Stack Rotatable Cuboids for Maximum Height

### Problem Choose a subset of `cuboids` and orient each by permuting its three dimensions. A cuboid may sit on another only when its width, length, and height are each no larger than the corresponding dimension below it. Return the greatest possible sum of the chosen vertical heights. ```cpp class Solution { public: int maxHeight(vector<vector<int>>& cuboids); }; ``` ### Examples **Example 1:** ```text cuboids = [[1,2,3],[2,3,4]] Output: 7 Explanation: Orient the smaller cuboid with height 3 on the one with height 4. ``` **Example 2:** ```text cuboids = [[2,4,6],[6,2,4],[1,1,1]] Output: 13 Explanation: The first two are equal after rotation; stack both at height 6 with the unit cube above. ``` ### Constraints - `n == cuboids.length`, with `1 <= n <= 100`. - Every cuboid has three integer dimensions, each in `[1,100]`. ### Clarifying Questions The source placement rule compares all three dimensions, including height. Equal dimensions are allowed, and each input cuboid is used at most once. ```hint Inspect the structure After an orientation is chosen, which of the three corresponding dimensions must satisfy the placement rule? ``` ### Reasoning checkpoints - Validate every chosen orientation and all three placement inequalities. - Prove maximum total height while allowing equal cuboids and using each physical occurrence at most once. ### Optional extensions Why would a variant comparing only base dimensions require a different orientation analysis?