gt;gt; Chapter One gt; "Sports and Pastimes, Done by Number": Mathematical Tricks, Mathematical Games gt; gt; You've probably played a mathematical game at one time or another. From the counting games we learn as children and the calculator tricks we play in the schoolyard to classics like Sprouts or Lewis Carroll's Game of Logic, there's a whole world of game playing to be had in the realm of numbers. Mathematicians used to be accused of doing magic (some still are), and while conjuring spirits or divining the future may be far from what most of us think of when we think of mathematics, there is a timeless innocent pleasure in the wool-over-the-eyes mathematical tricks of the kind that this chapter showcases. gt; The selections in this chapter cover the whole period from the middle of the sixteenth century to the end of the twentieth, and if they show one thing it is that tastes have not changed all that much. Some of the very first mathematics books to be printed in English contained "guess my number" tricks and questions about what happens when you double a number again and again and again: kinds that are still popular. gt; At the same time, there are some areas in which innovations in mathematics have opened up new ground for mathematical games and puzzles. Leonhard Euler, for instance (whom we will meet again in Chapter 7), did important work on the ways of traversing a maze or a set of paths, and this made it much easier for mathematical writers after him to set "route problems" with confidence. Rouse Ball's 1892 "recreations" included reports on this and on a mathematical problem—that of coloring a map—which was an unsolved problem in his day, and, like Alan Parr's open-ended family of Femto games in the final extract of this chapter, it shows how mathematical games can also be an invitation to explore, discover, and create for yourself. gt; gt; The Well Spring of Sciences gt; gt;Humfrey Baker, 1564gt; gt; Humfrey Baker was a teacher in sixteenth-century London, the translator of a book on almanacs, and the author of the very successful arithmetic primer, gt;The Welspring of Sciencesgt;, embodying its author's infectious enthusiasm for its subject (he once compared arithmetic to good wine, which needed no "garlande" to persuade buyers of its merits). gt; First published in 1562, gt;The Welspringgt; went into many editions down to 1670: the later versions were simply called gt;Baker's Arithmetickgt;. The final section of the book gave a selection of mathematical amusements, some of the first pieces of recreational mathematics to be printed in England. gt; Baker's dense prose is presented here in a simplified paraphrase. gt; Humfrey Baker (fl. 1557-1574), gt;The Welspring of Sciences, Which teacheth the perfecte worke and practise of Arithmetic both in vvhole numbers & fractions, with such easie and compendious instruction into the saide art, as hath not heretofore been by any set out nor laboured. Beautified vvith most necessary Rules and Questions, not onely profitable for Marchauntes, but also for all Artificers, as in the Table doth plainely appere. Novv nevvely printed and corrected. Sette forth by Humfrey Baker Citizen of London. (London, 1564), 158gt;vgt;-162gt;rgt;.gt; gt; gt; If you would know the number that any man doth think or imagine in his mind, as though you could divine ... gt; Bid him triple the number. Then, if the result be even, let him take half of it; if it be odd, let him take the "greater half" (that is, the next whole number above half of it). Then bid him triple again the said half. Next, tell him to cast out, if he can, 36, 27, 18, or 9 from the result: that is, ask him to subtract 9 as many times as is possible, and keep the number of times in his mind. And when he cannot take away 9 any more, tell him to take away 3, 2, or 1, if he can, so as to find out if there is anything left besides the nines. gt; This done, ask how many times he subtracted 9. Multiply this by 2. And if he had any thing remaining beside the nines, add 1. gt; For example, suppose that he thought of 6. Being tripled it is 18, of which a half is 9. The triple of that is 27; now ask him to subtract 18, or 9, or 27, and again 9. But then he will say to you that he cannot; ask him to subtract 3, or 2, or 1. He will say also that he cannot; thus, considering that you have made him to subtract three times 9, you shall tell him that he thought of 6, for 3 times 2 makes 6. gt; If he had thought of 5, the triple of it is 15, of which the "greater half" is 8. The triple of that makes 24, which contains two nines. Two times two makes four, and since there is something remaining we add 1. This makes 5, which is the number that he thought of. gt; gt; If someone in a group has a ring upon his finger, and you wish to know, as though by magic, who has it, and on which finger and which joint ... gt; Ask the group to sit down in order, numbering themselves 1, 2, 3, etc. Then leave the room, and ask one of the onlookers to do the following. Double the number of the person that has the ring, and add 5. Then multiply by 5, and add the number of the finger on which the ring is. Then ask him to append to the result the figure (1, 2, or 3) signifying which joint the ring is on. (Suppose the result was 89 and the ring was on the third joint; then he will make 893.) gt; This done, ask him what number he has. From this, subtract 250, and you will have a number with at least three digits. The first will be the number of the person who has the ring. The second will be the number of the finger. And the last will be the number of the joint. So, if the number was 893, you subtract 250, and there will remain 643. Which shows you that the sixth person has the ring on the fourth finger, and on the third joint. gt; But note that when you have made your subtraction, if