The Puzzle Instinct: The Meaning of Puzzles in Human Life

3 Puzzling Pictures

To the naked eye, the opposite angles formed when two straight lines intersect appear to be equal. But this appearance, as obvious as it may seem, does not establish the angles as necessarily or always equal. In his great textbook of mathematical theory and method, known as Elements , the Greek mathematician Euclid (who lived around 300 B.C.) went beyond the evidence of eyesight and proved, beyond any iota of doubt, the equality of such angles. Using an argument that is remarkably easy to follow, Euclid demonstrated that the opposite angles formed when two straight lines intersect not only look equal to the eye, they are necessarily so. On the next page is a diagram similar to the one used by Euclid in his demonstration. The intersecting lines are labeled AB and CD, and two of the four opposite angles formed by their intersection are labeled x and y. The angle between x and y on the top part of the diagram is labeled z. The goal is to show that x and y, no matter what their size, will always be equal [see figure 3.1]. Euclid’s proof can be given in contemporary algebraic notation as follows. He started by reminding his readers that a straight line is in fact an angle of 180°. Since CD is composed of two smaller angles, x and z, 3 PuzzlingPictures O I, M,  O V M-B Mathematics would certainly have not come into existence if one had known from the beginning that there was in nature no exactly straight line, no actual circle, no absolute magnitude. —Friedrich Nietzsche (1844–1900) the sum of the two is 180°—a statement that in modern algebraic notation can be represented with the equation x + z = 180°. Also, since AB is composed of y and z, their sum, too, is 180°—a statement that can be represented with the equation y + z = 180°: (1) x + z = 180° (2) y + z = 180° Equations (1) and (2) can be reformulated as follows: (3) x = 180° − z (4) y = 180° − z Logically, since things equal to the same thing are equal to each other, we can deduce that x = y, since both variables are equal to the same thing, namely (180° − z). Euclid ended his proof with the declaration “Which was to be demonstrated,” a phrase expressed later with the Latin abbreviation QED, for Quod erat demonstrandum—the stamp of mathematical authority that Euclid’s method of demonstration came to represent. To the emerging mathematical way of thinking, Euclid’s QED method was incontestable, because it demonstrated by logical argument why a certain pattern perceived as noteworthy and regular by the eye is the way it is. Although the early builders and land surveyors measured fields and laid out right angles with strings and various instruments, they relied primarily on visual inference. Nonetheless, using the patterns that they perceived with their eyes, they were able to construct truly impressive Figure 3.1 72 ThePuzzleInstinct buildings and accomplish remarkable engineering feats. Those ancient architects and engineers suspected, of course, that many of the patterns they saw had an underlying logic. But unraveling such logic had to await the groundbreaking approach of Euclid, who systematized the attempts of mathematicians such as Pythagoras and Thales (c. 662–c. 545 B.C.) before him to reconcile sensory perception with rational understanding . As Kline (1959: 75) aptly puts it, “Euclid was the great master who arranged the scattered conclusions of his predecessors so that they all followed by deduction.” Needless to say, the relation between visual perception and accurate reckoning, to use Ahmes’ term, was not the exclusive jurisdiction of the ancient Greek geometers. As a matter of fact, it has always been of great interest to mathematicians and puzzlists alike, all over the world. The puzzles that the latter have left for posterity constitute playful experimentations with visual pattern and the QED method. On this leg of our journey through Puzzleland, we thus come to the region inhabited by the Carrollian character of the Caterpillar, where things, like the insect itself, are not always what they seem to be. Here puzzlists, with their many ingenious tricks and traps in hand, have been busily working away since antiquity to expose eyesight as sometimes unreliable—the very same objective that Euclid and other great geometers had when they established geometry as a deductive science. VisualTrickery We start our journey by considering...