Two Glass Balls, 100 Floors

A ball dropped from floor X or higher will break. From any floor below X, it will not break. The challenge is to determine X using the fewest drops possible in the worst case.

Optimal worst-case answer: 14 drops
Interactive Simulator
Choose a hidden threshold floor X and watch the optimal strategy find it.
42
Experiment status 0 drops
Ball 1 is ready. First drop: floor 14.
Drop History
No drops yet.
Optimal First-Ball Schedule
Drop the first ball at floors with decreasing gaps: 14, 27, 39, 50, ...
With 13 drops
91 floors
13 + 12 + ··· + 1 = 91
Not enough ✗
With 14 drops
105 floors
14 + 13 + ··· + 1 = 105
Enough ✓
Find the smallest k such that
k(k + 1) / 2 ≥ 100
The smallest such value is k = 14.
Why the decreasing-gap strategy works
The trick is to make every worst-case branch use at most the same number of drops.

1) First drop at floor 14

If it breaks, then you only need to test floors 1 through 13 one-by-one with the second ball.

2) Next drop at floor 27

If the first ball survives 14, you have used one drop, so the next jump can be at most 13 floors.

3) Keep shrinking the gap

Then go +12, +11, +10, and so on, so every scenario stays within 14 total drops.

Schedule of first-ball drops: 14, 27, 39, 50, 60, 69, 77, 84, 90, 95, 99, 100.