Problem-solving ideas of Nim game(classical problem)

Nim game's rule:

You are playing the following Nim Game with your friend: There is a heap of stones on the table, each time one of you take turns to remove 1 to 3 stones. The one who removes the last stone will be the winner. You will take the first turn to remove the stones.

Both of you are very clever and have optimal strategies for the game. Write a function to determine whether you can win the game given the number of stones in the heap.

For example, if there are 4 stones in the heap, then you will never win the game: no matter 1, 2, or 3 stones you remove, the last stone will always be removed by your friend.

idea of Nim game

There are some situations in this game.

Let's think about what will happend while stones is few:

If there are 4 stones in the heap,you will lose the game in every matters.

If there are less than 3 stones,you will win the game in one step.

So,when there are more than 4 stones,you should remove some stones and let your friend getting 4 stones.In this situation,we can simplify this problem to that how to leaving 4 stones.And you will find it is the same as initially problem.

Now, what you should do is leaving multiple of 4 stones in every matters.So,if the number of stones is not the muiltiple of 4,you will win this game.

Finally,we can use a formula to express this problem:


you will win.