HAKMEM

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.