The Magic of Numbers, Or, Curious Tricks With Figures

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London : ALDINE PUBLISHING Co., 9, Red Lion Court, Fleet S treet, B.C.

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1, T H E T R O U B L E S O M E B R O T H E R S , T o m B r ig g s “ M a j o r ” a n d P h il B r i g g s “ M in o r 2 O U R SC H O O L, AND A L L A B O U T T H E B O Y S i / 3. S A M S K Y L A R K , o r th e M id d ie s o f th e G u n R o o m . 4, F R I E N D L E S S F R E D , a S t o r y o f L o n d o n S t r e e ts * - I ■ - 5. T O M , T H E M ID S H IP M A N , o r H o n o u r fo r th e B r a v e Artv^ntures-on th e H ig h w a ys o f O ld L o n d o n 6 . T H E C A V A ! lcr,:> -T'ADTAIN 7. F O U G H T F< 8. G E N E R A L * 9. L I F E F O R I MORGAN, ‘ 1 1 TRACKED ’ 1 2. R E C K L E S S 1 3. N I C E B O Y S '1 4. T H E W JL D 1 5. B I L L Y BR IC 1 6 . SAM SLA B 1 7. T H E S L A S t 1 8. D O C T O R C 1 9. T H E K I N G ' 20 T H E W IL D 21 IN S T E E L / 22. T H E M ER C 23. T H E B A N K 24. T H E C A N A 25. IR O N C L A D 26. T H E S O L D 27. T H E D IA M i 28. T H E O C AE 29. T H E Y O U N 30. T H E H A L F 31. T H E IR O N 32. T H E F R O N 33. T H E H U N T 34. D E A D W O O 35. D O U B L E D 36. B U F F A L O 37. W IL D IV A N 3 8. T H E P H A N 39. D IC K IN D ‘4 0. D E A D W O O 41. C A L A M IT Y 42. D IC K D A R I 4 3. C O R D U R O 44. T H E S E A < M.I.M.C. (LONDON) 45. R O V E R S 0 Mile 4 6. D E A D W O O 4 7. D E A D W O O 4 8 . D E A D W O O l^ _______________ _______ w vw — 4 9 . B L O N D B IL L , o r D ead w o o d D ick's D isco very 50. A G A M E O F G O L D , o r D ead w o o d D ick ’s B ig S t r ik e 51. T H E M P IC K E D P A R T Y ,” o r D ead w ood D ick o f D ead w o o d 52. D E A D W O O D D IC K ’S D R E A M . A M in in g T a l e o f T o m b s to n e 53. T H E D W A R F A V E N G E R , o r D ead w ood D ic k ’s W a r d 54. D E A T H N O T C H T O W N , o r D ead w ood D ic k ’s D o o m 55. G I P S Y J A C K O F JI M T Q W N , o r D ead w o o d D ic k in D u r a n g o 5 6. S U G A R ‘C O A T E D S A M , o r th e B la c k G o w n s o f G r im G u ic h 57. G O L D - D U S T D IC K . A R o m a n c e o f R o u g h s a n d T o u g h s 58. T H E S P I R I T O F S W A M P - .L A K E , o r D ead w o o d D ic k ’s D iv id e 5 9 . D E A D W O O D D IC K ’S D E A T H T R A I L F R O M O C E A N T O O C E A N 60. D E A D W O O D D IC K ’S B IG D E A L , o r th e G o ld B r ic k o f O re g o n 61. D E A D W O O D . D IC K ’S D O Z E N , o r th e F a k ir o f P h a n to m F la t s 62. D EA D W O C ^ § j*ttf2 K ’S D U C A T S , o r R a in y D ays a t th e D ig g in g s 6 3. D E A D W O O W * C K S E N T E N C E D , o r th e T e r r ib le V e n d e tta 64. D E A D W O O D D IC K ’S D E M A N D , o r th e F a ir y F a c e o f F a r o F ia ts . 65. D E A D W O O D D IC K IN D E A D C IT Y 66 . D E A D W O O D D IC K ’S DIA M O -N D S, o r th e M y s te ry o f J o a n P o r t e r 67, D E A D W O O D D IC K IN N E W Y O R K , o r a C u te C a s e 68 .. D E A D W O O D D I C K S D U S T , o r th e C h a in e d H a n d 6 9. D E A D W O O D D IC K , J U N IO R , o r th e S ig n o f th e C r im s o n C r o s s 70. N IC K E L - P L A T E D N E D 71. S U N F L O W E R S A M O F S H A S T A 72. F L U S H F A N T H E F E R R E T 73. P H IL O F L Y O F P H E N IX , o r D ead w o o d Dick, J u n io r 's , R a c k e t a t C la im N o . 10.

