Egg Dropping Puzzle

The Egg Dropping Puzzle is a famous mathematical and logic challenge that tests reasoning, optimization, and strategic problem-solving. You are given two identical eggs and a 100-story building. Your goal is to find the highest floor from which you can drop an egg without it breaking, using the fewest number of drops possible.

At first glance, it seems like trial and error. But with a smart strategy, you can minimize the risk and find the solution efficiently. This puzzle is often used in interviews, math competitions, and algorithm training because it perfectly blends logic with optimization.

Egg Dropping Puzzle Setup and Rules

Here are the key rules of the puzzle:

You have two eggs and a 100-floor building.
There exists a threshold floor (N) such that:
- If you drop an egg from floor N or below, it will not break.
- If you drop an egg from any floor above N, it will break.
Once an egg breaks, you can’t use it again.
Your objective is to find N while minimizing the worst-case number of drops.

So, what’s the most efficient way to find that exact floor?

Objective of the Egg Dropping Puzzle

The goal is simple yet tricky:
Determine the minimum number of drops required to identify the highest safe floor from which you can drop an egg without it breaking.

But the challenge is balancing risk and efficiency.
If you test too high, you might break the first egg early.
If you test too low, you waste too many drops.

The key is to find a method that minimizes the maximum number of drops needed, no matter where the critical floor is.

How to Solve the Egg Dropping Puzzle?

This puzzle is solved using a systematic dropping pattern that balances the number of drops between both eggs. The approach ensures that even in the worst-case scenario, you don’t exceed the minimum possible number of total drops.

Step 1: Establish a Strategic Dropping Pattern

To start, you need a strategy for the first egg.
If it breaks too early, you’ll rely on the second egg to pinpoint the exact floor one by one.
Therefore, your first egg’s drop sequence must be chosen wisely to reduce the remaining possible floors.

The optimal first drop is from the 14th floor.

Why 14?
Because the sum of the sequence 14 + 13 + 12 + ... + 1 = 105, which just covers 100 floors.
This means no matter where the egg breaks, the total drops never exceed 14.

Step 2: Continue with Decreasing Intervals

After the first drop from the 14th floor, you continue dropping the first egg from floors that are increasing but with decreasing intervals.

Here’s the pattern:

Drop Number	Floor Number	Interval
1	14	—
2	27 (14 + 13)	13
3	39 (27 + 12)	12
4	50 (39 + 11)	11
5	60 (50 + 10)	10
6	69 (60 + 9)	9
7	77 (69 + 8)	8
8	84 (77 + 7)	7
9	90 (84 + 6)	6
10	95 (90 + 5)	5
11	99 (95 + 4)	4
12	100	1

At each step, the interval decreases by one, ensuring that if the first egg breaks, the number of floors left to check with the second egg is minimal.

Step 3: Handle the Break – The Second Egg Strategy

If the first egg breaks, you already know the threshold floor lies between the previous safe floor and the floor where it broke.
Now, you’ll use the second egg to check each floor sequentially in that range.

For example:
If the first egg breaks at the 39th floor, you’ll test floors 28 to 38 one by one with the second egg.

This guarantees you’ll find the exact critical floor without exceeding your 14-drop limit.

Step 4: Understanding the Worst-Case Scenario

The worst-case scenario happens when the first egg doesn’t break until the very last interval.
For instance, if it breaks at floor 99, you’d already have dropped the first egg 10 times, and then check up to 4 more floors (96–99) with the second egg.

So, in the worst case, you’ll need 14 drops total, no matter where the threshold is.

Final Answer: Minimum Drops Required

The minimum number of drops required to find the highest safe floor with two eggs is 14 (in the worst case).

This means, with the best possible strategy, you’ll never need more than 14 drops to identify the correct floor in a 100-story building.

Why 14 is the Optimal Answer?

The logic behind using 14 comes from solving this inequality:

x + (x-1) + (x-2) + ... + 1 ≥ 100

The smallest integer x that satisfies this condition is 14, because:

14 + 13 + 12 + ... + 1 = 105 ≥ 100

Thus, 14 is the minimum number of trials required to guarantee success in all possible cases.

Mathematical Expression of the Egg Dropping Puzzle

For a building with n floors and k eggs, the problem can be solved using the recurrence:

T(k, n) = 1 + min { max(T(k-1, x-1), T(k, n-x)) }

where:

T(k-1, x-1) → when the egg breaks
T(k, n-x) → when the egg doesn’t break

For the 2-egg, 100-floor case, this formula confirms T(2, 100) = 14.

Why the Egg Dropping Puzzle is Popular?

This puzzle is not just fun, it’s highly practical. It’s used to teach optimization algorithms, binary search logic, and worst-case analysis in computer science and engineering.

You’ll find it in:

Coding interviews (Google, Amazon, Microsoft)
Algorithm courses (Dynamic Programming problems)
Logic puzzle competitions and IQ tests

It’s an elegant example of how math, reasoning, and strategy combine to find the best possible solution under constraints.

Egg Dropping Puzzle