Alice and Bob take turns playing a game, with Alice starting first.
Initially, there is a number n
on the chalkboard. On each player’s turn, that player makes a move consisting of:
- Choosing any
x
with0 < x < n
andn % x == 0
. - Replacing the number
n
on the chalkboard withn - x
.
Also, if a player cannot make a move, they lose the game.
Return true
if and only if Alice wins the game, assuming both players play optimally.
Example 1:
Input: n = 2 Output: true Explanation: Alice chooses 1, and Bob has no more moves.
Example 2:
Input: n = 3 Output: false Explanation: Alice chooses 1, Bob chooses 1, and Alice has no more moves.
Constraints:
1 <= n <= 1000
Code
關鍵在於奇數的因數不會有偶數!
class Solution {
public:
bool divisorGame(int n) {
return n % 2 == 0;
}
};
證明:
If N is even.
We can choose x = 1.
The opponent will get N - 1, which is a odd.
Reduce to the case odd and he will lose.
If N is odd,
2.1 If N = 1, lose directly.
2.2 We have to choose an odd x.
The opponent will get N - x, which is a even.
Reduce to the case even and he will win.
So the N will change odd and even alternatively until N = 1.