<p>Given a positive integer <code>n</code>, return <em>the <strong>punishment number</strong></em> of <code>n</code>.</p>

<p>The <strong>punishment number</strong> of <code>n</code> is defined as the sum of the squares of all integers <code>i</code> such that:</p>

<ul>
	<li><code>1 &lt;= i &lt;= n</code></li>
	<li>The decimal representation of <code>i * i</code> can be partitioned into contiguous substrings such that the sum of the integer values of these substrings equals <code>i</code>.</li>
</ul>

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

<pre>
<strong>Input:</strong> n = 10
<strong>Output:</strong> 182
<strong>Explanation:</strong> There are exactly 3 integers i that satisfy the conditions in the statement:
- 1 since 1 * 1 = 1
- 9 since 9 * 9 = 81 and 81 can be partitioned into 8 + 1.
- 10 since 10 * 10 = 100 and 100 can be partitioned into 10 + 0.
Hence, the punishment number of 10 is 1 + 81 + 100 = 182
</pre>

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

<pre>
<strong>Input:</strong> n = 37
<strong>Output:</strong> 1478
<strong>Explanation:</strong> There are exactly 4 integers i that satisfy the conditions in the statement:
- 1 since 1 * 1 = 1. 
- 9 since 9 * 9 = 81 and 81 can be partitioned into 8 + 1. 
- 10 since 10 * 10 = 100 and 100 can be partitioned into 10 + 0. 
- 36 since 36 * 36 = 1296 and 1296 can be partitioned into 1 + 29 + 6.
Hence, the punishment number of 37 is 1 + 81 + 100 + 1296 = 1478
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 1000</code></li>
</ul>

class Solution:
    def punishmentNumber(self, n: int) -> int:
        def partition_string(s):
            if len(s) == 0:
                return [[]]
            partitions = []
            for i in range(1, len(s) + 1):
                prefix, suffix = s[:i], s[i:]
                suffix_partitions = partition_string(suffix)
                for p in suffix_partitions:
                    partitions.append([prefix] + p)

            return partitions
    
        valid = []
        for i in range(1, n+1):
            x = i * i
            ps = partition_string(str(x))
            for p in ps:
                if sum(int(s) for s in p) == i:
                    valid.append(i)
                    break
        return sum(x**2 for x in valid)
1933