Some time ago I wrote the following in response to a David Gale message. It seems (not surprisingly?) not to have been completed. For what it is worth, I belatedly send it. R. Yes, someone else is biting. I'll do anything for money. Perhaps Don or Thane or Scott will give me 10% of the takings, if they push through the rest of this. I wonder if David will pay up for a complete STRATEGY, or does he want a complete ANALYSIS ? 1 non-empty heap. P-position. nim-value 0 2 non-empty heaps, sizes a and b. N-position. STRATEGY: Take whole of either heap. ANALYSIS: nim-value [(a-1)nim(b-1)] + 1 3 non-empty heaps, sizes a, b, c. STRATEGY: Pretend that you can't see the last bean in each heap. If you're forced, or stupid enough, to take a whole heap, see `2 non-empty heaps' above. Otherwise, play ordinary Nim on the heaps that you can `see'. I.e., you aim to `finish' the game by leaving (1,1,1). ANALYSIS: P-positions are just those with (a-1)nim(b-1)nim(c-1) = 0. However the general nim-value of (a,b,c) seems to be chaotic (though, as David Gale and others have shown, the value for fixed a, b is ultimately arithmetico-periodic). Examples of P-positions: (n,n,1), {2n+2,2n+1,2) (4n+3,4n+1,3), (4n+4,4n+2,3), ... 4 non-empty heaps, sizes a, b, c, 1. STRATEGY: Play ordinary Nim on heaps a, b, c. If your opponent takes the 1-heap, then pretend as above. Note that on parity grounds (a,b,c) and (a-1,b-1,c-1) can't both be P-positions. 4 non-empty heaps, all of size 2 or more. STRATEGY: If two heaps are of equal size, equalize the other two. If two pairs of equal heaps, hope that it's your opponent's turn to play. Note that you are playing MIS`ERE Nim on the first of the two pairs to vanish. That is, if your opponent goes to 1 or 0 in a heap, then you go to 0 or 1 respectively in the previously equal heap. So, assume that all heap sizes are distinct and at least 2. If any three heaps form a three-heap P-position, take the fourth heap. E.g., (2,3,4,n) is an N-position ---> (2,3,4) The smallest non-trivial P-position comprises the Fibonacci numbers (2,3,5,8). Here are good replies to all moves from this. 1358-->1356 358-->357 2258-->2255 2158-->2157 258-->256 2348-->234 2338-->2332 2328-->2323 2318-->2311 (mis`ere!) 238-->234 2357-->357 2356-->256 2355-->2255 2354-->234 2353-->2323 2352-->2332 2351-->2311 235-->234 In general the P-positions are those for which (a-1)nim(b-1)nim(c-1)nim(d-1) = 0, with the caveat that none of the heaps is 1. On such 1-heaps we are playing mis`ere Nim.