Stat 155 Fall 2009: Solutions to Homework 3
(was due September 24, 2009)
1. This is a sum of two subtraction games. Using the notation from class, we
can call the subtraction sets
S
4
and
S
5
. Then we know that if the two piles
have
n
and
m
chips, then
g
4
(
n
) =
n
mod 5 and
g
5
(
m
) =
m
mod 6. For
(
n,m
) to be a
P
position, we must have
g
((
n,m
)) =
g
4
(
n
)
⊕
g
5
(
m
) = 0.
This is true if and only if
g
4
(
n
) =
g
5
(
m
), that is:
n
mod 5 =
m
mod 6.
(Also see Example 2.13.)
2.
•
The loops in the ﬂower become stalks, and since 1
⊕
1
⊕
1 = 1, the
ﬂower has SG value 2.
•
In the case of the little girl, fuse the two vertices in her head, since
there are two edges, this circuit reduces to a single vertex, as does
the circuit that forms her skirt (since it has 4 edges). Proceeding in
this way, and using the Colon principle, we see that the SG value of
the girl is 3.
•
Keep in mind that the ground is considered a single vertex, so the
legs actually form a circuit with two vertices and two edges. Using
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This note was uploaded on 10/02/2009 for the course STAT 87531 taught by Professor Muralistoyanov during the Spring '09 term at Berkeley.
 Spring '09
 MURALISTOYANOV

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