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WILL ALMA

The State L ib ra ry of V ictoria

Cocker and P ilu o rth , 'W alkingam e and Yyse, I n th eir own sphere, by Bidder were outshone. T hey, w ith pen or pencil, problems so l\c d — H e, w ith no aid, b u t w ondrous memory ; They, w hen of years m ature acquired th e ir fam r, l i e “ lisped in num bers for the num bers cam e.”

B u t as th e principal o bject of th is volume is to enable th em to learn som ething in th eir sports, and to u n d er­ stan d w hat they are doing, wo shall, before proceeding to tho curious trick s H E delightful and valuable science an d feats cenucctod w ith th o scienco of num bers first arrived a t any of numbers, present th em w ith some degree of perfection in Europe amongarithm etical aphorisms, upon which m ost th e Greeks, who m ade use of tho letters of th e following examples arc founded. of th e alphabet to express thoir numbers. A p h o rism s o f N um ber. A sim ilar mode was followed by tho Homans, who, besides characters for each 1. If tw o even num bers bo ad d ed ra n k of classes, introduced others for to g eth er, or substracted from each other, five, fifty, and five hundred, which are th e ir sum or difference will be an even still used for chapters of books, an d some number. o th er u nim po rtant purposes. 2. If tw o uneven num bers bo added The common arithm etic, in which th e te n Arabic figures, 1, '2,3, 4, 5, G, 7, 8, 9, 0, or su b tracted th eir sum or differcnce'w ill are used, was unknow n to th e Greeks and bo. an even number. Romans. They came in to E urope by 3. The sum or difference of an even way of Spain from the Arabians, who are and an uneven liutnber added or sub­ believed to have received them from th e tra c te d will be an uneven number. an cient philosophers of India. 4. The product of tw o even num bers The Arabic system is supposed to have tak e n its origin from th e ton fingers of will bo an even num ber, and th e p ro d u ct1 th e hand, which were used in m aking of two uneven num bers will be an uneven calculations before arithm etic was brought number. a n art, and it is to this a rt th a t we G. The p roduct of an even and an uneven num ber will be an even num ber. in ten d to introduce our readers.

