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I saw this problem in the UKMT 2020 SMC

https://i.stack.imgur.com/k5EJg.png

The diagram above shows on the left an 8 × 8 square divided into four pieces which are rearranged to make the 13 × 5 rectangle shown on the right. However 8 × 8 < 13 × 5. Where has the additional area come from?

I tried to do this myself, but I couldn't figure out an explanation. Maybe the figure on the right is misleading and doesn't actually form a rectangle?

Blue
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Mike24
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    [You are correct!](https://en.wikipedia.org/wiki/Chessboard_paradox) – Andrew Chin Jul 12 '21 at 12:12
  • The triangle with legs 3,8 does not align with the triangle with legs $5,5+8$ since $5/13 \ne 3/8$. – lhf Jul 12 '21 at 12:17
  • On this site the "missing square" puzzle is solved at https://math.stackexchange.com/q/287/139123. This example is set up a bit differently, but it relies on the same disproportionate triangles, a right triangle with legs $5$ and $2$, a right triangle with legs $8$ and $3$, and a right triangle with legs $13$ and $5$. – David K Jul 12 '21 at 12:21
  • Thanks for the help! – Mike24 Jul 12 '21 at 12:22
  • Related https://math.stackexchange.com/a/743074/6460 – Henry Jul 12 '21 at 13:19

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