<p>Given <code>n</code> <code>cuboids</code> where the dimensions of the <code>i<sup>th</sup></code> cuboid is <code>cuboids[i] = [width<sub>i</sub>, length<sub>i</sub>, height<sub>i</sub>]</code> (<strong>0-indexed</strong>). Choose a <strong>subset</strong> of <code>cuboids</code> and place them on each other.</p>

<p>You can place cuboid <code>i</code> on cuboid <code>j</code> if <code>width<sub>i</sub> &lt;= width<sub>j</sub></code> and <code>length<sub>i</sub> &lt;= length<sub>j</sub></code> and <code>height<sub>i</sub> &lt;= height<sub>j</sub></code>. You can rearrange any cuboid&#39;s dimensions by rotating it to put it on another cuboid.</p>

<p>Return <em>the <strong>maximum height</strong> of the stacked</em> <code>cuboids</code>.</p>

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

<p><strong><img alt="" src="https://assets.leetcode.com/uploads/2019/10/21/image.jpg" style="width: 420px; height: 299px;" /></strong></p>

<pre>
<strong>Input:</strong> cuboids = [[50,45,20],[95,37,53],[45,23,12]]
<strong>Output:</strong> 190
<strong>Explanation:</strong>
Cuboid 1 is placed on the bottom with the 53x37 side facing down with height 95.
Cuboid 0 is placed next with the 45x20 side facing down with height 50.
Cuboid 2 is placed next with the 23x12 side facing down with height 45.
The total height is 95 + 50 + 45 = 190.
</pre>

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

<pre>
<strong>Input:</strong> cuboids = [[38,25,45],[76,35,3]]
<strong>Output:</strong> 76
<strong>Explanation:</strong>
You can&#39;t place any of the cuboids on the other.
We choose cuboid 1 and rotate it so that the 35x3 side is facing down and its height is 76.
</pre>

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

<pre>
<strong>Input:</strong> cuboids = [[7,11,17],[7,17,11],[11,7,17],[11,17,7],[17,7,11],[17,11,7]]
<strong>Output:</strong> 102
<strong>Explanation:</strong>
After rearranging the cuboids, you can see that all cuboids have the same dimension.
You can place the 11x7 side down on all cuboids so their heights are 17.
The maximum height of stacked cuboids is 6 * 17 = 102.
</pre>

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

<ul>
	<li><code>n == cuboids.length</code></li>
	<li><code>1 &lt;= n &lt;= 100</code></li>
	<li><code>1 &lt;= width<sub>i</sub>, length<sub>i</sub>, height<sub>i</sub> &lt;= 100</code></li>
</ul>

is_prev_smaller = lambda x, y: (x[0] >= y[0] and x[1] >= y[1] and x[2] >= y[2]) 

def dp_approach(cuboids, n):
    dp = [cu[2] for cu in cuboids]     
                                    # create dp array
    ans = max(dp)
    for i in range(1, n):                                              # iterate over each cube
        for j in range(i):                                             # compare with previous calculated cube
            if is_prev_smaller(cuboids[j], cuboids[i]):                             # update dp[i] if cube[i] can be put on top of cube[j]
                dp[i] = max(dp[i], dp[j] + cuboids[i][2])              # always get the maximum
        ans = max(ans, dp[i])   
                                                
    return ans

class Solution:
    def maxHeight(self, cuboids: List[List[int]]) -> int:
        for cub in cuboids:
            cub.sort()
        cuboids.sort(reverse=True)

        n = len(cuboids)

        return dp_approach(cuboids, n)
81