A hare jumps over a 20-square grid. It continues jumping to an adjacent square up or down, never revisiting a square. Its 5th, 10th, 15th jumps...Read on...

Imagine a hare jumping across a 20-square rectangular grid arranged in 4 rows and 5 columns. The hare starts on one square and continues jumping to an adjacent square (horizontally or vertically), counting each jump and never revisiting a square. After 20 jumps, it exits the grid.

During the 5th, 10th, and 15th jumps, the hare marks those squares with circles and writes the jump number inside. Your task is to determine where to place the numbers 5, 10, and 15 in the marked circles and chart the complete path of the hare across all 20 squares.

**Recommended time to solve:** 20 minutes.

This puzzle is challenging but solvable with not too much effort. It requires a strategic and analytical approach. Let’s walk through the solution step-by-step.

## Solution to the Jumping Hare on the 20-Square Grid Puzzle

### Part 1: Placing the Numbers 5, 10, and 15 in the Circles

#### Step 1: Define the Strategy

Focus on **WHAT WE KNOW:** Numbers 5, 10, and 15 must be placed in the marked circles.

Without knowing the exact starting or ending points, we need an efficient strategy that avoids unnecessary guesswork.

**Chosen Strategy:** Analyze the possible placements of 5, 10, and 15 by identifying patterns in the circle placements.

#### Step 2: Analyze Possible Placements

The puzzle figure will aid in our analysis.

**Identify Any Special Pattern:**

**Key Pattern:***The circles at Row 2, Column 3 and Row 3, Column 2 are separated by only 4 squares.*This short distance prohibits placing numbers 5 and 10 or 10 and 15 in these circles as each pair is separated by 5 jumps.**Result:**The optimal placement is to put 10 in the top-right corner circle (Row 1, Column 5) since it aligns with the grid's constraints. 5 and 10 can only be placed in the other two circles.

**Conclusion:** The feasible placements are:

**Number 10**in the**top-right corner circle**(Row 1, Column 5).**Number 5**in the**middle circle**(Row 2, Column 3). 5 cannot be placed at the bottom circle as it would be too far from 10 in top-right corner.**Number 15**in the**bottom circle**(Row 3, Column 2).

Below is the figure showing these placements.

### Part 2: Finding a Feasible Path of 20 Jumps

#### Step 1: Define the Strategy

**Special Pattern Insight:** Numbers 10 and 15 are positioned near the grid’s periphery. To keep central squares open for remaining paths, we will focus on peripheral paths.

**Goal:** Find feasible paths from 10 to 15 and 15 to 20 on the periphery, ensuring central squares are available for paths from 1 to 5 and 5 to 10.

#### Step 2: Find Peripheral Paths

**Path from 10 to 15:**There are two paths, marked with stars and hashes. The path marked with stars uses the periphery and aligns with the overall strategy.**Path from 15 to 20:**The peripheral path marked with stars continues the strategy and ensures a smooth solution.

**Result:** **Most promising path from 10 to 20 via 15 with stars is the chosen path.**

Below is the figure showing these paths.

#### Step 3: Final Solution with All 20 Squares Covered

With the most promising chosen path from 10 to 20, the remaining paths are straightforward:

**Path from 1 to 5:**Start from the bottom right corner: 1 → adjacent left 2 → adjacent left 3 → adjacent up 4 → adjacent up 5.**Path from 5 to 10:**Start at 5: adjacent right 6 → adjacent down 7 → adjacent right 8 → adjacent up 9 → adjacent up 10.

Below is the solution figure.

**Alternative Solution:** Consider exploring if a second solution exists for this engaging puzzle.

**Postscript:** The solution hinges on recognizing special patterns, defining strategies, and leveraging these insights effectively. One of the strategies playing a key role for the solution was: finding the most promising peripheral path bypassing unnecessary multiple path evaluations. This is spatial configuration awareness often used in positional board games like chess.

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