This preview shows pages 1–2. Sign up to view the full content.
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)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '09
 WEISBART

Click to edit the document details