Square Pairs Puzzle

The Problem

Arrange numbers 1 to 17 in a row such that the sum of every adjacent pair is a square number.

Strategy

Let’s map out possible neighbors for each number (1-17) that sum to a square (4, 9, 16, 25, 36).

• 1: 3, 8, 15
• 2: 7, 14
• 3: 1, 6, 13
• 4: 5, 12
• 5: 4, 11
• 6: 3, 10
• 7: 2, 9
• 8: 1, 17
• 9: 7, 16
• 10: 6, 15
• 11: 5, 14
• 12: 4, 13
• 13: 3, 12
• 14: 2, 11
• 15: 1, 10
• 16: 9 (only one neighbor!)
• 17: 8 (only one neighbor!)

The Sequence

Start with 17:

1. 17 must connect to 8 ($17+8=25$).
2. 8 connects to 17 (used) or 1 ($8+1=9$).
3. 1 connects to 8 (used) or 3 or 15.
   • If we pick 3: $1 \to 3 \to 6 \to 10 \to 15 \dots$
   • If we pick 15: $1 \to 15 \to 10 \to 6 \to 3 \dots$ Let’s trace from the other end (16) to meet in the middle.

Start with 16: 16 $\to$ 9 $\to$ 7 $\to$ 2 $\to$ 14 $\to$ 11 $\to$ 5 $\to$ 4 $\to$ 12 $\to$ 13 $\to$ 3 $\to$ 6 $\to$ 10 $\to$ 15 $\to$ 1 $\to$ 8 $\to$ 17.

Final Answer

16, 9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8, 17

(Check pairs: $16+9=25, 9+7=16, 7+2=9 \dots 8+17=25$. All squares!)