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From: mwm@ea.UUCP
Newsgroups: net.math
Subject: Re: game solution - why does it work?
Message-ID: <6900010@ea.UUCP>
Date: Thu, 27-Sep-84 17:34:00 EDT
Article-I.D.: ea.6900010
Posted: Thu Sep 27 17:34:00 1984
Date-Received: Mon, 1-Oct-84 03:43:19 EDT
References: <12@utecfc.UUCP>
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Nf-ID: #R:utecfc:-1200:ea:6900010:000:1157
Nf-From: ea!mwm    Sep 27 16:34:00 1984

I'm not sure this should go to the net, but...

Take the two points about the sum you already have:

/***** ea:net.math / utecfc!baines /  3:25 am  Sep 25, 1984 */
    Two (obvious?) notes: If you get at least one 'odd' column, you can subtract
    a number to make all columns even.
			  If all columns are 'even', whatever your opponent 
    takes away will make at least one column 'odd'.
    Anyone know why?              THANKS
					Ian
/* ---------- */

Now, consider the nim-sum if you have just won the game. It's even. Therefore,
any move that leaves the game in an odd state *can't* be a winning move. So
by leaving the game in any even state, you insure that your opponent can't
win. Since somebody has to win (no loops), and your opponent can't win,
you win.

Note that most take-away games can be won by a similar strategy.

I recommend the books "Winning Ways" (two volumes) by Berlekamp, Conway and
Guy. You can get them from Academic Press, but they are expensive.  The
first one ("Games in General") stands alone, and explains nim (among other
things). The second one ("Games in Particular") assumes that you've read
the first one.