Solutions to homework 3
5.5#1
Coin turning games
We analyze the following position under different coin turning game rules:
position
1
2
3
4
5
6
7
8
9
T
T
H
T
H
H
T
T
H
Using the decomposition principle we can represent the game as a (disjunctive) sum of 4 games each
exactly one head:
g
(TTHTHHTTH) =
g
(TTHTTTTTT)
⊕
g
(TTTTHTTTT)
⊕
g
(TTTTTHTTT)
⊕
g
(TTTTTTTTH)
=
g
(3)
⊕
g
(5)
⊕
g
(6)
⊕
g
(9)
.
Note, in part (a) we use this numbering, i.e., starting from 0, e.g.,
g
(TTH) =
g
(3), but in parts (b)(d)
we shift the indexes by one, e.g.,
g
(TTH) =
g
(2).
(a) A move in Turning Turtles is turning over a coin from heads to tails and, in addition, if desired,
turning over one other coin to the left of it. Note, this is essentially nim, the height of each pile is
represented by the location of the corresponding coin. Note, that
g
(
x
) =
x
. Hence, the SGvalue:
g
(TTHTHHTTH) =
g
(3)
⊕
g
(5)
⊕
g
(6)
⊕
g
(9) = 3
⊕
5
⊕
6
⊕
9 = 9
6
= 0
.
So, it is an Nposition, the
only winning move is to flip the coin on position 9.
(b) A move in Twins is to flip two coins, the rightmost being a heads to tails flip.
This is also
nim, but with the indexes shifted by one, i.e.,
g
(
x
) =
x
(with the new numbering). Therefore,
g
(TTHTHHTTH) =
g
(2)
⊕
g
(4)
⊕
g
(5)
⊕
g
(8) = 2
⊕
4
⊕
5
⊕
8 = 11. It is an Nposition, the winning
move is to flip the coins on positions 8 (from heads to tails) and 3 (from tails to heads).
(c) In a Subtraction game with subtraction set
S
=
{
1
,
3
,
4
}
, when exactly two coins must be turned
over a legal move is the following. Turn two coins over, one in position
x
(heads to tails) and the
other in position
x

1,
x

3 or
x

4. Note, if we always require each move to turn two coins
(even if
x
≤
4, compare with example in page 29) then the SGvalues for the first 9 positions are
the following:
x
0
1
2
3
4
5
6
7
8
g
(
x
)
0
1
0
1
2
3
2
0
1
.
Hence,
g
(2)
⊕
g
(4)
⊕
g
(5)
⊕
g
(8) = 0
⊕
2
⊕
3
⊕
1 = 0. This is a Pposition, no winning moves here.
(d) In Mock Turtles a legal move is to flip one, two or three coins, the rightmost being a heads to
tails flip.
Looking up the SGvalues from page 30 we have that:
g
(2)
⊕
g
(4)
⊕
g
(5)
⊕
g
(8) =
4
⊕
8
⊕
11
⊕
16 = 23
.
This is an Nposition, a winning move is to flip coin 8 from heads to tails