there is a zero in the tens-that is, in the second digit-you must take one from the hundreds digit. And that "one" will be worth ten tenths, signifying the tenth finger. So, if there remains 703, you must say that the sixth person has the ring on his tenth finger and on the third joint. gt; gt; In the same way, if a man casts three dice, you may know the score of each of them. gt; Ask him to double the score of one die, add 5, and then multiply by 5. Next, add the score of one of the other dice, and append to the result the score of the last die. Then ask him what number he has. Subtract 250, and there will remain 3 digits, which tell you the points of the three dice. gt; gt; Similarly, if three of your companions—say, Peter, James, and John—give themselves different names in your absence—for example, Peter would be called a king, James a duke, and John a knight—you can divine which of them is called a king, which a duke, and which a knight. gt; Take twenty-four stones (or any other tokens), and, first, give one to one of your friends. Next, give two to another of them, and finally give three to the last of them. Keep a note of the order in which you have given them the stones. Then, leaving the eighteen remaining stones before them, leave the room or turn your back, saying: "whoever calls himself a king, for every stone that I gave him let him take one of the remaining ones; he that calls himself a duke, for every stone that I gave him let him take two of them that remain; and he that calls himself a knight, for every stone that I gave him let him take four." gt; This being done, return to them, and count how many stones are left. There cannot remain any number except one of these: 1, 2, 3, 5, 6, 7. And for each of these we have chosen a special name, thus: gt;Angeli, Beati, Qualiter, Messias, Israel, Pietasgt;. Each name contains the three vowels a, e, gt;igt;, and these show you the names in order. gt;Agt; shows which is the king, gt;Egt; which is the duke, and gt;Igt; shows which is the knight, in the same order in which you gave them the stones. Thus, if there remains only one stone, the first name, gt;Angeligt;, shows by the vowels gt;a, e, igt; that your first friend is the king, the second the duke, and the third the knight. If there remain two stones, the second name, gt;Beatigt;, shows you by the vowels gt;e, a, igt; that your first friend is the duke, the second the King, and the third the knight. And so on for the other numbers and names. gt; gt; Mathematical Recreations gt; gt;Henry van Etten, 1633gt; gt; Henry van Etten's gt;Mathematicall Recreationsgt;, first published in French in 1624, collected together a wide variety of different material. Some of the "problems" were physical tricks or illusions, like "How a Millstone or other ponderosity may hang upon the point of a Needle without bowing, or any wise breaking of it." Others were numerical tricks like those in Baker's gt;Welspring of Sciencesgt;, above, and still others were optical effects or illusions. The extracts given below thus show some of the diversity in what could plausibly be called mathematics at the time: a diversity which is emphasized by the book's splendidly encyclopedic title. They include a remarkable early report of what Galileo had seen through his telescope, together with the cheery assertion that making a good telescope was a matter of luck ("hazard") as much as skill. gt; Van Etten was apparently a pseudonym of the French Jesuit Jean Leurechon (c. 1591-1670). The translation has been ascribed to various different people, but its real author remains a mystery. gt; gt; Henry van Etten (trans. anon.), gt;Mathematicall Recreations. Or a Collection of sundrie Problemes, extracted out of the Ancient and Moderne Philosophers, as secrets in nature, and experiments in Arithmetic, Geometrie, Cosmographie, Horologographie, Astronomie, Navigation, Musicke, Opticks, Architecture, Staticke, Machanicks, Chimestrie, Waterworkes, Fireworks, etc. Not vulgarly made manifest untill this time: Fit for Schollers, Students, and Gentlemen, that desire to knovv the Philosophicall cause of many admirable Conclusions. Vsefull for others, to acuate and stirre them up to the search of further knowledge; and serviceable to all for many excellent things, both for pleasure and Recreation. Most of which were written first in Greeke and Latine, lately compiled in French, by Henry Van Etten Gent. And now delivered in the English tongue, with the Examinations, Corrections and Augmentations.gt; (London, 1633), pp. 47-50, 98-102, 167, 208-209, 240. gt; gt; How to describe a Circle that shall touch 3 Points placed howsoever upon a plane, if they be not in a straight line gt; Let the three points be gt;A, B, Cgt;. Put one foot of the Compass upon gt;Agt; and describe an Arc of a Circle at pleasure; and placed at gt;Bgt;, cross that Arc in the two points gt;Egt; and gt;Fgt;; and placed in ITLITL, cross the Arc in gt;Ggt; and gt;Hgt;. Then lay a ruler upon gt;GHgt; and draw a line, and placing a Ruler upon gt;Egt; and gt;Fgt;, cut the other line in gt;Kgt;. So gt;Kgt; is the Center of the Circumference of a Circle, which will pass by the said three points gt;A, B, Cgt;. gt; Or it may be inverted: having a Circle drawn, to find the Center of that Circle. Make 3 points in the circumference, and then use the same way: so shall you have the Center, a thing most facile to every practitioner in the principles of Geometry. gt; gt; How to change a Circle into a square form gt; Make a Circle upon pasteboard or other material, and label the centre gt;Agt;; then cut it into 4 quarters, and dispose them so that gt;Agt;, at the center of the Circle, may always be at the Angle of the square. And so the four quarters of the Circle being placed so, it will make a perfect square, whose side gt;AAgt; is