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THE MAGIC OF NUMBERS; OK,

G. If tw o different num bers be divisible digits, as in the figure, and to denote any by auy one num ber, th eir sum and th eir figure, a sm all peg w as in serted into th e difference will also be divisible by th a t hole corresponding to it. If th e num ber consisted of several figures, m ore cubes num ber. 7. If several different num bers, divided were used one for each. A cipher was r e p ­ by 13, b:i added or m ultiplied together, resented by a peg of different shape from th eir sum and th eir product will also be th a t of th e others, and inserted in the central hole. divisible by 3. To perform any arith m etical process, a 8. If two num bers, divisible by 9, be added together, the sum of the figures in the square board was provided, jdivided by am ount will be either 9, or a num ber ridges into recesses of the sam e w idth as t h 3 cubes, and by th is the cubes were divisible by 9. ! 9. If auy num ber be m ultiplied by 9, retained in th e required horizontal or by any other num ber divisible by 9, and perpendicular lines. Suppose it was the am ount of the figures of the product necessary to add together th e num bers will be either 9, or a num ber divisible 763, 124, 859, th e cubes and pegs would be arranged th u s : by 9. 10. In every arithm etical progression, 0 0 0 0 0 0 0 0 r if the first and last term ba each m u lti­ 0 0 0 0 0 0 0 0 0 plied by the num ber of term s, and the sum 0 0 0 0 0 0 0 0 0 of th e two products be divided by 2, the quotient will be the sum of th e series. r. 0 0 0 c 0 0 0 0 11. In every geom etric progression, if 0 0 0 0 0 0 8 -0 0 auy tw o term s be m ultiplied together, 0 0 0 o' 0 0 0 0 0 their product will bo equal to th a t term , w hich answ ers to the sum of these two 0 0 0 0 0 0 0 0 0 indices. Thus, in the series— 0 0 0 0 £ 0 0 0 0 1 2 3 4 5 0 0 0 0 0 0 0 0 0 2 4 8 10 32 If th e th ird and fourth term s 8 and 10 e 0 0 0 0 0 0 0 0 0 0 0 be m ultiplied together, the product 128 0 0 0 0 0 0 0 0 0 0 0 0 will be the seventh term of the series. Tn j like m anner, if th e fifth term be m ultiplied 0 0 0 0 0 0 0 0 0 0 0 0 into itself ,the product will be the tw entieth The Abacus. term , and if th a t sum be m ultiplied into 1 IIIS in stru m en t is used for teaching itself, the product will be the tw entieth _ num eration, and th e first principles term . Thci'cfore, to find the last, or any other term of a geom etric series it is not of arithm etic. U pon a fram e necessary to continue th e series beyond are placed, wires a few of the first term s. parallel to oneanPrevious to the num erical recreations, o ther an d ateq u al wo shall here describe certain m echanical d is ta n c e s. Ten m ethods of perform ing arithm etical cal­ sm all balls are culations, such as are not only in th e m ­ strung upon each selves entertaining, b u t will be found w ir e , b e in g m ore or loss useful to the young reader. placed as in the Arithmetic. m a r g in . T he 1HE blind m athem atician, Dr. Saunderr i g h t w ire de­ son, adopted a very ingenious device notes units, the for perform ing arithm etical operations by I n ext tons, and so 1 3 7 8 1 the sense of touch. on, 7th wire being Small cubes of wood were provided, and the place of millions. In Using the in one face of each, nine holes wore abacus, all th e balls arc first ranged at pierced, thus : one end, and a num ber of th em are then 1 2 3 o o o m oved to th e o th er end of each wire, to 4 5 I! o o o correspond to the figures required. The 7 8 9 o o o exam ple given in the m argin is 15,781, These holes represented the nine .the height of M ount W ane.

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COMIC AND CURIOUS fcBOBLEMS iN ARITHMETIC.

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Progression. for th e third, six ; and so on, increasing F a hundred stones be placed in n by tw o, to th e hundredth. The num ber of yards, therefore, w hich straight line, a t the distance of a yard from each other, tho first being a t tho tho person m ust w alk will be equal to th e panic distance from a basket, how m any sum of th e progression, 2, 4, 0, &c,* th e yards m ust tho person w alk who engages la st te rm of w hich is 200 (22). B u t tho to pick them up, ono by one, and p u t sum of th e progression is equal to 202, th em into the basket ? I t is evident th a t th e sum of th e two extrem es, m ultiplied to pick up the first stone, and p u t it into by DO, or half the num ber of term s ; th a t th e basket, th e person m u st w alk tw o is to say, 10,100 yards, w hich m akes yards ; for the second, he m ust walk four ; m ore th a n 5J miles.

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The famous forty-five. OW can num ber 45 be divided into four of tho addition, tho rem ainder of tho sn ch p arts th a t, if to th e first p a rt yon subtraction, th e p roduct of th e m u ltip li­ add 2, from th e second p a rt yon su b tract cation, and tho quotient of tho division 2. tho th ird p a rt you m ultiply b y 2, and m u st be all equal ? tho fourth p a rt you divide by 2, th e sum is 10 The 1st is 8 ; to w hich add 2, th e sum 2, th e rem ainder is 10 The 2nd is 12; su b tract The 3rd is 0 ; m ultiplied by 2, th e p roduct is 10 2, th e quotient is 10 by Tho 4tll is 2 0 ; divided

