see the math reference in a previous section).
Repeating the experiment, and responding randomly to all questions, will probably result in a different pattern of correct and incorrect responses, and a different overall total of correct and incorrect answers.
However, repeating the experiment many times, and graphing the results as a histogram, will show that the probability of each different result can be computed using the binomial distribution.
For the entire SAT, the number of 5-choice questions is N = 160.
The probability of answering correctly is p = (1/5).
Thus, the binomial distribution indicates that the expected number of correct responses is (N * p) = 32 responses, with a standard deviation of Sqrt( N * p * (1 - p) ) = 5.05 responses.
The probability of answering all 5-choice questions correctly on the SAT is Pow(p,N) = 1.46 * 10^(-112), while the probability of answering all 5-choice questions on the SAT incorrectly is Pow((1-p),N) = 3.12 * 10^(-16).
When choosing responses at random, the expected raw score is: Nearest( (N * p) - ((N - (N*p))/4) ) = Nearest( N * (1/5 - ((1 - (1/5))/4)) ) = Nearest( N * (1/5 - ((4/5)/4)) ) = Nearest( 0 ) = 0.
This means that wild guessing is expected to lead to a raw score of zero.
Leaving all questions blank results in the same raw score of zero.
However, the difference between wild guessing and leaving the test blank is in the variance!
Wild guessing on all N = 160 5-choice questions will lead to a variance that corresponds to a standard deviation of 5.05, meaning that there is roughly a 68% chance that the specific observed number of correct answers will lie between (32 - 5.5) = 26.5 and (32 + 5.5) = 37.5, or between 16.5% and 23.4% of all 5-choice questions.
The corresponding raw scores would be: Nearest(26.5 - ((160-26.5)/4)) = -7 and Nearest(37.5 - ((160-37.5)/4)) = +7.
On the 2005.3.12 administration of the SAT, the test version corresponding to form code BWBA had the following distribution of responses for the 160 questions having a 5-choice response format: total{"A","B","C","D","E"} = {33,28,38,29,32}.
The differences of these totals from the expected mean of 32 are: total{"A","B","C","D","E"} - {32,32,32,32,32} = {+1,-4,+6,-3,0}.
These values are consistent with the standard deviation of (5.059...) correct responses expected for 68% of randomly-selected SATs.
[ The means and standard deviations are found by using a "multinomial distribution", with 5 bins, with populations {n1,n2,n3,n4,n5} such that (n1+n2+n3+n4+n5) = N = 160, and with relative probabilities {p1,p2,p3,p4,p5} such that (p1+p2+p3+p4+p5) = 1 and such that in this situation p1=p2=p3=p4=p5="p"=(1/5);
leading to mean values of (N*p1) = (N*p2) = (N*p3) = (N*p4) = (N*p5) = (N*p) = (160/5) = 32, and standard deviations equal to (in this case) Sqrt(N*p*(1-p)) = (5.059...). ]
Note that no procedure used by a test-taker to select responses that involves completely ignoring the questions on the SAT will have an advantage or disadvantage relative to any other such procedure, because in such a situation the distribution of correct responses is totally random.
So, there is no difference between the test-taker selecting responses at random and, for example, the test-taker always choosing a specific response, such as "C".
There is no logical connection between the test-taker's arbitrary procedure and the random procedure used by the authors of the SAT to select correct responses for the question type having a 5-choice response.
Now, if, over many SATs, one was able to demonstrate that one answer choice was more common than others, such as "E" being repeatedly more common than the other options {"A","B","C","D"}, then this bias could be exploited.
I think the College Board is very conscious of the need to avoid such a bias.

11.3 12700-choice questions

Each "student-produced response" question type can be answered by using one of a set of 12700 distinct values (in an integer or decimal format).
If a person chooses to pick one of these 12700 values at random as a response to a "student-produced response" question type, the probability of answering the question correctly is at least p = (1/12700) = (0.0000787...).
Some questions of the "student-produced response" question type have a range of correct values (with a range of corresponding acceptable choices among the set of 12700 distinct values having decimal-format encoding).
In such cases, the probability of answering the question correctly is higher than p = (1/12700).
For example, a range such as "0 < x < .375" corresponds to (376-2) = 374 distinct decimal values that can be encoded, leading to a probability of p = (374/12700) = (0.029...) of answering correctly at random (using only the set of 12700 distinct decimal values as described above).
To be assured of getting the lowest raw score possible for the set of ten (10) questions of the "student-produced response" question type, one can simply leave all corresponding responses blank.
This is because both blank "responses" and incorrect responses are given scores of zero points for the "student-produced response" question type.
But where is the challenge in that?
The situation with 12700-choice questions is complicated by the fact that the correct answers are very likely to NOT be evenly distributed among all of the 12700 choices.
The math questions on the SAT often, by their very nature, lead to numeric values close to unity (one; "1").
Therefore, unless the test-taker chooses a response totally at random from the self-imposed limited set of 12700 distinct decimal values, then the probability of answering correctly will be biased from the nominal p = (1/12700) value.
For example, if a test-taker always responds with ".5", for example, then the chance of getting each question correct is much higher than, for example, always responding with "9999".
( Note: Always answering ".5" would have led to one correct answer on the 2005.3.12 SAT with form code BWBA. )
Here are the ten correct answers to the "student-produced response" type of questions on the 2005.6.12 SAT with form code BWBA, sorted by value:
{ "0 < x < 1", 1/3 (.333), 5/8 (.625), 5/2 (2.5), 3, 12, 75, 89, 200, 1600 }.
( Note: "0 < x < 1" indicates that any value between zero and one, not including zero itself or one itself, is a correct response. )
While it is not a good idea to use the College Board's book "The Official SAT Study Guide: For the New SAT" as "experimental evidence", I think it is worth noting the distribution of correct responses in that book, as shown in the following graph.
The inverse logarithm of the average of the logarithms of the correct values is "13.1", indicating a value having the same order of magnitude as the average order of magnitude of all values.
It is evident from the graph that values below 0.1 and above 1000 are less common than values in the other ranges.
However, this "evidence" is based on a book of contrived SAT examples.
However, I believe it is the general character of SAT math problems that leads to this distribution.
Therefore, I believe that the phenomenon of there being fewer correct answers at the extreme ends of the possible orders of magnitude will persist (unless the College Board reads this analysis and ensures the slight bias is reduced).
There isn't much incentive to work very hard at this bias removal, because a person who relies on any such bias is already faced with a very tiny probability of guessing the correct answer.

11.4 Essay question

This section considers the probability of "guessing" the "correct answer" to the essay question on the SAT.
The person taking the SAT is given the following instructions regarding the essay question:
"Think carefully about the issue presented in the following excerpt and the assignment below";
"Plan and write an essay in which you develop your point of view on [the stated] issue. Support your position with reasoning and examples taken from your reading, studies, experience, or observations."
Leaving the essay response area of the answer sheet blank will result in an essay score of zero.
An essay that is not written on the assigned topic will receive a score of zero.
An essay that is deemed illegible, after several attempts have been made to read it, will receive a score of zero.
An essay that is written on the assigned topic, and is mostly or completely without spelling, grammar, and usage errors, and presents an opinion about the assigned topic in an organized, coherent, and convincing manner, could receive a total score of (6 + 6) = 12 points.
Suppose a person writes an essay on the answer sheet without opening the test booklet to actually read the assigned question.
What is the expected score?
The "guessed" essay need not be blank, so it can avoid the rule of a blank essay automatically receiving a score of zero.
If the "guessed" essay was in the form of a picture (sketch, diagram, realistic, impressionistic, abstract, surreal, or whatever), it would probably be deemed a non-essay.
If the College Board decision was challenged by someone claiming that an image (such as a diagram or a flowchart) is an "essay" in graphical form, then the College Board could make the argument that the "essay" in question was "illegible" or "off-topic".
In any case, such an "essay" would probably receive a score of zero.
So, a "guessed" essay should avoid being in the form of a picture or image.
If the "guessed" essay is without technical errors (errors of spelling, punctuation, grammar, and word usage), and successfully presents an opinion, then the score depends on whether or not the essay theme coincides with the assigned topic.
A person who memorizes a high-quality essay on an arbitrary topic has some tiny chance of matching the assigned topic and getting a high score.
However, if the essay topic does not match the assigned topic, then the high-quality essay is likely to receive a score of zero for being off-topic.
One interesting possibility is to research the distribution of assigned essay topics, and thus prepare an essay with the most-likely essay topic.
Actually, I have no doubt that some SAT coaching businesses encourage students to compose several fairly-generic blocks of text that can be used to build strong essays on any topic.
For example, the student would develop and refine miniature argument fragments based on concrete (and teacher ass-kissing) examples from literature held in high esteem by the cult of academia.
A student can convert a few events from, say, "The Canterbury Tales", "Hamlet", "Catcher in the Rye", "Great Expectations", or whatever, in to a seemingly spontaneous series of literary allusions that support the theme of the essay while warming the hearts of the readers (graders) with familiar and loved literary favorites.
I think the SAT should forbid the use of non-generic knowledge and examples, such as the use of analogies to events portrayed in literature or events of history.
Suppose a person wants to write an essay, without knowing the assigned topic, but nonetheless wanting to receive a non-zero score.
I think one strategy that would be successful is to imitate the vagueness and incoherence of the examples of "score 1" and "score 2" essays presented in College Board literature.
By writing an essay that is very vague, it will not be obvious that the essay is not on-topic.
Meanwhile, a clear essay with lots of detail can easily be recognized as off-topic.
The challenge is to write an essay that is without any technical errors, and demonstrates strong reasoning skills, but is, at the same time, very general and capable of plausibly being related to any assigned essay topic.
It would have a universal applicability, such as horoscopes and fortune-cookie messages.
The key is to exploit the reader's expectation that the essay will be on topic, and avoid contradicting that expectation.
The hope is that the vagueness would be attributed to the writer's lack of skill rather than being attributed to ignorance of, or disregard for, the assigned topic.
It would be hilarious to develop and perfect (though repeated field testing and experiments) the ultimate, best-scoring essay that had no actual topic.
This essay would be designed to consistently earn a particular score, regardless of the assigned essay topic.
People tend to invent coherence and relevance when a phenomenon fails to manifest the expected amount of coherence and relevance.
For example, some people are initially astonished by the apparent "intelligence" of crude imitations of intelligent behavior displayed by chat software agents ( ALICE, Eliza, ... ).
Another example of this effect is the common mistake of perceiving more intelligence or meaning in a creative work than was intended by its creator.
( Consider the film "Being There" (1979) featuring Peter Sellers. )
A "guessed" essay that has the goal of remaining vague might look like the following:
"There is no doubt in my mind that the answer is yes -- especially in this day and age. I know a lot about this because I see it every day at school. Let me share a personal story to illustrate."
[Insert an irrelevant, but generic, story here, letting the reader infer some possible connection to the assigned topic.]
"I think anyone who had an experience such as that would agree with my reasoning and my conclusion. I know this is just my personal opinion, and I would not criticize other people for disagreeing."
Each SAT test book on the 2005.3.12 testing date had one of the following following four essay assignments.
( I suppose the College Board has evidence to support the theory that any specific test-taker will be able to earn the same essay score regardless of the particular assigned topic. )
[1] "Is creativity needed more than ever in the world today?"
[2] "Is the opinion of the majority -- in government or in any other circumstance -- a poor guide?"
[3] "Are people better at making observations, discoveries, and decisions if they remain neutral and impartial?"
[4] "Is a person responsible, through the example he or she sets, for the behavior of other people?"
Anyhow, as a fun exercise, you can try writing a single essay that is plausibly relevant to all four assigned essay topics above.
The following assigned essay topics appear in the College Board book entitled "The Official SAT Study Guide : For the New SAT" (2005):
[1] "What motivates people to change?";
[2] "Do changes that make our lives easier not necessarily make them better?";
[3] "Is conscience a more powerful motivator than money?";
[4] "Can success be disastrous?";
[5] "Do we need other people in order to understand ourselves?";
[6] "Is the world changing for the better?";
[7] "Do you think that ease does not challenge us and that we need adversity to help us discover who we are?";
[8] "Should heroes be defined as people who say what they think when we ourselves lack the courage to say it?"

