This is an interesting puzzle that involves filling a grid with numbers such that the sum of any two neighboring numbers forms a perfect square. We already have the number 10 placed in the grid, and we need to fill the remaining positions with numbers from 1 to 15 without repeating any number.

Step 1: Identify valid sums

First, we need to find which sums of two whole numbers between 1 and 15 are perfect squares. The perfect squares up to 30 (since 1 + 15 = 16 is the largest sum we can have) are:

• 1, 4, 9, 16, 25

Thus, we need to identify pairs of numbers whose sums are one of these perfect squares. We will check combinations for possible valid pairs:

• $1 + 3 = 4$
• $1 + 8 = 9$
• $2 + 7 = 9$
• $3 + 6 = 9$
• $4 + 5 = 9$
• $5 + 11 = 16$
• $6 + 10 = 16$
• $7 + 9 = 16$
• $8 + 15 = 23$ (not a square)
• $9 + 7 = 16$
• $10 + 15 = 25$
• $11 + 14 = 25$
• $12 + 13 = 25$
• $13 + 12 = 25$
• $14 + 11 = 25$
• $15 + 10 = 25$

Step 2: Build the grid

Based on the valid pairs, we need to find a sequence where the sums of neighboring numbers form perfect squares. Without seeing the grid's specific layout, we would have to try several permutations and adjacent placements, but the strategy remains the same:

1. Start by placing 10 in a position.
2. Check all possible neighboring numbers from 1 to 15 that satisfy the perfect square sum condition.
3. Place these numbers next to the given number and continue filling out the grid.

Step 3: Attempting a solution

Without a visual representation of the grid, here's a conceptual example of how the numbers might be arranged (assuming we can form a linear sequence or a 2D arrangement). Using some of the pairs from earlier:

• $10 + 6 = 16$
• $6 + 3 = 9$
• $3 + 6 = 9$
• $6 + 9 = 16$

This continues as we go. You can test permutations with numbers that make these squares and adjust based on the specific grid structure.

Would you like me to provide a more detailed exploration of the grid filling process?