292. Nim Game

ou 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.

Solution

Case 1. If there are 4 stones, no matter how many stones you remove, the last one will be removed by your friend.

Case 2: But if there are 5 stones, you can remove 1, then your friend is going to remove first for the remaining 4, which you can always remove the last one. Similarly for 6 and 7, you can always remove some stones to only leave 4 stones. 

Case 3: But if there are 8 stones, and your friend is smart will always leave 4 stones and let you remove first and it becomes case 1

So we find that if n%4==0 which means you will lose the game for sure, otherwise you will be winner.

 public boolean canWinNim(int n) {  
    return n%4!=0;   
   }