The Puzzle Instinct: The Meaning of Puzzles in Human Life

5 Puzzling Numbers

As we saw in the opening chapter, some puzzles have played a significant role in the development of mathematics. Among the first works dealing with the basics of mathematical science one finds anthologies of puzzles. The Rhind Papyrus, as we saw, was most likely designed as an educational tool for teaching problem-solving, as a reference manual in practical mathematical theory, and as a source of brain-puzzling recreation all in one. And some of the greatest mathematicians of history have investigated theorems and methods through the medium of puzzles. One of these was Archimedes. Legend has it that he devised his Cattle Problem to take revenge on one of his adversaries, whom he was trying to dumbfound with his mathematical prowess. Nevertheless, the Cattle Problem stimulated the development of notational standards in mathematics. Cognizant of its potential impact on method, Archimedes dedicated it to his friend, the great Alexandrian astronomer Eratosthenes. The original statement of the puzzle is lost. Of the various versions that have come down to us, the one reproduced below, taken from the authoritative English-language edition of Archimedes’ works by T. L. Heath (1958: 319), contains the enigmatic extra conditions that follow the ellipsis: 5 PuzzlingNumbers M S, C,  O M R Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. —Bertrand Russell (1872–1970) If thou art diligent and wise, O Stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Thrinician isle of Sicily, divided into four herds of different colours, one milk white, another glossy black, the third yellow, and the last dappled. In each herd were bulls, mighty in number according to these proportions: understand, stranger, that the white bulls were equal to a half and a third of the black together with the whole of the yellow, while the black were equal to the fourth part of the dappled and a fifth, together with, once more, the whole of the yellow. Observe further that the remaining bulls, the dappled, were equal to a sixth part of the white and a seventh, together with all the yellow . These were the proportions of the cows: the white were precisely equal to the third part and a fourth of the whole herd of the black; while the black were equal to the fourth part once more of the dappled and with it a fifth part, when all, including the bulls, went to pasture together. Now, the dappled in four parts were equal in number to a fifth part and a sixth of the yellow herd. Finally the yellow were in number equal to a sixth part and seventh of the white herd. If thou canst accurately tell, O stranger, the number of cattle of the Sun, giving separately the number of well-fed bulls and again the number of females according to each colour, thou wouldst not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise . . . But come, understand also all these conditions regarding the cows of the Sun. When the white bulls mingled their number with the black, they stood firm, equal in depth and breadth, and the plains of Thrinicia, stretching far in all ways, were filled with their multitude. Again, when the yellow and dappled bulls were gathered into one herd they stood in such a manner that their number, beginning from one, grew slowly greater till it completed a triangular figure, there being no bulls of other colours in their midst nor none of them lacking. If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom. Today, Archimedes’ complicated puzzle is solved in as straightforward a manner as any other word problem in algebra. But it is mind144 ThePuzzleInstinct boggling to think how the ancient Greeks would have gone about solving it, possessing only the most rudimentary of representational techniques . Using modern algebraic notation, we start our solution by letting upper-case X, Y, Z, and T stand for the number of white, black, dappled, and yellow bulls, respectively; and lower-case x, y, z, and t for the...