HDU 3951 game coin game

Question:

One circle of coins, two people take turns. Each person can only take 1 ~ K. For example, 1, 3, 4.
How many coins and K are you going to win?

Solution:

Suppose we have 10 coins, K = 2. After the first player takes one, the second player takes the corresponding coins at the symmetric point of the circle to keep the remaining coins on both sides equal, in this way, no matter how the first player gets it, the second player only needs the same method on the other side to ensure that he is the last one to take the coin. it can also be known from the SG theorem that two games with the same SG value are defeated.

The same is true for the promotion to a larger extent. Therefore, the first player's victory may only be where N is an odd number and K is 1, or K> = n,

In other cases, the second player wins.