12. SAT registration

12.1 Introduction

This section contains highlights of my experience of "registering" to take the 2005.3.12 SAT.

12.2 SAT registration steps

On 2005.1.24, Monday, 12:23 A.M. PST, I typed "" in to the Mozilla "Firefox" Internet browser.
I clicked on the link "SAT(R) Registration".
"[x] Yes, and I use it almost every day."
Why are there no girls around me?! :-(
The scrolling text window contains 36 paragraphs of terms and conditions, with a few statements such as:
"All correlations are adjusted for restriction of range so that the full range of scores and high school GPA are the same as for the national college-bound seniors cohort."
Maybe subsequent SAT registration steps should randomly display questions about various topics mentioned in the agreement, and, if the person answers the surprise quiz question incorrectly, the person should be brought back to the text of the agreement for further study!
That would be annoying, but the customer would be protected from consenting to an agreement that the customer did not actually read, understand, or remember.
I think surprise license agreement questions would literally revolutionize how people think about agreements and contracts, in a hilarious manner!
Anyhow, reading the text of the terms and conditions reveals the following scandalous fact:
"The correlation between [...] combined verbal and math scores and freshman GPA is .52; [...]"
The following is an interpretation of such a correlation value.
We depend on the SAT to determine the rest of our lives, but the SAT has about as much predictive validity as a coin toss!
Yes, this hypothetical "coin" would have to be slightly biased according to one's high school academic record. :-)
I checked the following check box:
[x] I confirm that I have read and agree to the SAT Terms and Conditions above.
This is the admissions ticket.

13. SAT-day observations

Newport Harbor High School, Newport Beach, California; Students, waiting to enter the building to take the new SAT; 2005.3.12, SATurday
* Traffic was terrible near the high school -- stop signs, urgent parents
* Before the test, in the parking lot, two students, a guy and a girl, walking toward the school, had a converSATion that began with: "Are you ready for this sh*t? I can't believe I woke up for this!" (Both laughed in agreement.)
* Students really seemed casual about the SAT -- complaining only about having to wake up early on a SATurday.
Overhearing student chatter revealed that some students were not even aware of the sections or format of the (new) SAT.
* 8:03 A.M.: Three people carrying test booklets arrived and walked to one side of one of the buildings.
* 8:04 A.M.: The three people carrying test booklets walked past me again, going in a new direction.
[ Note: Soon after this observation I learned that one of these three people was the person supervising the testing in my designated testing room. ]
* My test room was "Sims 285". The rooms were assigned on the basis of alphabetical ranges of last names.
* One student was relocated in the room.
Another student, who was obviously a friend of the relocated student, joked that his friend was relocated because of the (brown) color of his skin.
Another student elaborated on the joke in a topical manner by explaining that the relocated student might have a bomb (presumably for jihad).
The supervisor heard these comments and apparently felt obligated to explain, directly and reassuringly to the relocated person, that the reason for the relocation, "has nothing to do with you."
* A test booklet apparently was missing or lost in our test room.
The supervisor was handing out test booklets, starting with the left side of the room (if the blackboard is considered the "forward" direction), when he stopped after two or three columns of students had received test books.
The supervisor asked if any of these students received more than one test book.
Nobody responded in the affirmative.
( By the way, I was on the far right side of the room, by the door, just in case anyone is wondering!
) The supervisor counted test books again, and decided to hand out the remainder of the test books to the rest of the people in the room.
After a couple of sections of testing, the supervisor asked all students to read their test book numbers aloud, and the supervisor wrote down the numbers.
There was a break in the sequence of numbers, on the left side of the room.
* One student asked another student, outside, during one of the "long" (5-minute) breaks, "Are you, like, totally bombing this test?" Response: "Yeah! [Laugh]" ( Note: I think the term "bombing" was intended as a metaphor, but we live in desperate times! LOL! )
* I wanted to go to a bathroom during one of the breaks, but I discovered that I had to exit the building because, as far as I could tell at the time, the only bathroom for males had its entrance on the outside of the ground level of the building.
The bathroom door was labeled "BOY'S ROOM", and suddenly I became very conscious of my age (35).
Did this room have an implicit upper-age limit?
Was it socially-unacceptable for me to use a urinal intended for boys between the ages of 14 and 19?
It wasn't a public bathroom, per se, because it was on the grounds of a high school, so one could argue that there was a de facto age range expectation.
I didn't know why I was anxious -- and the thought that only people with bad intentions or morally-corrupt imaginations would feel anxious made me more anxious.
Anyhow, I simply went in to the bathroom and tried to remain in the moment, without thoughts, focusing on completing the task as quickly as possible.
But I knew that if I saw something such as an unusually-small urinal, or a sink closer to the floor than sinks in most public bathrooms, then I would panic and have to run out of the bathroom.
Fortunately, the bathroom looked like any other public bathroom.
Even so, it bothered me that I was forced by circumstances to confront a distracting psychological and sociological problem in the middle of taking the SAT.
I don't have time for this, id! My super-ego is trying to fail the SAT.
* The whole school was very charming and had a casual, welcoming atmosphere.
Maybe actual students hate the place and can't wait to escape, but the halls and rooms seemed pleasant, relaxing, and visually rich and stimulating; very mellow and inviting.
All they need to do is carve out one of these rooms, transport it to a shopping plaza, and serve herbal tea; I would dig it!
Anyhow, I wondered if students and teachers thought this place was fun and like a home away from home.
I imagine that teachers in many public schools can really create unique inner spaces and environments that are powerful, passionate, and meaningful expressions.
I hadn't thought very much about the idea of classrooms as "art galleries", "exhibition spaces", or "performance venues", but I suppose a teacher has quite an opportunity for SATisfying any need for self-expression.
* The SAT administration ended at 12:48 P.M.

14. My goal of answering all questions incorrectly on the SAT

This section describes the results of my attempt to answer all questions incorrectly on the SAT on 2005.3.12.

