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Three Eggs (illustration)
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Three Eggs

Difficulty ★★★about 5 minLogic PuzzleCase analysisWorst case
A 100-story building and three eggs. What is the fewest drops needed, in the worst case, to find the lowest floor where an egg breaks?

Case Record

There is a 100-story building. An egg dropped from a certain floor or higher breaks, and from lower floors it does not. You want to find the lowest floor where it breaks.

You have only 3 eggs. An egg that does not break can be picked up and used again. A broken egg cannot be used.

Your question

In the worst case, what is the fewest number of drops that will always find the lowest floor where an egg breaks?

Case Files

Diagram

A 100-story building and 3 eggsFloor 100Floor 1Only 3 eggsA broken egg is goneAn unbroken egg can be reusedEggs break from some floor upward. Find that lowest floor with the fewestdrops, even in the worst case.
Diagram

Hints

If you get stuck, open the hints one at a time, in order.

Open hint 1

With one egg, how many drops does the worst case need?

Open hint 2

With three eggs, after the first drop, how many floors remain for two eggs?

Open hint 3

Count how many floors n drops can cover.

Solve It

Your question

In the worst case, what is the fewest number of drops that will always find the lowest floor where an egg breaks?

Choose the answer closest to the truth.

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The Truth

9

The answer is "9."

With 3 eggs and up to n drops, the number of floors you can cover is n + n(n−1)/2 + n(n−1)(n−2)/6. For n = 8 this is 92 floors; for n = 9 it is 129 floors. The smallest n that covers 100 floors is 9.

With 8 drops you can cover only 92 floors. 14 is the answer for two eggs.

Floors covered with 3 eggs and n dropsDrops nn + n(n−1)/2 + n(n−1)(n−2)/6Floors covered77＋21＋356388＋28＋569299＋36＋84129To cover 100 floors we need 9 drops (up to 129 floors). 8 drops reachonly 92 floors.
Diagram of the solution

Key Points

  • Count from small cases up
  • Split into the broken and not-broken cases
  • Add up the floors that can be checked

Why people get fooled

Because you reuse the answer for two eggs (14) as it is.

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