Remove Boxes

Remove Boxes

What is this problem about?

The Remove Boxes interview question is a challenging dynamic programming puzzle where you are given a sequence of colored boxes. You can remove a contiguous group of boxes that all share the same color in one move, and you earn points equal to the square of the number of boxes removed. Your task is to find the maximum total points achievable. The key difficulty is that removing one group can merge two previously separated groups of the same color, creating complex dependencies.

Why is this asked in interviews?

This problem appears frequently at companies like Microsoft, Meta, Amazon, and Google because it tests advanced dynamic programming intuition. It goes beyond simple 1D or 2D DP and requires a 3D memoization table. It assesses whether candidates can identify non-obvious subproblem definitions and handle "dependent" removals — a skill directly applicable to compiler optimization, text editing, and simulation tasks.

Algorithmic pattern used

The pattern is interval DP with memoization. The classic subproblem definition is dp[l][r][k] — the maximum points from boxes l to r, given that there are k extra boxes of the same color as boxes[l] appended to the left. You either remove the leftmost group immediately (earning (k+1)^2 points) or you find a position m within [l+1, r] where boxes[m] == boxes[l] and delay the removal to merge groups.

Example explanation

Consider boxes [A, B, A, C, A]. Rather than removing each A separately for 1 point each (total 3), you can plan removals so all three As merge into one group. Remove B first (1 point), then C (1 point). Now you have [A, A, A] — removing all three at once earns 3^2 = 9 points. Total: 1 + 1 + 9 = 11, far better than removing greedily.

Common mistakes candidates make

• Defining the subproblem as only dp[l][r], which misses the crucial extra k boxes context.
• Forgetting to handle the base case where l > r.
• Not memoizing properly, leading to exponential time complexity.
• Attempting a greedy approach — always removing the largest current group first — which does not yield the global optimum.

Interview preparation tip

The Remove Boxes coding problem is considered one of the harder interval DP problems. Before coding, spend time clearly explaining the dp[l][r][k] definition to your interviewer — getting this right is half the battle. Practice deriving the recurrence on paper first. Companies like Cisco, Sprinklr, and Bloomberg use this problem to filter candidates who can handle non-standard DP state definitions, so focus on articulating your reasoning step by step.