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45 R equired to su b tract 45 from 45, and leave 45 as a rem ainder 1 S o lu tio n -.—9 + 8 + 7 + G + 5 + 4 + 3 + 2 + l = 4 5 y 1+2+3+4+5+0+7+8+9=45 V / 8+0+4+1+9+7+5+3+2=45 7G542-24 Suppose for exam ple, the 24 figures p u t down arc 70542 ; ------------ these, added together, as nn7G518 its, m ake a to ta l of 24 ; miJes, fu riongs, ro d s , y a rd s , feet, inches. deduct 24 from th e first line, and F rom 1 0 0 0 0 0 7G518 re m a in ; if 5, tho centre figure T ake 0 7 ' 89 5 1 5 be stru ck out, th e to ta l will be 22. If 8, 0 0 0 0 0 1 th e first figure be struck out, 19 will be I n this problem , instead of borrowing th e to tal. 1 foot, we borrow j a foot=G inches, from I n order to ascertain w hich figure has w hich we take 5 inches, and 1 rem ains ; been stru ck out, you m ake a m en tal sum we th en carry J to 3, and borrow ing J a one m ultiple of 9 higher th a n th e total y a r d = l j feet, wo have 1J from 1 £ = 0 , given. I f 22 bo given as th e to tal, then and afterw ards proceed as usual. 3 tim es 9 are 27, and 22 from 27 show th a t 5 was stru ck out. If 19 bo The expunged figure. given, th a t sum deducted from 27 shows N th e first place desire a person to 8. w rite down secretly, in a line any Should th e to ta l bo equal m ultiples of num ber' of figures he m ay choose, and 9, as 18, 27. 30, th en 9 has been ex­ add them together as u n its ; having done punged. th is, tell him to su b tract th a t sum from W ith very little practice any person th e line of figures originally se t d o w n ; m ay perform this w ith rapidity, it is th e n desire him to strike out any figure therefore needless to give any further lie pleases, and add th e rem aining figures examples. The only way in w hich a in th e lino together as units, (as in person can fail in solving th is riddle is th e first instance,) and inform you of th e w hen eith er the num ber 9 or a cipher is result, when you will tell him th e figure struck out, as it th e n becomes impossible he has struck out. to tell w hich of th e tw o it is, th e sum of Subtraction. ,'ROM 1 m ile su b tract 7 furlongs, 39 rods, 5 yards, 1 foot, 5 inches.

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THE MAGIC OF ND3IEEES J

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h our a t w hich he has placed th e hand. F o r ex am p le:— Suppose th e hour a t w hich he intends to rise be 8, and th a t he has placed the The mysterious addition. hand a t 5 ; you will add 12 to 5, and tell T is required to nam e th e quotient of him to count 17 on th e dial, first reckon­ five or throe lines of figures—each ing 5, th e hour a t w hich th e index stands, line consisting of five or m ore figures — and counting backw ards from th e hour at only seeing the first line before th e other w hich he intends to rise ; and th e num ber lines are even p u t down. Any person 17 will necessarily end a t eight, which m ay w rite down the first line of figures shows th a t to be th e hour h^ chose. for you. H ow do you find th e quo­ To find the difference between two tien t ? E x a m p l e .—AVhen the first lino of fig­ numbers the greater of which is un­ ures is set down, subtract 2 from the last known. right-hand figure, and place it before the rilA K E as m any nines as there arc figures first figure of the line, and th a t is the JL in th e sm allest num ber, and su b tract quotient for five lines. F o r example, th a t sum from th e num ber of nines. Lot suppose th e figures given are 8G,214, the ano th er person add th e difference to the quotient will be 280,212. You m ay largest num ber, and taking aw ay th e first allow any person to p u t down th e figure of th e am ount add it to th e last two first and the fourth lines, bu t you m ust figure, and th a t sum will be th e difference always set down the th ird and fifth of th e tw o num bers. lines, and in doing so, always make F o r example : John, who is 22, tells lip 9 w ith the line above, as in th e follow­ Thom as, who is older, th a t he can dis­ ing example, cover th e difference of th e ir ages ; he 8G,214 Therefore in th e annexed therefore privately deducts 22 from 99 42,GS0 diagram y o u will see th a t y o u (his age consisting of two figures, he of 57,819 have m ade 9 in the th ird and course tak es tw o nines) ; th e difference, 62,85-1 fifth lines w ith the lines above w hich is 77, he tells T hom as to add to iJ7,145 them . If th e person desire his age, and to take aw ay th e first figure ---------to put down th e figures should from th e am ount, and add it to th e last Qt. 208,212 set down a 1 or 0 for the last figure and th a t will bo th e difference of figure, you m ust say we will th eir ages ; th u s,— have another figure, and a n ­ The difference betw een Jo h n ’s age other, and so on until he sets and 99 i s ................................................ 77 down som ething above 1 or 2. To w hich Thom as adding his ag e.. 85 In solving the puzzle w ith G7,85G three lines, you su b tract 1 The sum i s .................112 47,218 from the last figure, and place Then by taking away the first 52,781 it before the first figure, and figure 1, and adding it to the ---------m ake up the th ird line yourfigure 2, th e sum i s ...................... 13 Qt 167,855 self to 9. F o r example :— W hich add to J o h n ’s ag e................22 67,856 is given, and th e quo­ tie n t will be 167,855, as shown Gives th e age of T h o m as............... 85 in the above diagram. The Remainder. To tell at what hour a person very pleasing way to arrive at an a rith ­ intends to rise. m etical sum , w ithout th e use of E T the person set the hand of the dial either slate or pencil, is to ask a person . of a w atch at any hour he pleases and to th in k of a figure, th e n double it, then tell j’oii w hat hour th a t is ; and to the num ­ add a certain figure to it, now halve the ber of th a t hour you add in your m ind 12 ; whole . sum, and finally to su b tract from th en tell him to count privately th e n u m ­ th a t th e figure first thought of. You are b er of th a t am ount upon the dial, begin­ th en to tell th e thinker w hat is th e ning w ith the next hour to th a t on which rem ainder. ho proposes to rise, and counting back­ The key to this lock of figures is, th a t w ards, first reckoning the num ber of the h a l f , of w hatever sum you request to be th e figure iii the line being an even n u m ­ ber of nines in both cases.

