Answer: Tuesday Teaser #5
I like the idea that you’d have two laptops purely for the purposes of breaking!
There are two key parts to this puzzle’s answer. Firstly, imagine you only have a single laptop. In this scenario there is only one way of finding out the maximum height, namely to start from the bottom and work up floor-by-floor until the laptop breaks.
Secondly, because we have two laptops, we can optimistically skip some floors and then return to this simpler method as soon as the first laptops breaks. But how many floors should we skip? To answer this question suppose we start at the sixth floor. If the laptop breaks, we drop to the first floor and work upwards, so it could take as many as six drops. If it doesn’t break, we want to move upwards so that we don’t increase the potential number of drops, that is we’d drop from the eleventh floor so that if the laptop breaks there, we’d drop down to the seventh floor and work up. This scenario still leads to a maximum of six drops total, i.e. drops at floors 6, 11, 7, 8, 9 and 10. We then proceed in the same way, skipping one floor less each time until we reach the twentieth floor.
To clarify, if at each stage the first laptop survives, we’d drop at the 6th, 11th, 15th, 18th, 20th floors and if at any stage the laptop breaks, we simply drop down with the remaining laptop and work back up. This leads to a maximum of six drops.
Extending this to the general case, if we have a building with m floors, the minimum number of drops would be the smallest number n such that 1 + 2 + … + n is at least as big as m.
