{'fraction_non_alphanumeric': 0.020625415834996674, 'fraction_numerical': 0.001996007984031936, 'mean_word_length': 4.509157509157509, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Pythagoras in a Box

A diagram of the Pythagorean theorem displays a right triangle with squares constructed on each leg (of lengths a and b) and the hypotenuse (of length c). (See figure 1.) Arthur Carter, a versatile artist, businessman, and journalist, wrapped a bronze relief sculpture of the diagram of the Pythagorean theorem in a box: He surrounded it by the smallest rectangle that bounded all points in the diagram and that had at least one of its sides coincident with at least one of the sides of the squares (figure 2). In Carter’s wrapping, two adjacent sides of the bounding rectangle coincided with one side of each of the squares on the legs, while the other two sides of the rectangle touched corners of the square on the hypotenuse. A different way to wrap the Pythagorean diagram in a box (figure 3) is to make one side of the bounding rectangle coincide with one side of the square on the hypotenuse, while each other side of the rectangle touches one corner of one of the squares on the legs. I propose to call this way of wrapping the diagram of the Pythagorean theorem the Retrac wrapping, by contrast with the Carter wrapping. Suppose the cost of a wrapping is proportional to the area of the bounding rectangle. For a given right triangle, which wrapping is less expensive? Although the question is entirely elementary, the answer is not obvious from inspection of the diagrams. Let the two legs of the right triangle have positive lengths a and b and the hypotenuse have length