Spoiler for New Scientist Enigma 1773 “Cutting Corners” (Follow the link to see the puzzle.)
We need a triangle with one side 65 mm (we’ll work in millimeters to make everything an integer), another side a+b, and another side c+d such that the following are Pythagorean triples:
- 56, a, c+d
- 56, b, 65
- 60, c, a+b
- 60, d, 65
So immediately we have b = 33 and d = 25. Now we just scan a table of Pythagorean triples for ones with 56 as one leg and a as the other, where a+33 is the hypotenuse of a Pythagorean triple which has 60 as a leg. We find 56, 42, 70; 42+33 = 75 is the hypotenuse of the triple 45, 60, 75. So the triangle’s other two sides (in cm) are 7.0 and 7.5.
Is that all there is to it? Seems too easy.