Formatted question description:

464. Can I Win




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.

You can always assume that maxChoosableInteger will not be larger than 20 and desiredTotal will not be larger than 300.



maxChoosableInteger = 10

desiredTotal = 11




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.


Use depth first search. If the remaining total is 0 or less, then the result is true. Use state to store the numbers that have already been selected. Each time select a number that have not been selected. If the selected number is greater than or equal to the remaining total, return true. Otherwise, update state and do further search after selecting the number.

Use a map to store each state and the corresponding result. The map used here can reduce runtime if a state is already met before.

  • class Solution {
        public boolean canIWin(int maxChoosableInteger, int desiredTotal) {
            int maxSum = (1 + maxChoosableInteger) * maxChoosableInteger / 2;
            if (maxSum < desiredTotal)
                return false;
            Map<Integer, Boolean> map = new HashMap<Integer, Boolean>();
            return depthFirstSearch(0, maxChoosableInteger, desiredTotal, map);
        public boolean depthFirstSearch(int state, int maxChoosableInteger, int desiredTotal, Map<Integer, Boolean> map) {
            if (desiredTotal <= 0) {
                map.put(state, true);
                return true;
            if (map.containsKey(state))
                return map.get(state);
            boolean flag = false;
            for (int i = 1; i <= maxChoosableInteger; i++) {
                if (((state >> i) & 1) == 0) {
                    if (i >= desiredTotal) {
                        map.put(state, true);
                        return true;
                    } else {
                        int nextState = state | (1 << i);
                        flag = !depthFirstSearch(nextState, maxChoosableInteger, desiredTotal - i, map);
                    if (flag)
            map.put(state, flag);
            return flag;
  • Todo
  • class Solution(object):
      def canIWin(self, maxChoosableInteger, desiredTotal):
        :type maxChoosableInteger: int
        :type desiredTotal: int
        :rtype: bool
        def helper(pool, target, visited):
          if pool in visited:
            return visited[pool]
          if target <= 0:
            return False
          if pool >= self.maxPool:
            return True
          mask = 0x01
          for i in range(0, maxChoosableInteger):
            if pool & mask == 0:
              newPool = pool | mask
              if helper(newPool, target - (i + 1), visited) == False:
                visited[pool] = True
                return True
            mask = mask << 1
          visited[pool] = False
          return False
        if (1 + maxChoosableInteger) * (maxChoosableInteger / 2) < desiredTotal:
          return False
        if desiredTotal == 0:
          return True
        self.maxPool = 0
        mask = 1
        for i in range(0, maxChoosableInteger):
          self.maxPool |= mask
          mask = mask << 1
        pool = 0
        visited = {}
        return helper(pool, desiredTotal, visited)

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