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Square pairs

Source: Maths Jam
Can you order the integers 1 to 16 so that every pair of adjacent numbers adds to a square number?
For which other numbers \(n\) is it possible to order the integers 1 to \(n\) in such a way?

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Tags: numbers, square numbers, sums
If you enjoyed this puzzle, check out Sunday Afternoon Maths LIX,
puzzles about numbers, or a random puzzle.

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