<p>Given an integer array <code>nums</code>, return <code>true</code> <em>if you can partition the array into two subsets such that the sum of the elements in both subsets is equal or </em><code>false</code><em> otherwise</em>.</p>

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

<pre>
<strong>Input:</strong> nums = [1,5,11,5]
<strong>Output:</strong> true
<strong>Explanation:</strong> The array can be partitioned as [1, 5, 5] and [11].
</pre>

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

<pre>
<strong>Input:</strong> nums = [1,2,3,5]
<strong>Output:</strong> false
<strong>Explanation:</strong> The array cannot be partitioned into equal sum subsets.
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 200</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 100</code></li>
</ul>

class Solution:
    def canPartition(self, nums: List[int]) -> bool:
        total_sum = sum(nums)
        if total_sum % 2 == 1:
            return False
        target = total_sum // 2

        # problem(n, target)  = problem(i - 1, target) or problem(i - 1, target - nums[i])
        memo: dict[tuple[int, int], bool] = {}

        def dp(index: int, k: int) -> bool:
            num = nums[index]
            if num == k or k == 0:
                return True
            
            if index == 0:
                return False

            if (index, k) in memo:
                return memo[(index, k)]
            
            if dp(index - 1, k):
                memo[(index, k)] = True
                return True
            
            if k - num >= 0 and dp(index - 1, k - num):
                memo[(index, k)] = True
                return True
            
            memo[(index, k)] = False
            return False

        return dp(len(nums) - 1, target)
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