Nim $($nim$)$ Nim is a two$-$player game, where a number of stacks of matches are placed in front of the players $($a finite number of stacks, and in each stack a finite number of matches$).$ Each player in turn chooses a certain pile and takes as many matches from it as he wants $($he must take one match$).$ The player who takes the last match on the table wins

A$.$ Does it follow from the von Neumann theorem that one of the players has a winning strategy, reasoning?

In the following steps we will see how to determine the identity of the leading player and his winning strategy. We will write in a column the number of matches in each stack in base $2.$ For example, if there are $4$ stacks and the number of matches in the stacks is $2,12,13,21$ we will write

$10$

$1100$

$1101$

$10101$

Now we will check if the number of $1\text{'}$s in each column is even or not. In the example above, in the first $($right$)$ and fourth columns there is an even number of $1$s$,$ and in the other columns an odd number of $1$s$.$ A game state is called a winning position if the number of $1$s in all columns is even. The game state in the example is not Victory position

b$.$ Prove that from any game situation that is not a victory position it is possible to reach a victory position in one move

In our example if we take $18$ matches from the high stack we get $2,12,13,3$ and in base $2$ we get

$10$

$1100$

$1101$

$11$

C$.$ Prove that every legal move leads from a winning position to a position that is not a winning position

D$.$ Explain why the endgame is a winning position

e$.$ Explain which of the players is able to force a win and describe is winning strategy.