Sell Diminishing-Valued Colored Balls

Sell Diminishing-Valued Colored Balls

What is this problem about?

The Sell Diminishing-Valued Colored Balls interview question gives you an array where each element represents the count of a specific color of ball. You can sell any ball, but the value of a ball of color i on the k-th sale equals its remaining count. You want to maximize total profit from exactly orders sales. This combines greedy selection with mathematical summation using heap or binary search.

Why is this asked in interviews?

Amazon and MathWorks ask this MEDIUM problem because it requires non-obvious greedy thinking: always sell from the color with the most remaining balls to maximize profit. The challenge is doing this efficiently — simulating each sale one by one is O(orders × log n), which is too slow. The optimized approach groups bulk sales using arithmetic series, bringing complexity down to O(n log n + log(max_val)).

Algorithmic pattern used

The pattern is greedy with heap or binary search. Greedy insight: always sell from the color with the maximum current count. Use a max-heap to efficiently get the highest-count color. Instead of selling one ball at a time, find how many balls you can sell at a given "level" (count value) before the heap's top drops to the next level. Use arithmetic series: if you sell t balls at level v down to level v-k, total value = t * (v + (v-k+1)) / 2. Process levels in bulk until orders is exhausted.

Example explanation

Inventory: [2, 5], orders = 4.

Sorted descending: [5, 2]. Sell from the top:

• Sell 3 balls from color B (5→4→3→2), values: 5+4+3 = 12. orders remaining = 1.
• Both colors now at 2. Sell 1 more from any → value 2.

Total profit: 14. (Under modulo 10^9+7 for large inputs.)

Common mistakes candidates make

• Simulating one sale at a time — O(orders) can be 10^9, making this approach TLE.
• Not handling the modulo operation correctly when computing large arithmetic sums.
• Off-by-one when computing how many balls can be sold at the current level before reaching the next heap top.
• Forgetting to sort the inventory or use a max-heap initially.

Interview preparation tip

For the Sell Diminishing-Valued Colored Balls coding problem, the heap greedy math interview pattern is the efficient approach. The key optimization is bulk-selling entire "levels" of ball counts using arithmetic series formulas instead of individual pops. Amazon interviewers test whether you identify the bottleneck (order-by-order simulation) and propose the bulk approach. Practice writing the arithmetic sum formula sum = count * (high + low) / 2 correctly and handling the modulo requirement for large values.