14.1 The challenge of answering all questions incorrectly

The challenge of answering all questions incorrectly on the SAT is comparable to, but less than, the challenge of answering all questions correctly.
This section describes some of the differences.
For the essay, it is trivial to get the lowest raw score possible.
The essay can be left blank, or written illegibly, or it can be off-topic.
Meanwhile, getting the maximum combined score of "12" is obviously a challenge.
For "student-produced response" math questions, getting the lowest raw score possible is trivial.
The response can be left blank, or filled with syntax errors, or filled in with values that are extremely unlikely to be correct.
Meanwhile, getting the correct answers is a challenge.
The non-trivial challenge for the goal of getting the lowest raw score possible on the SAT is in avoiding correct answers for the 160 questions with 5-choice responses.
A person trying to answer all questions correctly must actually identify all correct answers.
Meanwhile, a person trying to answer all questions incorrectly has two options: (1) find the correct answer; or, (2) find any of the four incorrect answers.
So, even if a question has a high difficulty, one need only identify a single incorrect response to be able to answer the question incorrectly.
Sometimes identifying one of the incorrect responses is much easier than identifying the single correct answer.
However, there are some SAT questions that are sufficiently difficult that answering correctly and answering incorrectly are equally-challenging tasks.
It is almost impossible to avoid having to actually find the correct answers to specific questions, to be assured of answering them incorrectly; this is almost always the situation for passage-based reading questions and the majority of the 5-choice math questions.
Here are funny ideas I was tempted to explore when writing my essay on the SAT:
(1) Writing in a language other than English, such as French, Aramaic, Sanskrit, Egyptian hieroglyphics, Ancient Mayan, cave drawings, mathematical equations, etc;
(2) "Writing" my "essay" in the form of a drawing, schematic, flowchart, floor plan, graph, sketch, cartoon, diagram, or decision tree;
(3) Writing in the form of code or encryption, such as ASCII codes in hexadecimal, rot13, SHA, PGP, pig Latin, rebus, Morse code, Huffman encoding, dictionary code, words written backwards (as if reflected by a mirror), etc.
(4) Writing my essay by using only exact groups of sentences from the Bible.
If the Holy, Authoritative, Unerring words of the Christian God failed to earn a perfect score of "12" on the SAT, it would be blasphemy!
Even more controversy could be fomented if a few friends took the SAT at the same time and constructed essays using text from "competing" holy books, such as the Qur'an or the Talmud.
Holy essay scores would be compared, and perhaps it would be evident that some gods are rejected by the College Board.
( Note: The essays would be as on-topic as possible, but would be constructed using only direct (memorized) text fragments from holy books. )
(5) Writing "my" essay by simply copying the short text before the essay question, directly from the SAT test book to my answer sheet.
If I get a non-zero score, the graders have been duped (as was my essay, if you will allow me to pun)!
(6) Writing "my" essay by selecting text from one of the critical reading sections on the SAT; choosing the text that seems closest to the essay question topic.
(7) "Writing" my essay by taking all of the interesting nouns and verbs from the essay question itself and associated short text, and using a totally-deterministic, formulaic, trivial procedure or algorithm to construct grammatical sentences that contain the interesting words but otherwise have only accidental meaning.

14.2 Results

14.2.1 An image of my essay was available online

The following image is a copy of my essay as I wrote it on my SAT answer sheet.
This image was made available through the College Board Internet site, along with an online score report.
A week before the SAT I telephoned my brother in Brooklyn, New York, and told him about my plan to take the new SAT on the following weekend.
I told him that I created and memorized an essay to write down for the essay section of the new SAT.
I told him that my planned essay began with:
"The instructions in my test book say that AN OFF-TOPIC ESSAY WILL RECEIVE A SCORE OF ZERO. [new paragraph:] Is this fair?"
I explained how I would then argue that it wasn't fair.
My brother is a genius, and had the following brilliant idea: write arguments in the essay that lead up to the conclusion that, "Yes, it is fair to get a zero for an off-topic essay!" Perfect.
If I were to get a non-zero score on such an essay, which begins with a direct quote of a rule that establishes that the essay should get a score of zero (in case the grader wasn't confident!), then the humor of such a violation of policy would make up for any disappointment about not getting a score of zero.
Also, I love the self-defeating insanity of crafting a really great argument for why I should receive a terrible grade.
Nobody in his or her "right mind" would ever do that.
I do not know when I thought of ending the essay with "Love, Colin" and "XOXO" and a heart with Cupid's arrow through it.
The "love" concept seemed like a good idea, because it violated the imaginary emotional barrier between the test-taker and the grader; the situation would just confuse people, and the significance and implications would be unclear.
Signing my first name, "Colin"", seemed like it would cause more trouble, but not enough to prevent getting my scores, so I went ahead with that plan, too.
Finally, I sprinkled Internet chat jargon ( "LOL" (laughing out loud), "OMG" (oh my god), and a smiley: ;^) ) in to the essay, because such elements are far outside accepted standard written English; and including such elements is my way of pretending to have the common failing of not "addressing one's audience".
Putting in Internet jargon in a formal essay (the SAT! critical to one's future!) seemed like the ultimate mockery of the new SAT and its intimidating essay.
I had a few extreme ideas about the composition of the essay.
One idea was to pay someone to write my essay.
I even offered to pay my brother some token amount to have him write my essay.
Thus, my essay would be memorized, plagiarized, and... purchased -- bringing the ugliness of capitalism in to the mix.
Another idea I had was to "open source" my essay, allowing anyone with access to the Internet to revise "my" essay in the weeks and months before I would take the SAT.
Like other examples of Internet collaboration (open-source software, Wikipedia, etc), people would edit the "ultimate essay", sentence by sentence, debating its structure and logic, perfecting it, until, finally, it was time for me to memorize the result and write it on the SAT answer sheet on test day.
I like to search for the humorous and bizarre aspects of things, and I think it would be amusing (and interesting from a philosophical perspective) if a collaborative essay, or a an "out-sourced" (contracted; paid-for) essay, received a non-zero score when graded by SAT essay readers.
There would be no actual significance or scandal if such a thing happened, because the graders would be unaware of the origins of the essay.
However, I think such a result would be amusing, because it suggests (falsely) some sort of impropriety.
I like result that inspire broader thinking about subjects.

14.2.2 Mailed score report

This section describes the normal, mailed score report.
Student Score Report : envelope
Student Score Report : page 1
Student Score Report : page 2

14.2.3 Mailed Question and Answer Service (QAS) report

This section describes a special mailed report named the "Question and Answer Service (QAS)" report.
The "Question and Answer Service" is only available on a few SAT administration dates each year.
Luckily for me, this service was offered for the debut of the new SAT, administered on 2005.3.12.
This detailed report includes a reproduction of the original test book (excluding the "variable" section), various tables, a list of all correct answers, and a list of all of a student's responses.
The "Question and Answer Service (QAS)" report is fascinating.
Many of the interesting parts of this document were based on data found in the QAS report.
Question and Answer Service (QAS) : envelope
Question and Answer Service (QAS) : question booklet
Question and Answer Service (QAS) : answer sheet

14.2.4 The only question I answered correctly on the SAT

I answered one question correctly on the SAT.
The question was a "passage-based reading" question in the critical reading (CR) division of the SAT.
Specifically, it was question #20 in section 6.
The question had a difficulty rating of "5", which is the highest difficulty on a "1" through "5" scale.
The following is an image of the text and the question I answered correctly (unintentionally).
The text above introduces characters and themes at a higher rate than most of the literature I have encountered in my limited range of reading experience.
I was facing 12 questions associated with this text, and I was in a panic about running out of time.
I did not have the calmness required to carefully scan backwards for the most recent characters that could resolve the possessive adjective "their".
I selected a response totally at random, and I accept that the outcome was not what I wanted.
Oh, and by the way, if you see the SAT this weekend, be sure and tell it... "SATan! SATan! SATan!"(*)
[ * homage to a lyric from a song with the title "Sweet Loaf" by a post-punk nihilistic group named "The Butthole Surfers". ]

14.3 Correct multiple-choice answers

14.3.1 Graphical representation


14.3.2 Experimental music representations of the correct answers

The following audio file (MP3) was composed by assigning musical notes ( A-440 Hz, B-493 Hz, C-523 Hz, D-587 Hz, E-659 Hz ) to the sequence of 160 correct multiple-choice responses on the 2005.3.12 SAT (form code BWBA).
The following audio file (MP3) was composed by appending speech recordings of the letters "A", "B", "C", "D", "E", according to the sequence of 160 correct multiple-choice responses on the 2005.3.12 SAT (form code BWBA), to an audio file.
The following audio file (MP3) is closer to the original speech recordings of the letters "A", "B", "C", "D", "E".
It is not as interesting as the previous composition, but I wanted to offer a more intelligible composition of the letter answers to the SAT.

15. A brief history of the SAT

The SAT was created by the College Entrance Examination Board (CEEB), and was administered for the first time on 1926.6.23 to 8040 people.
In 1994, various aspects of the SAT were modified.
Rules were changed to allow the use of approved electronic calculators during the test.
Antonym questions were eliminated.
The number of reading-comprehension questions was increased.
Ten math questions requiring "student-produced responses" (12700-choice instead of 4-choice or 5-choice) were added to the SAT.
This version of the SAT included 3 hours of testing time, and offered a maximum combined total of 1600 scaled points.
The most recent administration of this version of the SAT had an admission price of USA $29.50 (2004).
On 2005.3.12, a new version of the SAT was administered for the first time.
This version has three divisions: math (M), critical reading (CR), and writing (W).
The writing division includes an essay.
This version of the SAT includes 3 hours and 45 minutes of testing time, and offered a maximum combined total of 2400 scaled points.
The first administration of this version of the SAT had an admission price of USA $41.50 (2005).

