You have 2 eggs and 100 floors. Eggs will break if dropped from a certain floor. You can throw eggs from any floor. How do you minimize the number of throws to find the critical floor?

Question

Accepted Answer

This is the classic 2 eggs and 100 floors problem, which is a well-known algorithmic puzzle. The goal is to minimize the worst-case number of throws needed to find the highest floor (critical floor) from which you can drop an egg without breaking it.

Key points:

You have 2 eggs.

An egg that breaks cannot be reused.

You want to minimize the maximum number of throws in the worst case.

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Optimal Strategy (Minimize Maximum Number of Throws)

Let’s define the strategy:

You don't want to drop the first egg in a linear way (1, 2, 3, ...) because that might take 100 tries in the worst case.

You also can't use binary search, because you only have 2 eggs — once the first breaks, you must go one-by-one with the second.

Instead, we use a decreasing step strategy:

Start dropping the first egg from floor x, then go to floor x - 1, then x - 2, ..., until the egg breaks or you reach floor 100.
This way, the worst-case number of throws is minimized.

Let’s say the first drop is from floor x. Then the next is x + (x-1), then x + (x-1) + (x-2), and so on.

So you want to find the minimum x such that:

x + (x-1) + (x-2) + ... + 1 >= 100

This is the sum of the first x natural numbers:

x(x + 1)/2 ≥ 100

Solving:

x^2 + x - 200 ≥ 0

Using quadratic formula:

x = [-1 + sqrt(1 + 800)] / 2
x = [-1 + sqrt(801)] / 2 ≈ (28.3 - 1)/2 ≈ 13.65

So, x = 14 is the smallest integer satisfying the condition.

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Final Strategy

Drop the first egg from floors:

14 → if it doesn't break, go to

27 (14 + 13) → then

39 (27 + 12)

50 (39 + 11)

60 (50 + 10)

69 (60 + 9)

77 (69 + 8)

84 (77 + 7)

90 (84 + 6)

95 (90 + 5)

99 (95 + 4)

100 (99 + 1)

Once the egg breaks, say at floor 84, then you go linearly from 77+1 to 83 with the second egg.

Result:

Maximum number of drops in the worst case = 14

Let me know if you want a simulation or code for this.

Accepted Answer

First, throw the first egg from the 14th floor. 
If this egg breaks, throw the rest of the eggs from floors 1 to 13. 
The breaking floor will be where the second egg breaks. 
This way, there will be no more than 14 drops

You have 2 eggs and 100 floors. Eggs will break if dropped from a certain floor. You can throw eggs from any floor. How do you minimize the number of throws to find the critical floor?

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