. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. CH. XII] MISCELLANEOUS PROBLEMS 257 a right-angled triangle. In fact, however, if we are given any two squares we can always by three cuts divide them into five pieces which can be put together to make a square. There are various solutions. Here are two which answer the purpose*. The first of these is as follows: Place the larger and smaller squares AG and CE side by side as in figure i below, and take AB equal to CD. Then a cut BH and a cut BE ( a cut BJ o
. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. CH. XII] MISCELLANEOUS PROBLEMS 257 a right-angled triangle. In fact, however, if we are given any two squares we can always by three cuts divide them into five pieces which can be put together to make a square. There are various solutions. Here are two which answer the purpose*. The first of these is as follows: Place the larger and smaller squares AG and CE side by side as in figure i below, and take AB equal to CD. Then a cut BH and a cut BE ( a cut BJ on the larger square and another cut JE on the smaller square) will divide the squares into five pieces which can be put together to make one square of which BH and BE are sides. A more symmetrical, though less simple, five-part solution (made up of the smaller square together with a four-part division. \c> s\ B A\ E a\ C D Figure i. Figure ii. of the larger square) can be effected as follows. Place the larger square (denoted by the letters A,B,C, D) next the smaller square (denoted by E), as in figure ii, with their bases in the same line. Bisect this line in one point a and on their common side take another point b whose distance above their bases is half the sum of their sides. Through these points draw lines perpendicular to each other, crossing in the centre of the larger square, and terminating in its sides. This divides the larger square into four equal parts A,B,C, D. Produce the two lines drawn through a and b for half their length beyond the common base and side of the squares, and through their extremities draw two other lines perpendicular to them. These four lines will form another * H. Perigal, Messenger of Mathematics, 1873, vol. ii, pp. 103—106; H. E. Dudeney, Amusements, London, 1917, p. 32. 17 B. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of
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