How can In break a heap into two heaps in the Grundy’s game?
What about breaking a heap into any number of heaps (no two of them being equal)?
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How can In break a heap into two heaps in the Grundy’s game?
What about breaking a heap into any number of heaps (no two of them being equal)?
Games of this type are analyzed in great detail in the book series “Winning Ways for your Mathematical Plays“. Most of the things you are looking for are probably in volume 1.
You can also take a look at these links: Nimbers (Wikipedia), Sprague-Grundy theorem (Wikipedia) or do a search for “combinatorial game theory”.
My knowledge on this is quite rusty, so I’m afraid I can’t help you myself with this specific problem. My excuses if you were already aware of everything I linked.
Edit: In general, the method of solving these types of games is to “build up” stack sizes. So start with a stack of 1 and decide who wins with optimal play. Then do the same for a stack of 2, which can be split into 1 & 1. The move on to 3, which can be split into 1 & 2. Same for 4 (here it gets trickier): 3 & 1 or 2 & 2, using the Spague-Grundy theorem & the algebraic rules for nimbers, you can calculate who will win. Keep going until you reach the stack size for which you need to know the answer.
Edit 2: The website I was talking about in the comments seems to be down. Here is a link of a backup of it: Wayback Machine – Introduction to Combinatorial Games.