Date: Tue, 5 Dec 1995 09:50:59 -0500
From: gls@atpsol.cvu.att.com (G. L. Sicherman)
To: rusin@math.niu.edu
Subject: coinage

	Date: Tue, 5 Dec 95 00:21:55 CST
	From: rusin@math.niu.edu (Dave Rusin)
	To: gls@hrcms.att.com
	Subject: Re: Sylver Coinage - {8,26,30} has an odd winning move

	1. Could you summarize what the problem is? I don;t think I've ever
	heard of it except in this newsgroup.

Sylver Coinage is J. H. Conway's game of denomination.  Two players take
turns naming denominations of coins.  No new coin may be expressible as
a sum, with multiples, of previous coins.  The player who names 1 loses.

Example:
		 4
		  	5 (not a multiple of 4)
		11	  (not a sum of 4's and 5's)
		        7 (not a sum of 4's, 5's, and 11's)
		 6	  (etc.)
			2
		 3
			1 (loses)

The game must end (why?), but need not have a time limit at first.

	4, 99999, 99998, 99997, 99995, ...
	4, 999999999, ...

Any prime greater than 3 is a winning first move.  See _Winning Ways_
for proof.  Some other safe positions are {2,3} (no moves left but 1!),
{4,5,11}, {4,6}, {6,9}, {8,12}, {8,14}, and {10,26}.  The chief question
in Sylver Coinage is which positions are safe, especially with g.c.d. > 1.

The literature is sparse.  Vol. 2 of _Winning Ways,_ occasional updates
by R. K. Guy in the _Monthly,_ Nowakowski's 1991 paper in _Combinatorial
Games,_ and my unpublished paper of the same year.

[deletia - djr]

					--Col. G. L. Sicherman
					  gls@hrcms.att.com