16. The purpose of the SAT

The purpose of the SAT is to predict how well a high school student would perform as a freshman at a college or university in the United States of America (USA).
Using the SAT for anything other than its intended purpose is an error.
The SAT is not a test of intelligence.
The SAT is not a test of academic merit, potential, or ability.
The SAT is not designed to predict how well a person would perform in a job.
Although the College Board might make an effort to compose test questions for which members of various American subcultures, and members of different genders, have no particular advantage or disadvantage, it is not the purpose of the SAT to be unbiased with respect to culture or gender.
The SAT is not designed to measure actual knowledge or understanding of mathematics.
The SAT is not designed to measure the ability to read, comprehend, analyze, and compose American English.
The purpose of the SAT is to predict how well a high school student would perform as a freshman at a college or university in the United States of America (USA).
The implications of that simple statement of purpose are profound.
A study of the results of the first administration of the new SAT indicated that longer essays generally received higher scores than shorter essays.
Part of that phenomenon is due to the simple fact that proficient and skillful writers can form and expand an essay more quickly than people with less proficiency and skill.
It might be that short and concise essays can express complete ideas in a compelling way, but the SAT is not failing in its purpose.
The educators who grade the essays are part of a culture of educators -- a culture which includes professors in colleges and universities.
Shorter essays might receive lower scores on the SAT, but, given the culture of educators, shorter essays will receive lower scores at colleges and universities, too!
Thus, the SAT is true to its purpose.
Any objection to the scoring of essays on the SAT is, in effect, an objection to the criteria for academic success at a college or university in the United States of America (USA); the SAT is designed to reward current criteria for college success, in accordance with its purpose as a predictor of success in college.
Another implication of the purpose of the SAT is that taking preparation courses that focus on the exact, narrow content of the SAT, no more and no less, is not "morally wrong".
If a person does better on the SAT as a result of specific training or coaching, then, in essence, the SAT has indicated that the person has a quality (such as initiative, motivation, or money) that correlates with good performance in a college or university in the United States of America (USA).

17. Predicting college performance

The purpose of the SAT is to predict how well a high school student would perform as a freshman at a college or university in the United States of America (USA).
Human beings are obviously not "closed systems" (systems with no inputs or outputs, totally isolated from any environment).
When it became apparent in the 1990s that most students had continuous, easy access to electronic calculators in college, the SAT conditions were changed to permit the use of electronic calculators.
Although the only purpose of this change was to better predict performance in college, in accordance with the purpose of the SAT, it is apparent that the SAT essentially adapted to reflect the increasing capabilities of humans augmented by commonplace resources.
Eventually, to maintain predictive accuracy, or even relevance, the SAT might adapt to allow the use of other resources, such as notebook computers with Internet access, mathematical calculation software, and even mobile phones (to call friends), because today's college students have continuous, easy access to all of these resources, helping them to complete college assignments.
Excluding these resources from tests designed to predict success in college is artificial and might lead to error in predictions that will only become more inaccurate over time.
Anticipating the shift in the set of skills necessary for college success, the College Board recently developed a test of basic computer skills (such as the skill of composing search queries).
Some computer skill is already a requirement for success in today's colleges and universities.
One controversial extension of this idea is that "cheating" on the SAT is acceptable, assuming the cheater can sustain a pattern of cheating throughout his or her college career.
If a person has the ability to pay someone to take the SAT on his or her behalf, and the person can sustain a pattern of paying others to complete assignments and take tests during the person's college career, then the SAT has properly predicted this person's success in college.
If this same person could pay others to perform all tasks in the person's professional job and in the person's social life, then in a very practical sense, the person does have great ability to be productive and successful.
Incidentally, I believe that schools should totally abandon any definition of "cheating" or "academic integrity".
The only rules that should apply in school should be the laws of the government.
Thus, a student could do anything to accomplish an academic task, assuming no laws were violated in the process.
Thus, the conditions of school would be in harmony with the conditions of society and "real life".

18. Why go to college?!

18.1 Buy a diploma!

Scenario: You're ambitious and driven to succeed, but you are convinced that going to college would be a waste of time, delaying your future of successful, if incomprehensible, reverse-discombobulating antidisestablishmentarianism.
Meanwhile, small-minded people in roles of power insist on seeing college diplomas before letting you do just about anything!
What can you do?!
Solution: Go to the Internet and buy a college diploma!
Receive it instantly, electronically, and in a format that is ready to print or send to the various bureaucracies currently obstructing your destiny!
Get a college diploma, without the college! [from an unsolicited e-mail message]
The following are just some of the many fine institutions selling diplomas.
[1] Almeda University ( )

      * Offers degree verification upon any inquiries

[2] Ashwood University ( )

      * $499 Ph.D. (for "life experience" in specific field)

      * Offers degree verification upon any inquiries

[3] Universal Life Church ( )

      Degrees that can be purchased online in the ULC virtual store
      without passing any exams:

        * Dr. of Divinity (D.D.)      : $35
        * Dr. of Metaphysics          : $35
        * Dr. of Biblical Studies     : $60

        * Doctor of Motivation        : $40

      The Frequently-Asked Questions (FAQ) section has this gem:

               "Can I preside over my own wedding?"

      I like to interpret this as asking if a person can marry
      himself or herself to someone else.

[4] Universal Life Church ( )

      Okay, comparison shopping pays off; this Universal Life Church
      has better deals than the other Universal Life Church ([3]).

         * Doctor of Divinity          : $29  ($6 less!)
         * Doctor of Metaphysics       : $29  ($6 less!)
         * Doctor of Biblical Studies  : $59  ($1 less!)

         * Doctor of Universal Life    : $29
         * Doctor of Religious Science : $40

      Also, confessions can be made online!

        Example:   "I confess...I bought my Doctorate of Divinity
                    from the other Universal Life Church!"
I think the idea of simply buying a degree, without actual school or testing, is hilarious.
I like to imagine college degrees becoming so devalued that they come as surprise gifts in cereal boxes or in McDonald's "Happy Meals(TM)".
People could receive so many college degrees in the mail each day that lawmakers would establish a national "Do Not Confer Degree" list, so that people could opt-out of the rampant bestowing, conferring, and honoring.
Bizarre degrees could top the New York Times best-degree list, such as: Doctorate of Pornography; Reverend of Recreation; Professor of Ignorance; Jack, of All Trades; Master of None; and Lord of The Rings.
Degrees could be received by accident, just by glancing at a television screen while a degree-conferring advertisement is playing, or overhearing a radio or loudspeaker that is conferring degrees to all within earshot.
In a world where college degrees, and other titles of distinction, can be purchased, standardized testing is valuable.
The decline in the credibility of college degrees might be the best thing that has ever happened to the standardized testing business after the invention of college itself!

18.2 College is obsolete!

I do not know the purpose of conventional colleges and universities in our society.
I know that if a person wants to start a professional career in medicine, engineering, journalism, or law, etc, then it is currently necessary to acquire a relevant kind of degree from a college or university.
So colleges and universities prepare people for vocations.
However, students who enroll in colleges and universities are forced to participate in some variety of courses, often named "breadth" courses or "core" classes, in subjects such as: foreign languages, history, anthropology, economics, sociology, psychology, philosophy, English, biology, physics, and mathematics.
I do not know why colleges and universities require students to complete "core" classes to receive degrees.
If it is only to protect the reputation of the college or university, by producing graduates who won't say extremely ignorant things about subjects beyond their core training, then students, as consumers of services, should reject current colleges and universities.
I am reminded of the ethically complicated practice of hospitals forcing patients to pay for, and undergo, various tests to rule out unlikely pathologies just to protect the hospital from accusations of negligence.
Colleges and universities should accept that students can graduate with honors in a particular subject and be totally ignorant with respect to other subjects.
Breadth courses account for two or more years of a typical undergraduate degree requirement.
For some, these two years are a total waste of time, and a total waste of two years of tuition money (the the cost of a new car and many years of food and rent); the college or university is holding a degree ransom, attaching undesirable, irrelevant requirements to the contract.
While I like the idea of professionals and experts delivering lectures to the public, and developing paths of study to help others achieve some level of understanding of a subject, I wonder if the appeal is enough to justify the continued existence of colleges and universities.
Is society too attached to the college concept?
College has burdened a lot of students and parents with a lot of debt.
What is the ultimate cost to society?
Is college, as currently manifested, actually hurting our economy, channeling an enormous amount of money in to a phenomenon that does not best serve the public?
The research efforts currently underway at universities should be made logistically independent from the academic aspects of a university.
Non-profit research activities funded by the government (federal grant programs such as the NSF, state research budget, etc), should be considered as a separate activity from the education "service" a university sells to undergraduate students.
I think the greatest benefit of colleges and universities to society is the sustained mixing of young people.
Students are exposed to peers with differing backgrounds, opinions, personalities, etc, gradually allowing the students to comprehend the full spectrum of the society in which they live, and also allowing students to perceive and move toward international social norms.
I think "home schooling" or "Internet learning" can mostly satisfy any education goals our society has (with the exception of mechanical knowledge or laboratory experiences), but I also think that society needs to invent a new social institution to provide the kind of peer exposure currently accomplished by colleges and universities.

19. My SAT fantasies

19.1 Introduction

This section describes some of the many fantasies I had regarding the SAT.

19.2 Alternative "calculators"

An "approved" calculator might be used during the math sections of the SAT.
Perhaps the College Board will eventually approve of the commonplace "calculating" devices shown in this section.

19.2.1 Abacus


19.2.2 Slide rule


19.2.3 Multi-sided dice


19.2.4 "Magic 8-Ball"

"Yes"... What does that mean? "Ask again later"?! Damn it!

