For each move of
G
2
+
G
3
there is an associated move in
G
1
+
G
3
:
If
one of the players moves in
G
3
when playing
G
2
+
G
3
, this corresponds to
the same move in
G
3
when playing
G
1
+
G
3
.
If one of the players plays
in
G
2
when playing
G
2
+
G
3
, say by moving from a Nim pile with
y
chips
to a Nim pile with
z < y
chips, then the corresponding move in
G
1
+
G
3
would be to move in
G
1
from a position with Sprague-Grundy value
y
to a
position with Sprague-Grundy value
z
(such a move exists by the definition
of the Sprague-Grundy function).
There may be extra moves in
G
1
+
G
3
that do not correspond to any move
G
2
+
G
3
, namely, it may be possible to
play in
G
1
from a position with Sprague-Grundy value
y
to a position with
Sprague-Grundy value
z>y
.
When playing in
G
1
+
G
3
, player A can pretend that the game is really
G
2
+
G
3
. If player A’s winning strategy is some move in
G
2
+
G
3
, then A can
play the corresponding move in
G
1
+
G
3
, and pretends that this move was
made in
G
2
+
G
3
. If A’s opponent makes a move in
G
1
+
G
3
that corresponds
to a move in
G
2
+
G
3
, then A pretends that this move was made in
G
2
+
G
3
.
But player A’s opponent could also make a move in
G
1
+
G
3
that does not
correspond to any move of
G
2
+
G
3
, by moving in
G
1
and increasing the
Sprague-Grundy value of the position in
G
1
from
y
to
z>y
. In this case,
by the definition of the Sprague-Grundy value, player A can simply play in
G
1
and move to a position with Sprague-Grundy value
y
. These two turns
correspond to no move, or a pause, in the game
G
2
+
G
3
. Because
G
1
+
G
3
is
progressively bounded,
G
2
+
G
3
will not remain paused forever. Since player
A has a winning strategy for the game
G
2
+
G
3
, player A will win this game
that A is pretending to play, and this will correspond to a win in the game

1.1 Impartial games
27
G
1
+
G
3
. Thus whichever player has a winning strategy in
G
2
+
G
3
also has
a winning strategy in
G
1
+
G
3
, so
G
1
and
G
2
are equivalent games.
We can use this theorem to find the
P
- and
N
-positions of a particular
impartial, progressively bounded game under normal play, provided we can
evaluate its Sprague-Grundy function.
For example, recall the 3-subtraction game we considered in Example 1.1.10.
We determined that the Sprague-Grundy function of the game is
g
(
x
) =
x
mod 4. Hence, by the Sprague-Grundy theorem, 3-subtraction game with
starting position
x
is equivalent to a single Nim pile with
x
mod 4 chips.
Recall that (0)
∈
P
Nim
while (1)
,
(2)
,
(3)
∈
N
Nim
. Hence, the
P
-positions
for the Subtraction game are the natural numbers that are divisible by four.
Corollary 1.1.1.
Let
G
1
and
G
2
be two progressively bounded impartial
combinatorial games under normal play. These games are equivalent if and
only if the Sprague-Grundy values of their starting positions are the same.