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Grand Piano

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Jack and Jill are moving into a new flat and their grand piano presents a potential problem. Fortunately, it will just pass round the corridor without being tipped or disassembled.
Given that its area, looking down from above, is the largest possible which can be passed around the corner, what is the ratio of its length to its width?

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If you enjoyed this puzzle, check out Sunday Afternoon Maths LXII,
puzzles about remainders, or a random puzzle.
© Matthew Scroggs 2018