A frog is sitting on the first of 10 stones that are placed in a line. The frog can jump either to the next stone or skip one stone and jump to the second stone ahead. The frog wants to reach the last stone (the 10th stone), but it wants to do so in as many different ways as possible. The question is: How many different ways can the frog reach the 10th stone?

Objective:
Determine the total number of distinct ways the frog can reach the 10th stone by making jumps of either one or two stones at a time.

Solution:

This puzzle can be approached using a method similar to the Fibonacci sequence, where each step depends on the sum of the previous steps.

Step-by-Step Solution The Frog and Stones Puzzle:

Step 1: Understand the Jump Possibilities

• From any given stone, the frog can jump to the next stone (1-stone jump) or skip one stone and land on the second stone ahead (2-stone jump).
• For instance, if the frog is on Stone 1, it can jump to Stone 2 or Stone 3.

Step 2: Calculate the Number of Ways to Reach Each Stone

To find the total number of ways to reach Stone 10, you need to calculate the number of ways to reach each stone up to the 10th one.

• Stone 1: The frog starts here, so there’s only 1 way to be on Stone 1.
• Stone 2: The frog can only come from Stone 1, so there’s 1 way to reach Stone 2.
• Stone 3: The frog can come from either Stone 1 (1-stone jump) or Stone 2 (2-stone jump), so there are 2 ways to reach Stone 3.

Continue this pattern for each stone:

• Stone 4: The frog can come from Stone 2 or Stone 3, so the number of ways to reach Stone 4 is the sum of the ways to reach Stone 2 and Stone 3.
• Stone 5: The frog can come from Stone 3 or Stone 4, so the number of ways to reach Stone 5 is the sum of the ways to reach Stone 3 and Stone 4.

Step 3: Calculate for Stone 10

By continuing this method, you calculate the number of ways to reach each subsequent stone until you reach Stone 10.

The sequence looks like this:

• plaintext

  Stone 1: 1 way
  Stone 2: 1 way
  Stone 3: 2 ways
  Stone 4: 3 ways
  Stone 5: 5 ways
  Stone 6: 8 ways
  Stone 7: 13 ways
  Stone 8: 21 ways
  Stone 9: 34 ways
  Stone 10: 55 ways
  

Final Answer

The frog can reach the 10th stone in 55 different ways.