Solutions to homework 1
1.5#1
Mis`
ere version of the takeaway game. There are 21 chips, we can remove 1, 2, or 3.
Last player to move loses, hence position 1 is a Pposition, from positions 2,3, and 4 we can move to
1, hence these are Npositions. Now, 5 is a Pposition again, since we can only move to Npositions
from it, and so on.
.. We can guess the pattern by the ﬁrst couple sample points: if
x
≡
1 mod 4 then
it is Pposition, else an Nposition. We can prove this by showing that (i) the terminal position is an
Nposition (mis`
ere); (ii) If we make a move from a Pposition (
x
≡
1 mod 4), then
x

1
6≡
1 mod 4,
x

2
6≡
1 mod 4, nor
x

3
6≡
1 mod 4; (iii) Similarly, if
y
6≡
1 mod 4, then one of
y

1,
y

2,
y

3
will be congruent to 1 mod 4.
Since, 21
≡
1 mod 4, the second player can always move to Pposition during the game, and force the
ﬁrst player to lose.
1.5#4
Subtraction games
(a)
S
=
{
1
,
3
,
5
,
7
}
. In this case
P
=
{
0
,
2
,
4
,...,
2
k,.
..
}
=
{
2
k

k
∈
N
}
, the even numbers. To prove
this we only have to note that using a step from
S
we always jump from even to odd numbers.
(b)
S
=
{
1
,
3
,
6
}
. After computing a couple of positions, we guess that
P
=
{
x

x
≡
0
,
2
,
or 4 mod 9
}
.
We need to check that
p
+
s /
∈
P
,
∀
p
∈
P
,
∀
s
∈
S
, and also that
∃
s
∈
S
such that
∀
n /
∈
P
:
n
+
s
∈
P
.
(c)