Question 29 of 29advanced✨ CreateShort Answer4 marks

Can you arrange numbers from 1 to 32 (without repetition) in a circle such that the sum of every adjacent pair of numbers is a square?

Correct Answer

Problem Analysis: This puzzle requires finding a Hamiltonian cycle in a graph where:

  • Vertices are numbers 1-32
  • Edges connect pairs summing to perfect squares

Key Constraint: Each number must have exactly two neighbors in the circle.

Example: Number 16 can only pair with 9 (16 + 9 = 25) and 20 (16 + 20 = 36) to form squares, creating specific constraints.

Conclusion: This puzzle may not have a solution for all ranges of numbers, as the circular arrangement requires both a Hamiltonian path AND the first and last numbers must also sum to a square.

Exercise: Square Pairs! | Q: 2 | (Page Page 18)
For More Understanding

Explanation

The chapter extends the square pairs puzzle to a circular arrangement. This requires finding a Hamiltonian cycle rather than just a path. The problem becomes more constrained because the first and last numbers must also sum to a square.

Solution Steps

  1. Step 1: List all pairs from 1-32 that sum to perfect squares

  2. Step 2: Note that squares possible are: 4, 9, 16, 25, 36, 49

  3. Step 3: Create connections based on these sums

  4. Step 4: Attempt to find a cycle that visits each number exactly once and returns to start

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