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COillC AND CURIOUS PROBLEMS IN ARITHMETIC.

added during th e working of the sum is th a t none of th e th ree wives m ay be t h e r e m a in d e r . In the example given, found in th e com pany of one or two m en five is th e half of ten, the num ber re ­ unless h er husband bo p resen t? This m ay be effected in two o r three quested to he added. Any am ount m ay he added, h u t th e operation is simplified by w ay s; the following m ay be as good as giving only even num bers, as they will any :—L e t A and wife go over—le t A r e ­ tu rn —let B ’s and C’s wives go over—A’s divide w ithout fractions. E xam ple. wife retu rn s—B and 0 go over—B and Think o f................................................. 7 . wife retu rn , A and B go over—C’s wife Double i t ............................................... 14 re tu rn ’s, and A ’s and B ’s wives go over— Add 10 to i t ...........................................10 th e n O comes back for his wife. Sim ple as th is question m ay appear, it is found H alve i t ............................................... 2)24 in th e works of Alcuin, who flourished a thousand years ago, hundreds of years b e­ "Which will leav e................................... 12 fore th e a rt of p rin tin g was invented. S ubtract th e num ber thought o f. . . . 7 The false scales. The B emainder w ill b e .................... 5 C H E E S E being p u t in to one of th e A person having an equal number scales of a false balance, w as found of counters or pieces of money in ea§h to w eigh 10 lbs., and w hen p u t in to th e hand, to find how many he has alto­ oth er only 9 lbs. W hat is th e tru e w eight ? gether. The tru e w eight is a m ean proportional T ) equest the person to convey any num- betw een th e tw o false ones, and is found JLij ber, as 4, for example, from th e one b y extracting th e square ro o t of th eir p ro ­ hand to th e other, and th en ask how m any duct. T h u s lG x 9 = 1 4 4 ; and square ro o t tim es the less num ber is contained in th e 144 = 1 2 lbs., th e w eight required. greater. L e t us suppose th a t ho says the one is the triple of the o th e r; and in,.this The apple woman. case, m ultiply 4, th e num ber of th e coun­ rO O H woman, carrying a basket ters conveyed, by 3, and add to th e p ro­ of apples, was m et by th ree boys, duct the sam e num ber, w hich will m ake 1C. L astly, take 1 from 8, and if 10, be th e first of whom bought half of w hat she divided by the rem ainder 2, the quotient had, and th en gave h e r back 10; th e will he th e num ber contained in each second boy bought a th ird of w h at re ­ hand, and consequently the whole n u m ­ m ained, and gave h er back 2 ; and th e th ird bought half of w hat she h ad now left ber is 1G. This curious problem deserves another and retu rn ed h er 1; after w hich she found exam ple. L e t us again suppose th a t 4 she had 12 apples rem aining, AVhat n u m ­ counters are passed from one h and to th e b e r h ad she a t first ? F rom th e tw elve rem aining, deduct 1, other, and th e less num ber is contained in the g reater 2J tim es. I n th is case we and 11 is th e ntim ber she sold th e la st m u st, as before, m ultiply 4 by 2J, w hich boy, w hich w as half she had ; h er n u m ­ will give 9 J ; to w hich if 4 be added, we b er at th a t tim e, therefore, was 22. F rom shall have 13J, or : if 1, th en , be 22 deduct 2, and th e rem aining 20 w as § tak en from 2J, the rem ainder will be 1$, of h e r prior stock, w hich w as therefore 30. or by w hich, if 43° be divided, th e quo­ F ro m 30 deduct 10, and th e rem ainder 20 tie n t 10 will be the num ber of counters is half h er original stock; consequently she had a t first 40 apples. in each hand.

