Solve the Challenging 20-Square Number Grid Puzzle
A hare jumps over a 20-square grid, never revisiting a square. Can you solve on which three circled squares it lands in its 5th, 10th, and 15th jumps?
The 20-Square Number Grid Puzzle
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 needs a strategic and analytical approach with the requirement of key pattern identification and use. Let’s walk through the solution step-by-step.
Solution to the Jumping Hare on the 20-Square Number 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 possibilities of 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: Only 4 squares separate the circles at Row 2, Column 3 and Row 3, Column 2. This short distance prohibits placing numbers 5 and 10 or 10 and 15 in these circles as exactly 5 jumps separate each pair. The separation of these two circles falls short by 1 jump. This pair of circles at the bottom and in the middle are too near for both 5, 10 and 10, 15.
- Result: 10 being common to both the pairs 5, 10 and 10, 15 prohibited for placement in the bottom and the middle circles, the only option is to place it at the top right circle (Row 1, Column 5). We must then place the pair 5 and 15 in the remaining two circles nearest to each other (the bottom and the middle circles).
Question remains, in which circle to place 5 and in which 15? Again, spatial analysis of the longest distance between the bottom and the top right circles settles this placement.
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), because 5 cannot be placed in the bottom circle, which would be too far from 10 in the top-right corner.
- Number 15 in the bottom circle (Row 3, Column 2).
Below is the figure showing these placements.
Part 2: Finding the Most Promising Paths Between the Three Circles
Step 1: Define the Strategy
Special Pattern Insight and Strategy Adopted: Numbers 10 and 15 are near the grid’s periphery (the outermost rows and columns). To keep central squares (the middle area of the grid) open for the remaining paths, we will focus on peripheral paths.
Goal: Find feasible paths from 10 to 15 and 15 to 20 (the terminal jump) on the periphery, ensuring central squares are available for paths from 1 (the starting point) to 5 and 5 to 10.
Step 2: Find the Most Promising Path Using the Peripheral Squares
- 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 to cover 11 squares. Placing the bulk of numbers on the squares in this long path is easy. The terminal number 20 will occupy the top left square in this most favorable scheme.
In contrast, the path marked by hashes blocks the central squares, most probably making solution impossible. In fact, at this stage, I am sure it will. We will move on with the promising path marked with stars and see where it leads us.
Result: The most promising path from 10 to 20 via 15 with stars is the chosen path.
Below is the figure showing these paths.
Part 3: Final Solution of the Number Grid Puzzle with All 20 Squares Covered
As expected, with the most promising path from 10 to 20 chosen, discovering the remaining paths are now 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.
In fact, as soon as I identified the path marked with stars, the final solution also became clear to me.
Below is the solution figure.
Any Alternative Solution? Explore whether a second solution exists for this engaging puzzle. It sure will be an interesting experience.
Postscript
The solution depends heavily on recognizing special patterns, defining strategies, and using these effectively. The strategy 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.
This honorable member in the class of number grid puzzles is difficult to solve without key pattern identification, use of the patterns, spatial configuration awareness and strategic approach throughout.
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