19.2.5 "Furby"

The electronic, intelligent, talking "Furby" speaks an obscure language. Learn this language, and "Furby" can help you during the SAT.
Another electronic, sort-of intelligent, talking companion is the notorious "Talking Barbie".
Talking Barbie was removed from store shelves for her proclamation, "math is hard!"
Duh, Barbie.
Even the (male) president(*) of Harvard University knows that girls can't do math.
But, damn it, Barbie is going to have fun in college and graduate with a GPA that is a lot higher than the males who studied math, physics, or engineering.
[ *...Lawrence H. Summers, president #27 of Harvard University (, appointed in 2001.6, said, in 2005.1, that innate differences between men and women might be one reason why fewer women than men succeed in math and science careers. ]

19.2.6 "Lex Luthor" and "Brainiac"

"Lex Luthor" and "Brainiac". Gay?
Lex Luthor and Brainiac are the most intelligent beings in the Universe.
They are also members of the "Legion of Doom", whose goal is to spread evil throughout space and time.
Lex Luthor and Brainiac might offer to help you DESTROY!, OBLITERATE!, ANNIHILATE!, ... err, I mean, "pass", the SAT... ...THE KEY TO ULTIMATE DOOM!!!

19.2.7 "Ouija board"

The spirit world loves multiple-choice!
Avoid summoning malevolent entities.

19.2.8 "mood ring"

A "mood ring" allegedly indicates a person's "mood" by the changeable color of the stone mounted on the ring.
If the mood ring turns black while you are taking the SAT, you are anxious and you are going to fail the SAT and be a miserable failure in life.

19.2.9 "security blanket"

Sometimes what a person needs most to answer SAT questions is CONFIDENCE!
A "security blanket" (or "blankie") might be just the thing to help solve those tough SAT problems.
Adult security blankets would be great for: corporate board meetings, job interviews, courtroom cross-examination, stock trading, blind dates, surprise audits, etc -- providing soft, gentle warmth, while blocking the bad, scary things outside.
Just hide yourself in the fuzzy, sky-blue blankie, and peek out at the SAT question.
Is it still there?
Make it go bye-bye!
All gone!

19.3 Crazy problems

Just to really rattle a high school student's nerves, pushing those borderline breakdown cases over the edge, the sadistic authors of the SAT might throw in questions such as these.
(1) Refer to figure (I).
Point O is at the center of a circle.
Line segment AB is the diameter of a second circle.
Right-triangle AOB is inscribed in the circles as shown.
What is the ratio of triangle area T to lune area L?
(2) Refer to figure (II).
Given that the area of square S is equal to the area of regular pentagon P, what is the ratio of area R to area S?

19.4 Essay in a multiple-choice format

What if the essay section required entering the essay by filling in bubbles, just like all other parts of the SAT?
Or, alternatively, the following multiple-choice format could be used.

19.5 Testing other skills on the SAT

Many changes to the SAT over its history were the result of various academics lobbying to have new skills tested.
What other skills will be tested in the future as a result of special-interest groups pressuring the College Board?

19.5.1 "artistic" ability

I saw advertisements analogous to the one shown above inside matchbook covers.
I guess the theory was that someone who opens a matchbook, to light a cigarette or start a suspicious fire, will see the art school advertisement, attempt the challenging sketch, and realize a hidden, undeveloped talent for art.

19.5.2 Ability to see stereogram images

I don't know what special-interest group would lobby the College Board for increased testing for the ability to see stereogram images, but it could happen -- so get your stereogram SAT test preparation book today!
The answer to the following question appears in the following stereogram.
What is the answer?

    (A) A
    (B) B
    (C) C
    (D) D
    (E) E

19.5.3 Optical illusions

SAT questions might be tricky, but they don't currently deceive the senses.
Humanity might one day be thrown in to a world of illusion, and only those who can see through the subterfuge will survive.
The SAT can prepare us for such a future.
(1) Which line is longer?

    (A) Line I
    (B) Line II
    (C) Both lines I and II
    (D) Neither
    (E) The question cannot be answered with the information given

(2) What is shown in figure III?

    (A) A vase
    (B) Two faces
    (C) A vase
    (D) Two faces
    (E) Wow, the SAT is so cool!

19.5.4 Interpreting cell-phone text messages

Text messaging through mobile devices such as cellular phones requires a new variety of vocabulary, grammar, and logic.
This new SAT critical reading question format will ensure that the SAT remains relevant for a few more years.
Questions 1-2 are based on the following text.

[This excerpt is from a text message received
 on a cellular phone.]

         1   BTW I
         2   LUV U
         3   LOL

1. Line 3 of this text probably indicates that
the author is

        (A) J/K
        (B) ROTFL
        (C) 1337
        (D) GF
        (E) FBI

2. The proper response to this message is most

        (A) :)
        (B) OMG!
        (C) <G>
        (D) (H)
        (E) IMD14U

[The passages for this test have been adapted from
published material.  The ideas contained in them do
not necessarily represent the opinions of the
College Board or Educational Testing Service.
Laugh Out Loud!!!1!SHIFT-ONE!!! ;-)]

19.5.5 Consumer survey

This new section of the SAT would be a consumer survey, part of a market analysis done by the College Board on behalf of multi-national corporations.
(1) Which of the following character storylines is most likely to become a successful franchise, leading to blockbuster films, a series of video games and novels, action figures, lines of clothing, brands of house wares, specialty colognes and perfumes, fast food meals, cereals, snacks, plush toys, and various psychotropic medications?
(A) "Pepsi's Island" : After a ship carrying a cargo of Pepsi-related products wrecks on an uncharted island during a tropical storm, the castaways establish a Pepsi-centric community on the island.
Sometimes their remote island is visited by amusing and attractive celebrities on behalf of various affiliates of the Pepsi company.
These guests integrate their products in to the story, creating a hilarious cross-marketing experience for the whole target demographic!
(B) "Card Consumers" : Cha Ching is a boy living in an anime world, filled with an ever-increasing amount of playing cards and associated merchandise.
Rather than challenge other boys (or creepy adults!) for their playing cards, which is a time-consuming and inefficient proposition, Cha Ching spends his time repeatedly learning how to purchase playing cards online using his parents' credit cards.
Each episode of the planned television show begins and ends with Cha Ching buying new playing cards online, showing the exact process of sneaking the credit cards and entering the information in to the Internet site or cell phone payment portal.
Catchy musical lyrics promote the idea of automatic payment.
(C) "Captain RIAA" : In this animated series, five ordinary kids, from five different continents, are selected by the Recording Industry Association of America (RIAA) to defend our planet from the evil of multimedia piracy.
Each kid is given a unique power ring to combat the forces of copyright infringement in a distinct and spectacular manner.
Although the five kids function as a team to fight crime, sometimes they need more than their group talents to stop extreme cases of file-sharing or to overcome pesky privacy protection laws.
Luckily, the kids can combine their powers to summon the recording industry executive's greatest champion: "Captain RIAA"!
Captain RIAA quickly shuts down file-sharing Internet sites, infiltrates peer-to-peer (P2P) networks, issues lawsuits, and sometimes even proposes legislation or new hardware and media format standards to slow down the rate of casual piracy.
Each episode will feature a contemporary, chart-topping musician or band, and the musician or the members of the band will demonstrate their suffering in vivid detail as a result of the piracy all around them.
The kids will stop the pirates, and the musician or members of the band will express their sincere, heartfelt gratitude for ending the crushing misery caused by the theft of their hard labors.
Each episode will have an intermission with a lesson about identifying friends or family members who might be unwittingly or casually violating copyright laws.
Each episode will end with RIAA contact information to "recommend friends for free anti-piracy advice", removing the stigma of giving authorities tips.
Viewers are eligible for Captain RIAA badges and a pad of violation "tickets" (with real legal force!), and cool software that ensures your personal computer isn't "damaged" by pirated music, videos, or games!

19.5.6 Paranormal abilities

Although mainstream academic thinking rejects the concept of paranormal phenomena, some fringe groups might promote wider testing for paranormal abilities, such as telepathy, using Zener cards.
(1) Of which shape am I thinking?

19.5.7 Paradox resolution

People are constantly faced with paradox and contradiction, especially in advertisements, Zen koans, political speeches, quantum mechanics, and women.
Those who excel in this world are masters of paradox.
The SAT should test this ability.
(1) Will you get this question wrong?

    (A) Yes
    (B) No
    (C) No. ... Yes. ... I mean, no. .... Damn it!
    (D) Ask again later, when I have my score report
    (E) I don't know

(2) Which came first?

    (A) Chicken
    (B) Egg

(3) Which of the following is an acceptable use of a time-travel device?

    (A) Going back in time and preventing the inventor of the
        time-travel device from inventing the device;

    (B) Giving instructions on how to build the time-travel device
        to someone who predates the birth of the inventor of the
        time-travel device;

    (C) Starting a newspaper delivery route in the space-time continuum
        that involves everyone receiving next week's newspaper;

    (D) Going in to the time-travel device, going slightly back in time
        and preventing yourself from entering the time-travel device;

    (E) Deposit money in an interest-bearing account in the present,
        and then make numerous trips to the future to withdraw some
        of the accrued interest, bringing back money to the present to
        add to the initial investment, a process that exponentially
        increases present money until there is more money than money!

19.5.8 Scam-ability

Scams are everywhere, especially after the invention of the credit card and the Internet.
This SAT question format will be sponsored by credit reporting agencies, such as Equifax and TRW.
Answering such questions "incorrectly" (being deceived by the scam) will have a negative impact on one's actual credit score!
Questions 7-9 are based on the following text.

[The following text is taken from an e-mail message received
 in 2005.  Response e-mail addresses have been omitted.]


        I am the Bill and Exchange (assistant) Manager of the
        Bank.Mr.Martins Dike. In my department I discovered
  10    an abandoned sum of USD$ US$7 million dollars. In an
        account that belongs to one of our foreign customer
        who died along with his wife and two children in
        November 1999 in a plane crash. Since we got
        information about his death, we have been expecting
  15    his next of kin to come over and claim his money
        because we cannot release it unless somebody applies
        for it as next of kin or relation to the deceased,
        as indicated in our banking guidelines but
        unfortunately we learnt that all his supposed next
  20    of kin or relation died alongside with him in the
        plane crash leaving nobody behind for the claim.