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■The three jealous Husbands. The Graces and Muses. /T1HBEE jealous husbands, A, B, 0 , w ith H E th ree G races, carrying each an JL th eir wives, being ready to pass by equal num ber of oranges, were m et night over a river, find a t the w ater side a b o at w hich can carry b u t tw o a t a tim e, b y th e nine Muses, w ho asked for some and for w ant of a w aterm an tliey are com ­ of th em : and each G race having given to pelled to row them selves over th e river at each Muse th e sam e num ber, it was th e n several tim es. The question is how those found th a t th e y h ad all equal shares, six persona shall pass, tw o a t a tim e, so H ow m any h ad th e Graces at first ?

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THE MAGIC OP NUMBERS ; OB,

Tho least num ber thas ,ttll answ er this question is tw elv e; for if we suppose th a t each Grace gave one to each Muse, the la tte r would each have three, and there

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would rem ain th ree for each Grace. (Any m ultiple of 12 will answ er th e conditions of th e question.)

The Jesuitical Teacher. three ladies each, b u t no two ladies were to bo allowed to walk together tw ice d ur­ ing th e week. H ow could tlicy be ar. ranged to su it th e above conditions ? FRT. s Kt . THU. WED. TUB. k n a e ] a h 0 a f r a i m b 1 0 b f m b i p b d n b g k i c g n 0 d k c h 1 c 0 o c f d h m d i 0 e m n 0 i k d 1 p 0 g h h k P f S 1 S m 0 h f n

T E A C H E R , having fifteen young ladies under her care, wished them to take a walk each day of th e week.

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SUN. a b c a e f g h i k 1 m n

MON. d L e h 0 m P f k o 1 n

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Atithmetical Puzzle. f from C you take 9, and from 9 you take 10 ; and if 50 from 40 be taken, there will just half a dozen rem ain.

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ANSW ER.

P rom SIX I F rom IX I F rom X L Take IX I Take X | Take L S

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The philosopher’s pupils.

Rem s.

The' money game.

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th e o th er th e odd num ber, or piece of silver. Tho sam e operations m ay th en be perform ed in regard to these two persons, as are perform ed in regard to th e tw o hands of th e same person, calling th e one privately tho right, and th e other tho left.

