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From: monta@cmu-cs-g.ARPA (Peter Monta)
Newsgroups: net.math
Subject: MasterMind, Jotto, entropy
Message-ID: <246@cmu-cs-g.ARPA>
Date: Fri, 8-Mar-85 02:19:37 EST
Article-I.D.: cmu-cs-g.246
Posted: Fri Mar  8 02:19:37 1985
Date-Received: Sun, 10-Mar-85 05:42:25 EST
Organization: Carnegie-Mellon University, CS/RI
Lines: 33

> Guessing "wrong" answers in Mastermind can indeed get you to the answer
> sooner than guessing only right answers.
>   - Joel McCormack {ihnp4 decvax ucbvax allegra}!decwrl!joel

This is an important point, which took me some time to realize.  I worked
on a near-isomorph of MasterMind called Jotto as part of a cognitive
psychology course---I finally wrote a program very similar to the
description of the exhaustive table-generating program.

Jotto is 26-color, 5-position MasterMind, duplicate colors permitted,
guess feedback only the number of pegs (no distinction between black
and white).  Of course, the colors are letters, guesses are five-letter
words, and feedback is a number from 0 to 5.  Guesses are restricted to
valid five-letter words, as are targets.

My thought is that the right thing to maximize is the *information*
obtained from each guess.  Suppose you guess a word.  The act of guessing
partitions the set of all possibilities into six blocks---those with
0 letters in common with the guess, those with 1 letter in common, etc.
When the opponent tells you the disposition of your guess, he gives you
information in the amount of the entropy of this partition.  Assuming
that the targets are equally probable, the entropy is the sum over blocks
of P*log P, where P is the cardinal of the block.

The program would simply iterate through its dictionary, guessing the
entry that maximized the entropy of the partition it induced.  The technique
should be applicable to any MasterMind-like game; in fact, it should do
even better with black and white pegs, since this would increase the number
of blocks.

Peter Monta
ARPA: monta@cmu-cs-g
UUCP: ...!rochester!cmu-cs-pt!cmu-cs-g!monta