Can I Win

Description

In the "100 game" two players take turns adding, to a running total, any integer from 1 to 10. The player who first causes the running total to reach or exceed 100 wins.

What if we change the game so that players cannot re-use integers?

For example, two players might take turns drawing from a common pool of numbers from 1 to 15 without replacement until they reach a total >= 100.

Given two integers maxChoosableInteger and desiredTotal, return true if the first player to move can force a win, otherwise return false. Assume both players play optimally.

 

Example 1:

Input: maxChoosableInteger = 10, desiredTotal = 11
Output: false
Explanation:
No matter which integer the first player choose, the first player will lose.
The first player can choose an integer from 1 up to 10.
If the first player choose 1, the second player can only choose integers from 2 up to 10.
The second player will win by choosing 10 and get a total = 11, which is >= desiredTotal.
Same with other integers chosen by the first player, the second player will always win.

Example 2:

Input: maxChoosableInteger = 10, desiredTotal = 0
Output: true

Example 3:

Input: maxChoosableInteger = 10, desiredTotal = 1
Output: true

 

Constraints:

• 1 <= maxChoosableInteger <= 20
• 0 <= desiredTotal <= 300

Solution(python)

class Solution(object):
    def canIWin(self, maxChoosableInteger, desiredTotal):
        """
        :type maxChoosableInteger: int
        :type desiredTotal: int
        :rtype: bool
        """
        if (1 + maxChoosableInteger) * maxChoosableInteger/2 < desiredTotal:
            return False
        self.memo = {}
        return self.helper(range(1, maxChoosableInteger + 1), desiredTotal)


    def helper(self, nums, desiredTotal):

        hash = str(nums)
        if hash in self.memo:
            return self.memo[hash]

        if nums[-1] >= desiredTotal:
            return True

        for i in range(len(nums)):
            if not self.helper(nums[:i] + nums[i+1:], desiredTotal - nums[i]):
                self.memo[hash]= True
                return True
        self.memo[hash] = False
        return False