        It is therefore upon this discovery that I in my
        department now decided to make this business
  25    proposals to you and release the money to you as next
        of kin or relation to the deceased for safety and
        subsequent disbursement since nobody is coming for it
        and we don't want this money to go into the bank
        treasury as unclaimed fund. The banking law and
  30    guidelines here stipulates that if such money remains
        unclaimed after five years, the money will be
        transferred into the bank treasury as unclaimed fund.

        The request of foreigner in this transaction is
  35    necessary because our late customer was a foreigner
        and a Nigerian cannot stand as next of kin to a
        foreigner. We agree that 45% of this money will be
        for you as foreigner partner, in respect to the
        provision of a foreign account, 10% will be set aside
  40    for expenses incurred during the business and 45%
        would be for me. After I shall visit your country for
        disbursement according to the  percentages indicated.
        Therefore to enable the immediate transfer of this
        fund to you as arranged, you must apply first to the
  45    bank, and send your account number, your private
        telephone and fax number for easy and effective
        communication and location where the money will be

  50    Don't worry, since i work with the bank, i will
        always give you further information's and also an
        underground assistance to ensure that everything
        works out fine for both of us.

  55    And be also informed that any required documents that
        the remittance department will be procured by an
        attorney which i am going to direct you to contact
        for his services in securing all the required
        documents of claims on your behalf
        Upon receipt of your reply, I will send to you by
        fax or e-mail the text of the application which you
        shall retype and fax to our foreign remittance
        arrangement for easy execution of the transaction.
        I will not fail to bring to your notice that this
        transaction is 100% hitch-free on both sides. As all
        required arrangement have been made for the transfer.
        You should contact me immediately as soon as you
  70    receive this letter.

        Please when replying this confidential letter, I
        advise you to reply me through my alternative email
        address: ( xxxxxxxx@xxx.xx ) or (
  75    xxxxxxxxxxxxx@xxx.xx )

        Trusting to hear from you immediately. I expect your
        urgent reply.

  80    Best Regards.


        7. The person who received this e-mail message was

            (A) The luckiest damned bastard alive!
            (B) Just one of MILLIONS of lucky bastards
            (C) Me
            (D) You
            (E) All of the above

        8. Lines 23-29 describe:

            (A) A business proposal
            (B) A way to prevent a faceless bureaucracy
                from absorbing funds it can do without
            (C) A crime involving, at the very least,
                false identity and theft
            (D) The best thing that ever happened to me
            (E) All of the above

        9. It can be inferred from the text that
           $700K US will be "set aside for expenses
           incurred during the business".

           How could transferring money from one account
           to another possibly cost $700K US?

            (A) Who cares?  My cut is $3.15M US!
            (B) Overnight delivery of fifteen tons of
            (C) Non-affiliated bank Automated Teller Machine
                  (ATM) fees!
            (D) Conversion from the metric system!
            (E) It takes money to make money!  Nothing
                  ventured, nothing gained!  Your fear and
                  cynicism will prevent you from ever
                  experiencing life and wealth.  Pathetic!

19.5.9 Paranoia

One glaring gap in the set of personal attributes assessed by the SAT is personality.
The SAT can more directly gather information on personality, to help build up a psychological profile.
Of course this new division of the SAT will have a scaled score ranging from 200 through 800, with "800" indicating super-human sanity, and "200" indicating infinite inner chaos.
Paranoia is one important component of personality.
Security firms might value some level of paranoia.
( Fun idea: Consider the challenge of convincing a paranoiac to take anti-paranoia pills! )
(1) Consider the sequence 1,4,9,16,25,36,49,...
    What is the next number in this sequence?

    (A) I swear I don't know.  Who told you I would know?

    (B) Shhhhhh!  Not here, man!  Did you bring the money?

    (D) What happened to choice 'C' ?  I'll tell you
           what happened.  It was ELIMINATED!

    (E) d0 00 2d a4 41 66 b5 e4 29 d9 76 f7 98 df b2 a2
        (AES key: a0f495e6c2574b2782fe0e9acf5324a4)

19.5.10 SAT : the text adventure!

To help the SAT "connect" with today's youth, the College Board can transform the SAT in to a text adventure, like a "choose your own adventure" novel.
This fun format turns the boring SAT in to an exciting, action-packed quest for "entering the dragon's lair", where glorious riches and a beautiful princess await (a metaphor for gaining admission to college, and the wealth and sex that inevitably ensue).
(1) You are in front of a house in the middle of a grassy field.
    There is a path leading east and south.  You see a bottle
    on the ground.

    (A) Enter house    (go to question (13)).
    (B) Walk east      (go to question (7)).
    (C) Walk south     (go to question (9)).
    (D) Pick up bottle (go to question (2)).

(2) The bottle has a label: "Potion of Doom!"

    (A) Put down bottle (go to question (1)).
    (B) Drink bottle    (go to question (3)).

(3) You have died.

    Your SAT score is 1000 experience points.

    (Please give your answer sheet to the supervisor
    and quietly leave the SAT gaming room so that others
    can continue their quests.)

19.5.11 Voting on government propositions

To combat voter apathy, especially among today's youth, the SAT will include state and federal propositions and referenda, for which high school students must vote or receive reduced overall SAT scores.
1. Proposition 35A calls for $3.2B of
   California state funds to be given to
   a private citizen, Colin P. Fahey,
   of Irvine, for the purpose of his
   advancement and outreach to the
   pretty ladies in his community.
   Independent analysis predicts this
   fiduciary appropriation will greatly
   enhance Colin P. Fahey's life, with
   minimal impact to the environment.

       (A) Yea
       (B) Nay

19.5.12 Driving license written exam

These SAT questions would be used by the Department of Motor Vehicles (DMV) to be able to issue driving licenses through the SAT program.
1. Which of the following signs indicates that you are a representative of a book publisher, on the longest, most-disturbing nighttime car trip of his entire life, to visit the reclusive author of the "insanely" popular book, "In the Mouth of Madness"?
Do you read Sutter Caine?
More people believe in his writing than the Bible!
He sees you.
He sees you!

19.5.13 Massively-parallel calculation

Use some SAT questions to get the student to work on a small part of a giant and important calculation, just as SETI@Home (Search for Extraterrestrial Intelligence), Einstein@Home (search for gravity waves), Folding@Home (protein folding), and GIMPS (Great Internet Mersenne Prime Search) use spare time on millions of personal computers to advance science for the benefit of all mankind.
Organizations can pay for computing time on the "million-student CPU" of the SAT.
Alternatively, some SAT questions could be sold to the highest commercial bidders, such as media and publishing corporations, so students can be employed to proofread thousands of articles, books, screenplays, journals, etc, very carefully, because their futures depend upon doing the work correctly.

19.5.14 Internal Revenue Service (IRS) tax audits

Allocate some questions on each of the millions of SAT booklets to the Internal Revenue Service (IRS), so that students will perform, unwittingly, tax audits on their fellow citizens.
1040. Consider the following hypothetical wages, tips, ordinary dividends,
      qualified dividends, business gains, capital gains, IRA distributions,
      pensions and annuities, rental real estate, royalties, partnerships,
      trusts, farm income, social security benefits, educator expenses,
      student loan interest deductions, health savings account deductions,
      and moving expenses.

      Compute the "adjusted gross income" using the tax laws summarized
      in the figure.

      If you pay someone to prepare the answer to this SAT question on
      your behalf, the preparer's signature must also appear below.

      By signing your name you declare, under penalty of perjury, that to
      the best of your knowledge and belief, your answer is correct, true,
      and complete.

      Failure to answer this question correctly carries a maximum penalty
      of a $100000 fine and 10 years in prison.

19.5.15 Analysis of national security intelligence

Given today's troubled world, some SAT questions can be allocated to serve national security intelligence analysis efforts.
To ease the burden of the staffs of the CIA and Interpol, each student will be given random pieces of intercepted communications, or fragments of foreign newspapers or broadcast transcripts, all in the guise of "literary" critical reading text, just like all of the other reading comprehension text on the SAT, but with more relevance and deadly seriousness.
[The following text was taken from (DELETED) on (DELETED) MHz at (DELETED)
in the (DELETED) region of (DELETED).  You are temporarily granted a GS-40
field agent security clearance for this page of the SAT only.]

      1   "Today's weather will be warm and sunny,
          with highs in the 90s, and lows in the
          mid 60s.  Wind of 15 miles per hour is
          expected from the West.  The humidity
      5   is 20%, and the barometer is falling.
          Stay tuned for the complete five-day
          forecast later in the program."

    1. Assess the threat level of the text above.

        (A) Low
        (B) Amber
        (C) A Clear and Present Danger
        (D) The Sum of All Fears
        (E) Go to "DefCon 1".

    2. What is the best characterization of the quality
        of actionable intelligence in the text?

        (A) Sloppy
        (B) Suspicious
        (C) Potentially-embarrassing
        (D) Boring
        (E) Arousing

    3. The author of the text is probably

        (A) Not a native English speaker
        (B) A head of state
        (C) In hiding
        (D) A drug czar
        (E) Really mad

    4. Please circle any appropriate bombing targets on
        the map provided.

[The text for this test have been adapted from classified top-secret
materials.  The ideas contained in them do not necessarily represent the
opinions of the College Board or Educational Testing Service.]

