In summary, the set of impartial combinatorial games as defined in Section 1.2 is
equivalent to the set of graph games with finitely progressive graphs. The Sprague-Grundy
function on such a graph exists if transfinite values are allowed. For progressively finite
graph games, the Sprague-Grundy function exists and is finite.
If the graph is allowed to have cycles, new problems arise. The SG-function satisfying
(1) may not exist. Even if it does, the simple inductive procedure of the previous sections
may not su
ﬃ
ce to find it. Even if the the Sprague-Grundy function exists and is known,
it may not be easy to find a winning strategy.
I – 17

Graphs with cycles do not satisfy the Ending Condition (6) of Section 1.2. Play may
last forever. In such a case we say the game ends in a tie; neither player wins. Here is an
example where there are tied positions.
a
b
c
d
e
Figure 3.4
A cyclic graph.
The node
e
is terminal and so has Sprague-Grundy value 0. Since
e
is the only follower
of
d
,
d
has Sprague-Grundy value 1. So a player at
c
will not move to
d
since such a move
obviously loses. Therefore the only reasonable move is to
a
. After two more moves, the
game moves to node
c
again with the opponent to move. The same analysis shows that
the game will go around the cycle
abcabc . . .
forever. So positions
a
,
b
and
c
are all tied
positions. In this example the Sprague-Grundy function does not exist.
When the Sprague-Grundy function exists in a graph with cycles, more subtle tech-
niques are often required to find it. Some examples for the reader to try his/her skill are
found in Exercise 9. But there is another problem. Suppose the Sprague-Grundy function
is known and the graph has cycles. If you are at a position with non-zero SG-value, you
know you can win by moving to a position with SG-value 0. But which one? You may
choose one that takes you on a long trip and after many moves you find yourself back
where you started. An example of this is in Northcott’s Game in Exercise 5 of Chapter 2.
There it is easy to see how to proceed to end the game, but in general it may be di
ﬃ
cult
to see what to do. For an e
ﬃ
cient algorithm that computes the Sprague-Grundy function
along with a counter that gives information on how to play, see Fraenkel (2002).
3.5 Exercises.
1.
Fig 3.5
Find the Sprague-Grundy function.
2. Find the Sprague-Grundy function of the subtraction game with subtraction set
S
=
{
1
,
3
,
4
}
.
I – 18
1
0
0
0

3. Consider the one-pile game with the rule that you may remove at most half the
chips. Of course, you must remove at least one, so the terminal positions are 0 and 1. Find
the Sprague-Grundy function.
4. (a) Consider the one-pile game with the rule that you may remove
c
chips from a
pile of
n
chips if and only if
c
is a divisor of
n
, including 1 and
n
. For example, from a
pile of 12 chips, you may remove 1, 2, 3, 4, 6, or 12 chips. The only terminal position is 0.