Problem 139

Problem 139

Pythagorean tiles

Let (a, b, c) represent the three sides of a right angle triangle with integral length sides. It is possible to place four such triangles together to form a square with length c.

For example, (3, 4, 5) triangles can be placed together to form a 5 by 5 square with a 1 by 1 hole in the middle and it can be seen that the 5 by 5 square can be tiled with twenty-five 1 by 1 squares.

However, if (5, 12, 13) triangles were used then the hole would measure 7 by 7 and these could not be used to tile the 13 by 13 square.

Given that the perimeter of the right triangle is less than one-hundred million, how many Pythagorean triangles would allow such a tiling to take place?


用(a, b, c)表示边长为整数的直角三角形的三边,可以将四个这样的三角形放在一起,使其外框构成边长为c的正方形。

例如,边长为(3, 4, 5)的三角形可以构成一个5x5的正方形,中间留有一个1x1的洞。而这个5x5的正方形又恰好可以用25个1x1的小正方形组成。

然而,如果我们用(5, 12, 13)的三角形,则中间的洞将会是7x7大小,不能用来组成13x13的大正方形。