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Today Puzzle #620

Puzzle No. 620– Wednesday 27 November

A puzzle is composed of 5 shaped pieces, which when fitted together forms a square. Three of the pieces are triangles, one of which has an area half the total area, one whose area is an eighth, and one whose area is a sixteenth. The forth piece is a smaller square whose area is a quarter the size of the total area. What is the missing piece of the puzzle?

Dr Geoff Evatt, School of Mathematics, University of Manchester

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A triangle whose area is a 16th the overall size. Working out the area is easy (simply subtract the area of the other pieces from the full square). However to find the shape is much more fun. To see how the pieces can fit together, first draw a square and then see the only possible combinations for fitting the largest triangle and square in; it quickly becomes apparent the only way is to draw a diagonal across the square, where one half becomes the largest triangle. One can then see that the only place to fit the square is then in the corner of the larger square. This leaves two infilled triangular spaces, both of which are an eight the overall area, thus taking up another one of the triangular pieces. The remaining triangle slots in the other space leaving room for the missing piece, which is a triangle whose area is also a sixteenth the overall a size.

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