19.5.16 Converting the SAT to a game of chance (gambling)

The College Board could transform the dull SAT in to an exciting game of chance!
A person taking the SAT can let fate decide if he or she's gets a question correct or incorrect, just by scratching "instant-win" bubbles on the answer sheet.
If a person is not SATisfied with the instant scores, he or she can buy more "answer sheets" from the "test supervisor".
Also, the College Board will conduct random drawings, a lottery, to award perfect scores.
Strange how fate works, but, hey, the sports jock who made fun of nerds is going to Harvard!
That lucky bastard!
Certain questions on the SAT will be labeled "double or nothing", raising the stakes, and the thrill, of SAT gaming.
It is inevitable that people will write books on SAT gaming strategy, such as "How to break the College Board".
There will be rumors of some students winning millions of scaled-score points, enough to get admitted to Harvard thousands of times, on full scholarship.
Obviously, the College Board will be taken over by organized-crime bosses.
Rival standardized-testing companies (example: ACT) will be eliminated in brutal "turf wars".

19.6 Selling out the SAT to commercial interests

Eventually all institutions are corrupted by the influence of commerce.

19.6.1 Advertising space

The SAT could be used as a totally-uncontrolled, "free-for-all" dash for a student's money, even if it means compromising the integrity of the entire testing process!
Sell advertising space in the test booklet; Coke, Pepsi, hard liquor, anti-depressants, guns, airplane tickets, whatever!
Also, have the test supervisors read any paid advertising announcements before, during, and after the test:
"[...] If you finish a section early, then you may use the remaining time to review your answers. Do not turn to any other section of the test."
"And, uh, although I can see that all of you in the room today are males, I am contractually obligated to read the following promotional announcement:"
"'You make plans for the weekend, but then comes...your "period"!"
[An advertisement for a feminine hygiene product begins here]
"We will start the feature presentation (i.e., testing) soon, but:"
"Sony, Taco Bell, Pepsi, AOL, and a clandestine multi-national consortium named "The Great Hand of Power", would like to "personally" invite you to see "Revenge of the Something About Meeting the Sleepless Cider House Bridges of Last Summer, Actually, Interrupted, 3, Returns", nominated as best movie of its type by financially-compensated marketing firms."
"Combining elements from all of your favorite films, and featuring performances by more celebrity actors than any other film in history, and deafening you with a soundtrack of the same music that is impossible to avoid on the radio and music-video stations, and costing $1.2 billion to produce, this epic masterpiece will saturate every aspect of your life as a consumer for many salary cycles!"
"Are there any questions about the promotions you heard today? Okay."
"Now open your test booklets to the first page."

19.6.2 Question opt-out fees

This idea involves allowing the students to pay for cellular phone "lifeline" calls to customer service personnel at the College Board.
For $8 per minute, students can get basic coaching on difficult questions during the test.
For a higher fee, a student can ask the customer service person to eliminate two wrong answer choices.
For an even higher fee, the customer service person will indicate which response is believed to be the correct answer by the majority of college-bound students.
Also, each question on the SAT will be clearly marked with a price, in dollars, in exponential relation to the difficulty of the question.
A student can "opt-out" of a question, and receive full credit on the question, by paying the dollar amount indicated by the question.
Here is an example:
(13) Which of the following is not a prime number?

     (A) (( 34790 !  ) - 1)
     (B) ((2^24036583) - 1)
     (C) 3215031751
     (D) 378163771
     (E) I accept a charge of __$120__ to opt-out of this question

[ Clue: The number is the only number less than 2.5 * 10^10 that
is falsely-identified as a prime by the strong k-pseudoprime test. ]

19.6.3 Products and services offered during actual SAT testing

(1) Calculator rentals or purchasing;
(2) Pencils or sharpeners -- at outrageous prices;
(3) Neck and shoulder massage;
(4) Food and Beverages: such as soda, popcorn, hot dogs, pretzels, and candies -- and students will be frisked and searched to ensure they don't bring in snacks during the "feature presentation"; beer or cocktails, for the over-21 crowd, will cost a fortune and will come in tiny bottles, like on the airlines.

20. Children and parents, taking the SAT together!

The book entitled "The Official SAT STUDY GUIDE: For the New SAT" (2004), published by The College Board, has a section entitled "Who Develops the SAT?" in the Getting Started chapter.
In that section appears the statement, "We're not talking about your parents' SAT here."
But, in fact, parents can register to take the SAT on the same date and at the same testing center as their children.
If there is only one testing room at the testing center, or if test takers are assigned testing rooms according to the typical convention of grouping by alphabetical ranges of family (or last) names, then a child's parents can spend four hours watching their son or daughter take the SAT.
Depending on the personalities or ethical beliefs of the parents, all kinds of things might happen.

21. Conclusion

Perhaps I will print the following text on a tee-shirt: "SAT M 200 CR 200 W 200".
However, there are still a few experiments I want to try on the SAT.
( I like the perverse role reversal here, with the SAT itself becoming the test subject instead of its normal role as tester.
) For example, the math problems requiring "student-produced responses" tempt me to try the division-by-zero encoding (example: " 1/0"), the zero-over-zero encoding (example: " 0/0"), and various unconventional encodings (examples: " 001", "1/01", ".000", ".0/9", ... ).
For the essay, I want to: write an essay without an actual topic; write an "on-topic" essay generated by a deterministic procedure at test time; write an on-topic essay constructed using only text quoted verbatim from the Bible or other holy book; and write essays using other principles described in this document.

Appendix: Mathematics used in this document


This section of the document introduces mathematical terminology as it is used elsewhere in this document.
Unfortunately, establishing the formula for the "standard deviation of a sub-population of a multinomial distribution", for example, to allow the reader to interpret some of the brief comments I make in this document, requires intermediate terms and procedures to be defined.
I expressed the information in a way that can be typed using ordinary characters, and I used a syntax similar to the various syntaxes used in common computer languages, making the information more accessible to people who are familiar with the syntax and semantics of popular computer languages ( C, C++, Java, C#, ... ).

Basic mathematical definitions


    Pi       = 3.14159 26535 ...;

    E        = 2.71828 18284 ...;   Base of the natural logarithms and exponents


    x != y : true if x is not             equal to y; otherwise false
    x >= y : true if x is greater than or equal to y; otherwise false
    x <= y : true if x is less    than or equal to y; otherwise false
    x *  y : value is "x multiplied by y"
    x += y : x = (x + y);
    x -= y : x = (x - y);
    x *= y : x = (x * y);


    Integer( x )
      return( integer part of x, found by ignoring the fractional part of x );

    Fraction( x )
      return( x - integer( x ) );

    Thus, x = Integer( x ) + Fraction( x )

    Sq( x )
      return( x * x );

    Sqrt( x )
      return( positive square root of x );       Defined for x >= 0

    Sum( initialization; test; iteration; summand )
      result = 0;
      while ( test )
        result += summand;
      return( result );

    Product( initialization; test; iteration; multiplicand )
      result = 1;
      while ( test )
        result *= multiplicand;
      return( result );

    Pow( x, y )
      return( x raised to the y exponent );

        Pow(x<0,y<0) = defined for integer y, some rational y, and no irrational y;
        Pow(x<0,y=0) = 1;
        Pow(x<0,y>0) = defined for integer y, some rational y, and no irrational y;
        Pow(x=0,y<0) = undefined (division by zero);
        Pow(x=0,y=0) = undefined (defining a value would lead to contradictions);
        Pow(x=0,y>0) = 0;
        Pow(x>0,y<0) = always defined;
        Pow(x>0,y=0) = 1;
        Pow(x>0,y>0) = always defined;

        Pow(x,y) = Pow((1/x),(-(y)));  "if x!=0"
        Pow(x,y) = Pow(x,(Integer(y))) * Pow(x,(Fraction(y)));

    Exp( x )
      return( Pow( E, x ) );   E = 2.71828..., the base of the natural logarithms

    Integral( variable; lower; upper; integrand )
      return( integration of integrand as variable goes from lower to upper );

      which can be approximated by:

        Sum( x=lower; x <= upper; x += dx;  (integrand(x) * dx)  )

      for "small" dx; in the limit as dx approaches zero from the positive
      side of zero.  One possible segmentation of the interval of integration is
      a uniform segmentation, with dx = ((upper - lower) / N), where N is the
      (possibly-large) number of segments.


    Let F have N elements: F[1], F[2], F[3], ..., F[N].
    Let G have N elements: G[1], G[2], G[3], ..., G[N].

    element operations
        (F+G) has N elements: (F[1]+G[1]), (F[2]+G[2]), ..., (F[N]+G[N])
        (F-G) has N elements: (F[1]-G[1]), (F[2]-G[2]), ..., (F[N]-G[N])
        (F*G) has N elements: (F[1]*G[1]), (F[2]*G[2]), ..., (F[N]*G[N])

    mean, average
        Mean(F) = (1/N) * Sum( i=1;i<=N;i+=1; F[i] );

        Var(F) =  (1/N)*Sum( i=1;i<=N;i+=1; Sq(F[i]-Mean(F)) );
        Var(F) = ((1/N)*Sum( i=1;i<=N;i+=1; Sq(F[i])         )) - Sq(Mean(F));

    standard deviation
        Sd(F) = Sqrt( Var(F) );

        Cov(F,G) = (1/N) * Sum( i=1;i<=N;i+=1; ((F[i]-Mean(F)) * (G[i]-Mean(G))) );

    properties of variance and covariance
        Cov(F,F) = Var(F);
        Cov(F,G) = Cov(G,F);
        Var(F+G) = Var(F) + (2 * Cov(F,G)) + Var(G);

        Corr(F,G) = Cov(F,G) / (Sd(F) * Sd(G));

    properties of correlation

        (-1) <= Corr(F,G) <= (+1);

        Corr(F,G) = (+1) ==> "F and G are positively correlated"
        Corr(F,G) = ( 0) ==> "F and G are not correlated"
        Corr(F,G) = (-1) ==> "F and G are negatively correlated"

        Non-zero correlation values close to zero indicate that
            F and G are weakly correlated.

