From fcfaaa6e2d67dd6a7de408937ff76308d1fe2eb9 Mon Sep 17 00:00:00 2001 From: Antti H S Laaksonen Date: Sat, 18 Feb 2017 17:13:06 +0200 Subject: [PATCH] Corrections --- luku25.tex | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/luku25.tex b/luku25.tex index 17bc3da..37ccbbd 100644 --- a/luku25.tex +++ b/luku25.tex @@ -209,7 +209,7 @@ is as follows: \end{tikzpicture} \end{center} -Surprisingly enough, in this game, +Surprisingly, in this game, all even-numbered states are winning states, and all odd-numbered states are losing states. @@ -256,7 +256,7 @@ $[10,12,5]$ is $10 \oplus 12 \oplus 5 = 3$, so the state is a winning state. But how is the nim sum related to the nim game? -We can explain this by studying how the nim +We can explain this by looking at how the nim sum changes when the nim state changes. ~\\ @@ -395,7 +395,7 @@ The \key{Grundy number} for a game state is \[\textrm{mex}(\{g_1,g_2,\ldots,g_n\}),\] where $g_1,g_2,\ldots,g_n$ are Grundy numbers for states to which we can move from the state, -and the mex function returns the smallest +and the mex function gives the smallest nonnegative number that is not in the set. For example, $\textrm{mex}(\{0,1,3\})=2$. If there are no possible moves in a state, @@ -787,10 +787,10 @@ For example, when $n=8$, the possibilities are $1+7$, $2+6$ and $3+5$, so \[f(8)=\textrm{mex}(\{f(1) \oplus f(7), f(2) \oplus f(6), f(3) \oplus f(5)\}).\] -In this game, the value of $f(n)$ is based on values +In this game, the value of $f(n)$ is based on the values of $f(1),\ldots,f(n-1)$. The base cases are $f(1)=f(2)=0$, -because it is not possible to divide heaps +because it is not possible to divide the heaps of 1 and 2 sticks. The first Grundy numbers are: \[