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经济代写|博弈论代考Game theory代写|CASEC513 Sums of Nim Heaps

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经济代写|博弈论代考Game theory代写|Sums of Nim Heaps

In this section, we derive how to compute the Nim value for a general Nim position, which is a sum of different Nim heaps. This will be the Nim sum that we have defined using the binary representation, now cast in the language of game sums and equivalent games, and without assuming the binary representation.

For example, we know that $* 1+* 2+* 3 \equiv 0$, so by Lemma $1.12, * 1+* 2$ is equivalent to $* 3$. In general, however, the sizes of the Nim heaps cannot simply be added to obtain the equivalent Nim heap, because $* 2+* 3$ is also equivalent to $* 1$, and $* 1+* 3$ is equivalent to $* 2$.

If $* k \equiv * n+* m$, then we call $k$ the Nim sum of $n$ and $m$, written $k=n \oplus m$. The following theorem states that the Nim sum of distinct powers of two is their arithmetic sum. For example, $1=2^{0}$ and $2=2^{1}$, so $1 \oplus 2=1+2=3$.

经济代写|博弈论代考Game theory代写|Finding Nim Values

In this section, we analyze some impartial games using the mex rule in Theorem $1.14$

A game similar to the Rook-move game is the Queen-move game shown in Figure $1.3$ where the rook is replaced by a Chess queen, which may move horizontally, vertically, and diagonally (left or up). The squares on the main diagonal are therefore no longer losing positions. This game can also be played with two heaps of tokens where in one move, the player may either remove tokens from one heap as in Nim, or reduce both heaps by the same number of tokens (so this is no longer a sum of two Nim heaps!). In order to illustrate that we are not just interested in the winning and losing squares, we add to this game a Nim heap of size 4 .

Figure $1.4$ shows the equivalent Nim heaps for the positions of the Queen-move game, determined by the mex rule. The square in row 3 and column 4 occupied by the queen in Figure $1.3$ has entry $* 2$. So a winning move is to remove two tokens from the Nim heap to turn it into the heap $* 2$, creating the losing position $* 2+* 2$. Because 2 is the mex of the Nim values of the options of the queen, these may include (as in Poker Nim) higher Nim values. Indeed, the queen can reach two positions equivalent to $* 4$, in row 3 column 1 , and row 0 column 4 . If the queen moves there this creates the game sum $* 4+* 4$ which is losing, so these are two further winning moves in Figure 1.3.

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经济代写博恋论代考Game theory代写| Finding Nim Values

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