Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: $Revision: 1.6.2.16 $; site ea.UUCP Path: utzoo!watmath!clyde!burl!ulysses!mhuxj!cbosgd!ihnp4!inuxc!pur-ee!uiucdcs!ea!mwm 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> Lines: 29 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.