Can I win

Time: O(N!); Space: O(N); medium

In the “100 game,” two players take turns adding, to a running total, any integer from 1..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 of 1..15 without replacement until they reach a total >= 100.

Given an integer maxChoosableInteger and another integer desiredTotal, determine if the first player to move can force a win, assuming both players play optimally.

Constraints:

  • maxChoosableInteger <= 20

  • desiredTotal <= 300

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 = 10

Output: True

Explanation:

  • The first player chooses 10 and wins the game.

[1]:

class Solution1(object):
    """
    Time: O(N!)
    Space: O(N)
    """
    def canIWin(self, maxChoosableInteger, desiredTotal):
        """
        :type maxChoosableInteger: int
        :type desiredTotal: int
        :rtype: bool
        """
        def canIWinHelper(maxChoosableInteger, desiredTotal, visited, lookup):
            # Base Case
            if visited in lookup:
                return lookup[visited]

            mask = 1
            for i in range(maxChoosableInteger):
                if visited & mask == 0:
                    if i + 1 >= desiredTotal or \
                       not canIWinHelper(maxChoosableInteger, desiredTotal - (i + 1), visited | mask, lookup):
                        lookup[visited] = True
                        return True
                mask <<= 1
            lookup[visited] = False
            return False

        if (1 + maxChoosableInteger) * (maxChoosableInteger // 2) < desiredTotal:
            return False

        return canIWinHelper(maxChoosableInteger, desiredTotal, 0, {})

[2]:

s = Solution1()

maxChoosableInteger = 10
desiredTotal = 11
assert s.canIWin(maxChoosableInteger, desiredTotal) == False

maxChoosableInteger = 10
desiredTotal = 10
assert s.canIWin(maxChoosableInteger, desiredTotal) == True