<p>In the &quot;100 game&quot; two players take turns adding, to a running total, any integer from <code>1</code> to <code>10</code>. The player who first causes the running total to <strong>reach or exceed</strong> 100 wins.</p>

<p>What if we change the game so that players <strong>cannot</strong> re-use integers?</p>

<p>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 &gt;= 100.</p>

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

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> maxChoosableInteger = 10, desiredTotal = 11
<strong>Output:</strong> false
<strong>Explanation:</strong>
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 &gt;= desiredTotal.
Same with other integers chosen by the first player, the second player will always win.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> maxChoosableInteger = 10, desiredTotal = 0
<strong>Output:</strong> true
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> maxChoosableInteger = 10, desiredTotal = 1
<strong>Output:</strong> true
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= maxChoosableInteger &lt;= 20</code></li>
	<li><code>0 &lt;= desiredTotal &lt;= 300</code></li>
</ul>

class Solution:
    def canIWin(self, maxChoosableInteger: int, desiredTotal: int) -> bool:
        # If the sum of all integers is less than the desired total, no one can win
        if (1 + maxChoosableInteger) * maxChoosableInteger // 2 < desiredTotal:
            return False
        
        # Memoization dictionary to store the results of subproblems
        memo = {}
        
        # Bitmask to represent the used numbers
        used = [False] * (maxChoosableInteger + 1)
        
        def backtrack(total):
            # Convert the used list to a tuple to use as a key in the memoization dictionary
            key = tuple(used)
            
            # If the result for this state is already computed, return it
            if key in memo:
                return memo[key]
            
            # Iterate through all the numbers
            for i in range(1, maxChoosableInteger + 1):
                if not used[i]:
                    used[i] = True
                    # If the current number is greater than or equal to the desired total,
                    # or if the opponent cannot win after choosing this number,
                    # then the current player can win
                    if total + i >= desiredTotal or not backtrack(total + i):
                        used[i] = False
                        memo[key] = True
                        return True
                    used[i] = False
            
            # If no number can lead to a win, store this result in the memoization dictionary
            memo[key] = False
            return False
        
        # Start the backtracking with an initial total of 0
        return backtrack(0)

# Example usage:
solution = Solution()
print(solution.canIWin(10, 11))  # Output: False
print(solution.canIWin(10, 0))   # Output: True
print(solution.canIWin(10, 1))   # Output: True
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