The Big Smile

March 13, 2008

4 1(5)2 can be drawn to resemble a smiley face. Extending the smile with 1(6)3 creates a position which I pondered Monday while sitting in St. Bart's Coffee Company in Fort Lauderdale. Not on spring break - I'm much too old for that - but just down there visiting my son.

But first, is 4 1(5)2 a switch? I found the positive assertion is some notes I wrote several years ago. And what do I mean by a switch? Switch is a concept of mine, but it has never caught on. I have actually forgotten my definition, but I suppose it is still in the WGOSA Glossary. That glossary, by the way, badly needs thorough revision. I will try to work on that as time allows.

But at least to some approximation a switch is a formation that, if played as a game, can be forced by the player to move to yield that player's choice of either an even or an odd number of survivors. So I'm thinking maybe that 4 1(5)2 is indeed a switch, with 3(6)5 limiting survivors to two. Please, someone correct me if I'm wrong. I have given the question a moment's thought.

Correct me, too, on my St. Bart's conclusions. I see the big smile also as a switch, and with at least one waiting move. I have 4 1(5)2 1(6)3 1(7)6[3] giving three survivors, and 4 1(5)2 1(6)3 4(7)5 yielding two survivors. My waiting move is 4 1(5)2 1(6)3 2(7)2 .

Jeff Peltier uses the following notation, which I think might be standard in the theory of mathematical games. n ^ m , for integers n and m, refers to normal and misere Sprague-Grundy numbers. Every formation has both a normal and a misere Sprague-Grundy number. The normal Sprague-Grundy number of a formation is the lowest nonnegative number not assigned to any possible direct descendent of that formation. I am not completely clear on misere Sprague-Grundy numbers. Perhaps someone could write an article on that. But I believe the definition is the same except that formations with no direct descendents are assigned 1's instead of 0's.

Anyway, I propose the following Sprague-Grundy numbers (and I'm really going out on a limb here):

4 1(5)2: 3^3
4 1(5)2 1(6)3 : 2^2
4 1(5)2 1(6)3 1(7)6[3] : 0^1
4 1(5)2 1(6)3 4(7)5 : 1^0
4 1(5)2 1(6)3 2(7)2 : 3^3

Another of my idiosyncratic concepts is switchman. A switchman is a survivor when we play by the rule that switches may not be reduced to nonswitches. (Let's call that railroad sprouts.) I see 4 1(5)2 1(6)3 2(7)2 yielding an even number of switchmen. Since that formation has an odd number of cannibals and will yield an even number of switchmen, I see it as 3^3 instead of 2^2.

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