The missing square

Question

Four pieces are cut from squared paper: a right triangle 8 wide and 3 tall, a right triangle 5 wide and 2 tall, and two L-shaped pieces of 7 and 8 squares. Arranged one way, they appear to form a right triangle 13 wide and 5 tall. Arranged another way, they appear to form the same triangle with one square missing from its interior.

Where does the extra square come from? The pieces have not changed.

Accepted Answer

Neither arrangement is a true triangle. The two small triangles have different slopes (3/8 and 2/5), so the "hypotenuse" of the big shape bends slightly: bowed outward in one arrangement and inward in the other. The difference between the two bent shapes is exactly one square. Count the area. The pieces have areas 12 (the 8 by 3 triangle), 5 (the 5 by 2 triangle), 7 and 8: total 32. A genuine 13 by 5 right triangle has area 13 × 5 / 2 = 32.5. So the pieces cannot form that triangle exactly. The first arrangement has area 32 and is slightly smaller than the true triangle: its long edge sags be…

The missing square

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