<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>


#include <algorithm>
#include <bitset>
#include <unordered_map>
#include <unordered_set>

using namespace std;
class Solution
{
public:
  bool canIWin(int maxChoosableInteger, int desiredTotal)
  {
    if (desiredTotal <= maxChoosableInteger)
    {
      return true;
    }

    if (maxChoosableInteger * (maxChoosableInteger + 1) / 2 <
        desiredTotal) // if total sum of all numbers still lesss than desired
                      // total no one can win
    {
      return false;
    }

    unordered_map<int, bool> memo;
    int state{};

    bool result = dp(state, maxChoosableInteger, desiredTotal, memo);
    return result;
  }

  bool dp(int state, int maxChoosableInteger, int remainingTotal,
          unordered_map<int, bool> &memo)
  {
    if (memo.find(state) != memo.end())
    {
      return memo[state];
    }

    for (int i = 0; i < maxChoosableInteger; ++i)
    {

      int mask = 1 << i;

      if (state & mask) /// NECESSARY this is so that
      {
        continue;
      }

      int newState = mask | state;

      if (remainingTotal <= i + 1 ||
          !dp(newState, maxChoosableInteger, remainingTotal - (i + 1), memo))
      {
        return memo[state] = true;
      }
    }

    if (memo.find(state) != memo.end())
    {
      return memo[state] = false;
    }

    return false;
  }
};
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