<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:
                
        """
        :type maxChoosableInteger: int
        :type desiredTotal: int
        :rtype: bool
        """
        
        if (1+maxChoosableInteger)*maxChoosableInteger/2 < desiredTotal:
            return False

        if maxChoosableInteger >= desiredTotal:
            return True

        if (1+maxChoosableInteger)*maxChoosableInteger/2 == desiredTotal:
            return (maxChoosableInteger % 2 == 1)
        
        # if (1+maxChoosableInteger)*maxChoosableInteger/2 - 1 == desiredTotal:
        #     return (maxChoosableInteger % 2 == 1)


        return self.canActionPlayerWin(
            integer_list=[maxChoosableInteger - x for x in range(maxChoosableInteger)],
            total_to_win=desiredTotal,
            memory_dict={}
            )

    def canActionPlayerWin(self, integer_list, total_to_win, memory_dict):
        """
        Given integer_list to choose from and total_to_win as the leftover value to the desiredTotal, 
        return True if the current action player can win, else False.

        Suppose integer_list is kept sorted in descending order.
        Suppose sum(integer_list) >= total_to_win always holds. 

        This is a correct solution but has exp runtime. 
        """

        if integer_list[0] >= total_to_win:
            return True

        evaluation_key = tuple(integer_list)

        if evaluation_key in memory_dict.keys():
            return memory_dict[evaluation_key]

        for i, x in enumerate(integer_list):
            if self.canActionPlayerWin(
                integer_list=integer_list[:i] + integer_list[i+1:],
                total_to_win=total_to_win - x,
                memory_dict=memory_dict
            ) is False:
                memory_dict[evaluation_key] = True
                return True
        memory_dict[evaluation_key] = False
        return False

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