        Absolute correlation values close to unity indicate that
            F and G are strongly correlated (positively or negatively).

        Note: The square of the correlation value indicates approximately
        what fraction of the one variable's value varies directly with
        changes to the other variables's value.  So, a correlation of 0.5
        indicates that approximately 25% of one variable's value varies
        directly with changes to the other variable.



        Factorial( P )
          = Product( i=1; i<=P; i+=1; (i) );   "for integer P >= 0"

    properties of factorial

        Factorial( 0 ) = 1;
        Factorial( 1 ) = 1;
        Factorial( P ) = P * Factorial( P-1 ); "for integer P >= 1"
        Factorial( P ) = 1 * 2 * 3 * ... * (P-2) * (P-1) * P;
                             "for integer P>0 (and with P sufficiently
                             large for all terms shown to be positive)"

    values of factorial
       P  =  0       1       2       3       4       5       6       7       8
             1       1       2       6      24     120     720    5040   40320

    large-number approximation to factorial

        Factorial( N ) asymptotically approaches, for large N:
                      --> ( Sqrt(2*pi*N) * Pow(N,N) * Exp(-(N)) )



          = Factorial(N) /
            (Factorial(K) * Factorial( N - K ));       "where 0 <= K <= N"

          = "total number of unordered subsets, with each unordered subset
             containing K items taken from a pool of N distinct items";

    values of combinations
       K  =  0       1       2       3       4       5       6       7       8
N = 0  :     1
    1  :     1       1
    2  :     1       2       1
    3  :     1       3       3       1
    4  :     1       4       6       4       1
    5  :     1       5      10      10       5       1
    6  :     1       6      15      20      15       6       1
    7  :     1       7      21      35      32      21       7       1
    8  :     1       8      28      56      70      56      28       8       1

    combinations example

    if we have N = 4 distinct items { A, B, C, D }, and consider all unordered
    subsets of size K = 2, then Combinations(N=4,K=2) = 6 gives us the total
    number of such subsets: {A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}.


          = Factorial(N) / Factorial( N - K );
          = "total number of ordered subsets, with each ordered subset containing
             K items from a pool of N distinct items";

    values of permutations
       K  =  0       1       2       3       4       5       6       7       8
N = 0  :     1
    1  :     1       1
    2  :     1       2       2
    3  :     1       3       6       6
    4  :     1       4      12      24      24
    5  :     1       5      20      60     120     120
    6  :     1       6      30     120     360     720     720
    7  :     1       7      42     210     840    2520    5040    5040
    8  :     1       8      56     336    1680    6720   20160   40320   40320

    permutations example

    if we have N = 4 distinct items { A, B, C, D }, and consider all ordered
    subsets of size K = 2, then Permutations(N=4,K=2) = 12 gives us the total
    number of such subsets:  {A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D},
                             {B,A}, {C,A}, {D,A}, {C,B}, {D,B}, {D,C}.

    generalized permutations

        GeneralizedPermutations( N; K[1], K[2], K[3], ..., K[M] )

          = Factorial(N) /
            ( Factorial(K[1]) * Factorial(K[2]) * ... * Factorial(K[M]) );

          = (Factorial(N) / Product( i=1; i<=M; i+=1; Factorial(K[i]) ))

          = "permutations, of N total, with K[1] alike, K[2] alike, ..., K[M] alike"


    binomial theorem

        Pow( (a+b), N )
          = Sum( i=0; i<=N; i+=1; (Combinations(N,i) * Pow(a,N-i) * Pow(b,i)) );


    binomial distribution

        N = number of trials, experiments, or events
        K = some number of successes or failures; 0 <= K <= N;
        P = probability of any trial succeeding

            = (Combinations(N,K) * Pow(P,K) * Pow(P,(N-K)));
            = "probability of exactly K successes (and (N-K) failures)
               for a random experiment lasting N trials"

            = (P * N);

            = (P * (1-P) * N);

            = Sqrt( BinomialVariance(P,N) )
            = Sqrt( P * (1-P) * N );

            = BinomialProbabilityOfExactTotalOfSuccesses(P,N,N)
            = Pow(P,N);

            = BinomialProbabilityOfExactTotalOfSuccesses(P,N,0)
            = Pow((1-P),N);

            = (1 - Pow((1-P),N));

            = Sum( i=K; i<=N; i+=1;
                BinomialProbabilityOfExactTotalOfSuccesses(P,N,i) );

    fraction of binomial distribution within standard deviations of the mean

      S = number of standard deviations spanned on each side of the mean
          such that the full interval spanned is:
             [ (BinomialMean(P,N) - S * BinomialStandardDeviation(P,N)),
               (BinomialMean(P,N) + S * BinomialStandardDeviation(P,N)) ]

      A = fraction of binomial distribution within S standard deviations on
          both sides of the mean, assuming large N;
        = 2 * Integral( x; 0; S; (Exp(-x*x/2)/Sqrt(2*Pi)) );
        = (2/Sqrt(2*Pi)) * Integral( x; 0; S; Exp(-x*x/2) );

 S = 0   0.6744  0.5     1.0     1.5     2.0     2.5     3.0     3.5     3.62
 A = 0   0.5000  0.3830  0.6826  0.8664  0.9544  0.9876  0.9974  0.9996  0.9998

       50.00% of the area within 0.6744 standard deviations of the mean;

       68.26% of the area within one    standard deviation  of the mean;
       95.44% of the area within two    standard deviations of the mean;
       99.74% of the area within three  standard deviations of the mean;

       99.98% of the area within 3.62   standard deviations of the mean;

       (These percentages are approached asymptotically as N increases;
        differences from these values for N = 10, or N = 20, etc, could
        be computed; I suspect that the percentages above are accurate
        to all digits shown for N >= 50.)

    multinomial distribution

        N = number of trials, experiments, or events
        M = number of distinct bins or classifications for the result of each trial

        { K[1], K[2], K[3], ..., K[M] }    = populations of each of the M bins
                                             [Note: (K[1]+K[2]+K[3]+...+K[M]) = N ]

        { P[1], P[2], P[3], ..., P[M] }    = probability of choosing
                                             each of the M bins
                                             [Note: (P[1]+P[2]+P[3]+...+P[M]) = 1 ]

        { mean[1], mean[2], ..., mean[M] } = expected mean (average) population
                                             of each of the M bins

        { var[1], var[2], ..., var[M] }    = expected variance for the population
                                             of each of the M bins
                                             [Note: variance is the square of
                                             the expected standard deviation]

        ( sd[1], sd[2], ..., sd[M] }       = expected standard deviation for
                                             the population of each of the M bins

        { cov[1,1], cov[1,2], cov[2,1], ..., cov[M,M] }
                                           = expected covariance between bins
                                             across all of the possible (and
                                             half-redundant) (M * M) bin pairs

    probability of a specific configuration

        P( K[1], K[2], ..., K[M] )

          =  GeneralizedPermutations( N; K[1], K[2], K[3], ..., K[M] ) *
                  Product( j=1; j<=M; j+=1; Pow(P[j],K[j]) );

          =  (Factorial(N) / Product( i=1; i<=M; i+=1; Factorial(K[i]) )) *
                  Product( j=1; j<=M; j+=1; Pow(P[j],K[j]) );

        [Note: This is the general form of any specific term in the algebraic
        expansion of Pow( (P[1] + P[2] + ... + P[M]), N ).]


          mean[i] = ( N * P[i] );


          var[i] = ( N * P[i] * (1 - P[i]) );

        standard deviation

          sd[i] = Sqrt( var[i] );


          cov[i,j] = ( - ( N * P[i] * P[j] ) );

    interesting properties of multinomial expansions

        Pow( (a + b), n )         has   (n+1)                       terms;
        Pow( (a + b + c), n )     has   (n+1)*(n+2)       / (2)     terms;
        Pow( (a + b + c + d), n ) has   (n+1)*(n+2)*(n+3) / (2*3)   terms;
        . . . .
        Pow( (a + b + c + d + ... + x), n ), with k variables total, has:
                    (n+1)*(n+2)*...*(n+(k-1)) / (Factorial(k-1))    terms;

        Or,( Factorial(n+(k-1)) / (Factorial(n) * Factorial(k-1)) ) terms;
        Or,          BinomialCoefficient( (n+(k-1)), n )            terms.

        This number also corresponds to the number of distinct ways of forming
        groups of n items, with up to k distinct kinds of items, where order is
        not important, and the number of instances of any of the k kinds of
        items can range from zero through n.

        This number also arises in statistical mechanics, for the microcanonical
        analysis of entities following Bose-Einstein statistics, giving the number
        of ways, Wj, of arranging the jth level for Nj bosons among Gj states:
        (Wj)BE = (Factorial(Nj+(Gj-1)) / (Factorial(Nj) * Factorial(Gj- 1))).
        (Whereas fermions only allow single occupation of states, leading to:
        (Wj)FD = (Factorial(    Gj   ) / (Factorial(Nj) * Factorial(Gj-Nj))).)
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