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Solutions to homework 2
4.5#2
Onepile game with the following rules:
You may remove (1) any number of chips divisible by three provided it is not the whole pile, or (2) the
whole pile, but only if it has 2 (mod 3) chips.
The terminal positions are zero, one, and three. They have SpragueGrundy value 0. From 2 we can
only move to 0, by move (2), hence
g
(2) = 1. From 4 we can only move to 1, by move (1), hence
g
(4) = 1. From 5 we can move to 2 and 0, by moves (1) and (2), respectively, hence
g
(5) = 2, and so
on. The ﬁrst 15 SG values are the following:
x
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
g
(
x
)
0
0
1
0
1
2
1
2
3
2
3
4
3
4
5
We can observe that there is a pattern. Disregarding the ﬁrst three positions (0, 1, 2) three consecutive
numbers are increasing by 1 and the next three are also increasing in the same fashion, but the ﬁrst
element is 1 more than that of the previous triple. Formally,
g
(0) = 0 and
∀
n >
0 we have that
g
(
n
) =
b
n
3
c 
1 + (
n
mod 3)
.
Here,
b
x
c
denotes the largest integer less than or equal to
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This note was uploaded on 08/24/2009 for the course MATH 262447221 taught by Professor Weisbart during the Spring '09 term at UCLA.
 Spring '09
 WEISBART

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