From: Robin Chapman
Subject: Re: The Golay Code and a Game in Winning Ways
Date: Tue, 12 Jan 1999 18:56:13 GMT
Newsgroups: sci.math
Keywords: The game of Mogul
In article <77g12u$t1o$1@nnrp2.dejanews.com>,
torquemada@my-dejanews.com wrote:
> I don't have access to the book Winning Ways so could someone please, please,
> please, help me out by posting the rules of the (heap) game Mogul apparently
> described on page 435 of that book.
I don't have the book to hand but here's the gist. You are given a row
of n coins, so each has a head or tail visible. We fix a number k. The
legitimate moves are to reverse between 1 and k coins with the proviso that
the left-most coin reversed must change from head to tail. One wins
by leaving all coins tail up.
We can model each position as a binary number between 0 and 2^k-1 with heads
being 1 and tails 0. Each move changes up to k digits, but reduces the number
in size. The winning positions are thus the least binary numbers with Hamming
distance at least k+1 from all previous winning numbers. The set of winning
numbers thus forms a lexicographic code. With n = 24 and k = 7 we get the
(extended) binary Golay code.
> I'm interested in that game because (remarkably!!!) the winning positions can
> be thought of as codewords in the extended binary Golay code. Does Winning
> Ways discuss this aspect of the game? I've been reading a paper on this by
> Sloane so I know what the game is up to isomorphism (in the sense of surreal
> numbers) - but I'm curious to know what the *actual* rules are.
I don't think they mention the code explicitly, but they do give the
original version of the Miracle Octad Generator (due to Curtis) for finding
the code words of weight 8.
Robin Chapman
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