equall to the diameter. Now here is to be noted that the square is greater than the Circle by the vacuity in the middle. gt; gt; With one and the same compasses, and at one and the same extent, or opening, how to describe many Circles concentrical, that is, greater or lesser one than another gt; It is not without cause that many admire how this proposition is to be resolved; yea, in the judgement of some it is thought impossible, who consider not the industry of an ingenious Geometrician, who makes it possible: and that most facile, sundry ways. For in the first place, if you make a Circle upon a fine plane, and upon the Center of that Circle a small peg of wood be placed, to be raised up and put down at pleasure by help of a small hole made in the Center, then with the same opening of the Compasses you may describe Circles Concentrical: that is, one greater or lesser than another. For the higher the Center is lifted up, the lesser the Circle will be. gt; Secondly, the compass being at that extent upon a Gibbous body, a Circle may be described, which will be less than the former, upon a plane, and more artificially upon a Globe, or round bowl. And this again is most obvious upon a round Pyramid, placing the Compasses upon the top of it, which will be far less than any of the former; and this is demonstrated by the 20th Proposition of the first book of Euclid's gt;Elementsgt;. gt; gt; Of spectacles of pleasure gt; ... Now I would not pass this Problem without saying something of Galileo's admirable Glass: for the common simple perspective Glasses give to aged men but the eyes or sight of young men, but this of Galileo gives a man an Eagle's eye, or an eye that pierceth the heavens. First it discovereth the spotty and shadowed opacous bodies that are found about the Sun, which darkeneth and diminisheth the splendour of that beautiful and shining Luminary; secondly, it shows the new planets that accompany Saturn and Jupiter; thirdly, in Venus is seen the new, full, and quartal increase, as in the Moon by her separation from the Sun; fourthly, the artificial structure of this instrument helpeth us to see an innumerable number of stars, which otherwise are obscured by reason of the natural weakness of our sight. Yea, the stars in the Milky Way are seen most apparently; where there seems no stars to be, this instrument makes apparently to be seen, and further delivers them to the eye in their true and lively colour, as they are in the heavens: in which the splendour of some is as the Sun in his most glorious beauty. This Glass hath also a most excellent use in observing the body of the Moon in time of Eclipses, for it augments it manifold, and most manifestly shows the true form of the cloudy substance in the Sun, and by it is seen when the shadow of the Earth begins to eclipse the Moon, and when totally she is overshadowed. gt; Besides the celestial uses which are made of this Glass, it hath another notable property: it far exceedeth the ordinary perspective Glasses which are used to see things remote upon the Earth, for as this Glass reacheth up to the heavens and excelleth them there in his performance, so on the Earth it claimeth preeminency. For the objects which are farthest remote, and most obscure, are seen plainer than those which are near at hand, scorning, as it were, all small and trivial services, as leaving them to an inferior help. Great use may be made of this Glass in discovering of Ships, Armies, etc. gt; Now the apparel or parts of this instrument or Glass is very mean or simple, which makes it the more admirable (seeing it performs such great service), having but a convex Glass, thickest in the middle, to unite and amass the rays, and make the object the greater ..., augmenting the visual Angle. As also a pipe or trunk to amass the Species, and hinder the greatness of the light which is about it (to see well, the object must be well enlightened, and the eye in obscurity). Then there is adjoined unto it a Glass of a short sight to distinguish the rays, which the other would make more confused if alone. As for the proportion of those Glasses to the Trunk, though there be certain rules to make them, yet it is often by hazard that there is made an excellent one, there being so many difficulties in the action, therefore many ought to be tried, seeing that exact proportion, in Geometrical calculation, cannot serve for diversity of sights in the observation. gt; gt; Of the Dial upon the fingers and the hand gt; Is it not a commodity very agreeable, when one is in the field or in some village without any other Dial, to see only by the hand what of the clock it is? which gives it very near, and may be practised by the left hand in this manner. gt; Take a straw, or like thing, of the length of the Index or the second finger. Hold this straw very tight between the thumb and the right finger, then stretch forth the hand and turn your back and the palm of your hand towards the Sun, so that the shadow of the muscle which is under the thumb touches the line of life, which is between the middle of the two other great lines, which is seen in the palm of the hand. This done, the end of the shadow will show what of the clock it is: for at the end of the great finger it is 7 in the morning or 3 in the evening; at the end of the Ring finger it is 8 in the morning or 4 in the evening; at the end of the little finger or first joint, it is 9 in the morning or 3 in the afternoon; 10 and 2 at the second joint; 11 and 1 at the third joint; and midday in the line following, which comes from the end of the Index finger. gt; gt;(Continues...)gt; gt; gt; gt;gt; gt;gt;gt; Excerpted from gt;A Wealth of Numbersgt; Copyright © 2012 by Benjamin Wardhaugh. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. 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