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May 5

Game Theory

I have a question about what I believe is called "combinatorial game theory". Anyway, the kind with perfect knowledge, no randomness, that kind of thing. Other than Nim, can you think of any impartial games it applies to that weren't invented for the purpose of applying it to them? About all I can find are examples of how you would analyze a game, if people actually played games that way. Also, can you think of any partisan games it's entirely solved, that weren't invented for that purpose? Black Carrot 04:59, 5 May 2007 (UTC)[reply]

If you are interested in this kind of stuff, I recommend Winning Ways for your Mathematical Plays. Our article on the book lists a number of games that can be analyzed using the Sprague–Grundy theorem, such as Wythoff's game, which dates back to 1907, predating the discovery of the theorem by 28 years. Among the partisan games analyzed in the book are Domineering, Col, and Snort, but I don't know if these are completely solved, nor who invented them when. --Lambiam ^Talk 06:31, 5 May 2007 (UTC)[reply]

Solved game.--Kirby♥time 11:35, 5 May 2007 (UTC)[reply]

Ah, that's a start. Thanks. Black Carrot 04:15, 6 May 2007 (UTC)[reply]

Another example that comes to mind is Go:Chilling gets the last point. As I recall it's a game theoretical analysis of Go endgames. Dugwiki 19:40, 7 May 2007 (UTC)[reply]

proof for arithmetic progression

\ a_{n}=a_{1}+(n-1)d,

always works, but I'm just wondering if there is a proof for that equation. Thanks. Imaninjapirate^{talk to me} 05:22, 5 May 2007 (UTC)[reply]

One possibility is to take this equation to be part of the definition of being an arithmetic progression. Then it is true by definition. Alternatively, you can take as the definition that, for some d, a_n+1 = a_n + d for n = 1, 2, .... Then you can use mathematical induction to prove the equation. --Lambiam ^Talk 05:38, 5 May 2007 (UTC)[reply]

Suggestions for a proof

I am convinced that if G = G₁ x G₂, then the Cayley table of G is the Kronecker product of the Cayley tables of G₁ and G₂. How would one prove this (not looking for a complete proof, please don't provide one)? Would it be relatively straightforward?

I've never heard the terms Cayley table or Kronecker product before, but looking at the definitions, and given that we order the elements of G in the obvious way (i.e. lexicographically), isn't this just completely obvious? i.e. write down definition of an element of the Kronecker product, write down definition of an element of the Cayley table, observe they coincide? Algebraist 12:22, 5 May 2007 (UTC)[reply]

Yeah, it is rather plain and obvious. Often I look at something and read too much complexity into proving it. 12:46, 5 May 2007 (UTC)