Unformatted text preview: 3. Note that (1 , 2 , 3) denotes 3 piles of “Lasker” heaps, not nimheaps. By the sum theorem, g (1 , 2 , 3) = g 1 (1) ⊕ g 2 (2) ⊕ g 3 (3), where g 1 ,g 2 ,g 3 denote the SpragueGrundy functions of the single pile Lasker’s Nim games. Using the deﬁnition of the SpragueGrundy function, the values of the g i can be worked out. The only one diﬀerent from regular nim is 3. Since, in addition to the usual nim moves, 3 can be split into the two smaller piles of sizes 2 and 1, we ﬁnd that the SpragueGrundy value of (3) is in fact 4. Thus we have g (1 , 2 , 3) = 1 ⊕ 2 ⊕ 4 = 7 and (1 , 2 , 3) is an Nposition for Lasker’s Nim. 4. The winning move is to split the pile with 3 chips into two smaller piles with 2 chips and 1 chip. This gives us the position (1 , 2 , 1 , 2) which has SpragueGrundy value 0....
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 Spring '09
 MURALISTOYANOV

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