Frog in a Well: A Classic Riddle and Its Solution

Have you heard of the classic riddle involving a frog in a well? This riddle presents an intriguing mathematical challenge that has been discussed and debated for many years. Let's delve into the details and explore the correct answer, which may surprise you.

The Problem

Imagine a well that is 10 meters deep. A frog can jump 2 meters at a time but, unfortunately, it slips back 1 meter during each jump. The question is: how many jumps will it take for the frog to escape the well?

The Mathematical Analysis

Many have approached this problem with various methods, leading to different conclusions. Let's break down the steps to find the right solution.

First Method: Some argue that each jump effectively gains the frog 1 meter (2 meters up minus 1 meter slip). Thus, the frog would need to gain 10 meters to escape. Therefore, it should take 10 jumps:

10 meters ÷ (2 meters jump - 1 meter slip) 10 meters ÷ 1 meter gain 10 jumps

Breaking It Down

Let's take a closer look at the jumps the frog makes:

1st jump: 1 meter (2 meters up - 1 meter slip) 2nd jump: 2 meters (3 meters total - 1 meter slip) 3rd jump: 3 meters (4 meters total - 1 meter slip) 4th jump: 4 meters (5 meters total - 1 meter slip) 5th jump: 5 meters (6 meters total - 1 meter slip) 6th jump: 6 meters (7 meters total - 1 meter slip) 7th jump: 7 meters (8 meters total - 1 meter slip) 8th jump: 8 meters (9 meters total - 1 meter slip) 9th jump: 9 meters (10 meters total - no slip needed to escape)

At the 9th jump, the frog should be at the 9-meter mark and will not slip back. Therefore, it only needs one more jump to escape, reaching the 10-meter mark.

Alternative Interpretations

Some have argued that if the frog reaches the 10-meter mark on its last jump, it does not slip back. This interpretation supports the idea that the frog needs 10 jumps to escape.

Others consider the well's edge more complex. If the frog must go over the 10-meter mark to escape, it could take 11 jumps, making the final calculation a bit tricky:

10 meters 1 meter (to go over the edge) 11 meters total, 11 meters ÷ 1 meter gain 11 jumps

Final Insight

The classic solution to the riddle, considering the well as a 10-meter deep pit, is that the frog needs to make 10 jumps to escape. This conclusion assumes that the frog does not slip back after reaching the 10-meter mark, which is usually the most common interpretation.

Understanding the mechanics of the riddle and the assumptions involved can help us appreciate the elegance of the solution. The frog's journey encompasses a series of gains and losses, ultimately leading to its freedom.