In this section we prove Theorem 1.2, and
we need several facts about the relations between numbers in base
, the nim-sum of numbers and the floor function.
By Lemma 2.4
that satisfies one of the following two conditions
satisfies and , then by the fact that for we have for , and hence by the fact that we have . Therefore we have
Let , then and . By the fact that for we have for , and hence by the fact that we have . Let , then we have .
Next we are going to define the function
for a state
is the set of all states that can be reached from the state in one step (directly).
Next we prove that if you start with an element of , then any move leads to an element of .
Next we prove that if you start with an element of , then there is a proper move that leads to an element of .
By Theorem 2.3 and 2.4 we finish the proof of Theorem 1.2. If we start the game with a state
, then by Theorem 2.3 any option by us leads to a state
. From this state
by Theorem 2.4 our opponent can choose a proper option that leads to a state in
. Note that any option reduces some of the numbers in the coordinates. In this way our opponent can always reach a state in
, and finally he wins by reaching
is the set of L states.
If we start the game with a state , then by Theorem 2.4 we can choose a proper option leads to a state in . From any option by our opponent leads to a state in . In this way we win the game by reaching . Therefore is the set of W states.