HAKMEM
 /hak'mem/, n.

    MIT AI Memo 239 (February 1972). A legendary collection of neat
    mathematical and programming hacks contributed by many people at MIT and
    elsewhere. (The title of the memo really is ?HAKMEM?, which is a
    6-letterism for ?hacks memo?.) Some of them are very useful techniques,
    powerful theorems, or interesting unsolved problems, but most fall into the
    category of mathematical and computer trivia. Here is a sampling of the
    entries (with authors), slightly paraphrased:

    Item 41 (Gene Salamin): There are exactly 23,000 prime numbers less than 2^
    18.

    Item 46 (Rich Schroeppel): The most probable suit distribution in bridge
    hands is 4-4-3-2, as compared to 4-3-3-3, which is the most evenly
    distributed. This is because the world likes to have unequal numbers: a
    thermodynamic effect saying things will not be in the state of lowest
    energy, but in the state of lowest disordered energy.

    Item 81 (Rich Schroeppel): Count the magic squares of order 5 (that is, all
    the 5-by-5 arrangements of the numbers from 1 to 25 such that all rows,
    columns, and diagonals add up to the same number). There are about 320
    million, not counting those that differ only by rotation and reflection.

    Item 154 (Bill Gosper): The myth that any given programming language is
    machine independent is easily exploded by computing the sum of powers of 2.
    If the result loops with period = 1 with sign +, you are on a
    sign-magnitude machine. If the result loops with period = 1 at -1, you are
    on a twos-complement machine. If the result loops with period greater than
    1, including the beginning, you are on a ones-complement machine. If the
    result loops with period greater than 1, not including the beginning, your
    machine isn't binary ? the pattern should tell you the base. If you run out
    of memory, you are on a string or bignum system. If arithmetic overflow is
    a fatal error, some fascist pig with a read-only mind is trying to enforce
    machine independence. But the very ability to trap overflow is machine
    dependent. By this strategy, consider the universe, or, more precisely,
    algebra: Let X = the sum of many powers of 2 = ...111111 (base 2). Now add
    X to itself: X + X = ...111110. Thus, 2X = X - 1, so X = -1. Therefore
    algebra is run on a machine (the universe) that is two's-complement.

    Item 174 (Bill Gosper and Stuart Nelson): 21963283741 is the only number
    such that if you represent it on the PDP-10 as both an integer and a
    floating-point number, the bit patterns of the two representations are
    identical.

    Item 176 (Gosper): The ?banana phenomenon? was encountered when processing
    a character string by taking the last 3 letters typed out, searching for a
    random occurrence of that sequence in the text, taking the letter following
    that occurrence, typing it out, and iterating. This ensures that every
    4-letter string output occurs in the original. The program typed
    BANANANANANANANA.... We note an ambiguity in the phrase, ?the Nth
    occurrence of.? In one sense, there are five 00's in 0000000000; in
    another, there are nine. The editing program TECO finds five. Thus it finds
    only the first ANA in BANANA, and is thus obligated to type N next. By
    Murphy's Law, there is but one NAN, thus forcing A, and thus a loop. An
    option to find overlapped instances would be useful, although it would
    require backing up N ? 1 characters before seeking the next N-character
    string.

    Note: This last item refers to a Dissociated Press implementation. See
    also banana problem.

    HAKMEM also contains some rather more complicated mathematical and
    technical items, but these examples show some of its fun flavor.

    An HTML transcription of the entire document is available at http://
    www.inwap.com/pdp10/hbaker/hakmem/hakmem.html.

HAKMEM

    /hak'mem/ MIT AI Memo 239 (February 1972).  A
   legendary collection of neat mathematical and programming
   hacks contributed by many people at MIT and elsewhere.  (The
   title of the memo really is "HAKMEM", which is a 6-letterism
   for "hacks memo".)  Some of them are very useful techniques,
   powerful theorems, or interesting unsolved problems, but most
   fall into the category of mathematical and computer trivia.
   Here is a sampling of the entries (with authors), slightly
   paraphrased:

   Item 41 (Gene Salamin): There are exactly 23,000 prime numbers
   less than 2^18.

   Item 46 (Rich Schroeppel): The most *probable* suit
   distribution in bridge hands is 4-4-3-2, as compared to
   4-3-3-3, which is the most *evenly* distributed.  This is
   because the world likes to have unequal numbers: a
   thermodynamic effect saying things will not be in the state of
   lowest energy, but in the state of lowest disordered energy.

   Item 81 (Rich Schroeppel): Count the magic squares of order 5
   (that is, all the 5-by-5 arrangements of the numbers from 1 to
   25 such that all rows, columns, and diagonals add up to the
   same number).  There are about 320 million, not counting those
   that differ only by rotation and reflection.

   Item 154 (Bill Gosper): The myth that any given programming
   language is machine independent is easily exploded by
   computing the sum of powers of 2.  If the result loops with
   period = 1 with sign +, you are on a sign-magnitude machine.
   If the result loops with period = 1 at -1, you are on a
   twos-complement machine.  If the result loops with period
   greater than 1, including the beginning, you are on a
   ones-complement machine.  If the result loops with period
   greater than 1, not including the beginning, your machine
   isn't binary - the pattern should tell you the base.  If you
   run out of memory, you are on a string or bignum system.  If
   arithmetic overflow is a fatal error, some fascist pig with a
   read-only mind is trying to enforce machine independence.  But
   the very ability to trap overflow is machine dependent.  By
   this strategy, consider the universe, or, more precisely,
   algebra: Let X = the sum of many powers of 2 = ...111111 (base
   2).  Now add X to itself: X + X = ...111110.  Thus, 2X = X -
   1, so X = -1.  Therefore algebra is run on a machine (the
   universe) that is two's-complement.

   Item 174 (Bill Gosper and Stuart Nelson): 21963283741 is the
   only number such that if you represent it on the PDP-10 as
   both an integer and a floating-point number, the bit
   patterns of the two representations are identical.

   Item 176 (Gosper): The "banana phenomenon" was encountered
   when processing a character string by taking the last 3
   letters typed out, searching for a random occurrence of that
   sequence in the text, taking the letter following that
   occurrence, typing it out, and iterating.  This ensures that
   every 4-letter string output occurs in the original.  The
   program typed BANANANANANANANA....  We note an ambiguity in
   the phrase, "the Nth occurrence of."  In one sense, there are
   five 00's in 0000000000; in another, there are nine.  The
   editing program TECO finds five.  Thus it finds only the first
   ANA in BANANA, and is thus obligated to type N next.  By
   Murphy's Law, there is but one NAN, thus forcing A, and thus a
   loop.  An option to find overlapped instances would be useful,
   although it would require backing up N - 1 characters before
   seeking the next N-character string.

   Note: This last item refers to a Dissociated Press
   implementation.  See also banana problem.

   HAKMEM also contains some rather more complicated mathematical
   and technical items, but these examples show some of its fun
   flavour.

   HAKMEM is available from MIT Publications as a TIFF file.

   <ftp://ftp.netcom.com/pub/hb/hbaker>.

   (1996-01-19)