HAKMEM /hak'mem/ n. MIT AI Memo 239 (February 1972)

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.