PERSO N having in one hand a piece of gold, and in th e other a piece of silver, you m ay tell m w hich hand he has th e gold, and in which th e silver, by tho following m e th o d : Some value, re p ­ resented by an even num ber, such as 8, m ust bo assigned to the gold ; and a value represented by an odd num ber, such as three, m ust be assigned to th e silver ; after which, desii'o the person to m ultiply the num ber in the right hand by any even num ber w hatever, such as 2, and th a t iu tho left by an odd num ber as 3 ; th en bid him add together the tw o p ro ­ ducts, and if the whole sum bo odd, tho gold will be in th e right hand, and the silver in tho left ; if the sum be even, the contrary will be tho case, To conceal the artifice b etter, it will be sufficient to ask w hether th e sum of the two products can be halved w ithout a rem ainder ; for in th a t case th e to ta l will be even, and in the contrary easo odd. I t m ay be readily seen, th a t the pieces, iustead of being in the two hands of tho sam e person, m ay be supposed to be in the hands of tw o persons, one of whom has the even num ber, or piece of gold, and

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O find a num ber of w hich th e half, 'fo u rth , and seventh added to three shall be equal to itself. This was a favourite problem among th o ancient G recian arithm eticians, who stated th e question in th e following m an ­ n e r : “ Tell us, illustrious Pythagoras, how m any pupils frequent th y school ? ” “ One half,” replied th e philosopher, “ study m athem atics, one fourth n atural philosophy, one seventh observe silence, and there are th ree fem ales besides.” Tho answer is, 28 : 1 4 + 7 + 4 + 3 = 2 8 . To discover a square number. SQUARE num ber is a num ber p ro­ duced by th e m ultiplication of any A num ber into it s e l f ; thus, 4 m ultiplied by 4 being th e square root from w hich it springs. The extraction of th e square root of any num ber takes some tim e ; and after all your labour you m ay perhaps find th a t th e num ber is n o t a square num ber, t o save th is trouble, it is w orth knowing th a t every square num ber ends either w ith a 1, 4, 5, 6, or 9, or w ith tw o cyphers, preceded by one of these num bers. A nother p roperty of a square num ber is, th a t if it be divided by 4, th e rem ainder, if any, will be 1—th u s, th e square of five

COMIC AND CURIOUS PROBLEMS IN ARITHMETIC.

o u t these four ; th e n proceed saying, 16, 15, 14, 13, rub out these lour ; and begin agaiu, 12, 11, 10, 9, and rub out th e s e ; and proceed again, 8, 7, 6, 5, th en rub ou t these ; The Sheepfold. and lastly say, 4, 3, 2, 1, w hen FA R M E R had a pen m ade of CO these fou>r are rubbed out. hurdles, capable of holding 100 sheep The whole tw en ty are rubbed only : supposing he w anted to m ake it out a t five tim es, and every sufficiently large to hold double th a t tim e an odd one, th a t is, 17th, num ber, how m any addional hurdles 13tli, 9th, 5th, and 1st. would he have occasion for ? This is a trick which, if once A nsw er.—Two. There were 24 hurdles seen, m ay be easily re tu rn e d ; on each side of the pen ; a hurdle a t the and th e puzzle a t first is, it top. and another a t th e bottom ; so th at, n o t occurring im m ediately to by m oving one of th e sides a little back, th e m ind to begin to rub them and placing an additional hurdle a t the out backw ards. I t is as sim ­ to p and bottom , th e size of th e pen ple as anything possibly can be. w ould be exactly doubled.

is 25, and 25 divided by 4 leaves a re ­ m ainder of 1 •, and again, 16, being a square num ber, can be divided by 4 w ithout leaving a rem ainder.

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Countrywoman and eggs. COUNTRYWOMAN carried eggs to a garrison, whore she had th ree guards to pass. She sold to th e first guard half th e num ber she had, and half an egg m ore ; to the second, th e half of w hat rem ained, and an half egg besides ; and to th e th ird guard she sold the h alf of the rem ainder, and half another egg. W hen she arrived a t th e m arket-place, she had th ree dozen still to s e ll; how was thisjpossible, w ithout breaking any of th e eggs ? I t 'would seem a t th e first view th a t th is is im possible, for how can half an egg be sold w ithout breaking any of th e eggs ? The possibility of all th is seem ing im possibility will be evident, w hen it is considered, th a ^ b y taking the g reater half of an odd num ber, we take th e exact half + J. W hen th e countryw om an passed the first guard she had 295 eggs ; by sel­ ling to th a t guard 148, w hich is th e half + J, she had 147 rem ain in g ; to th e second guard she disposed of 74, w hich is th e m ajor half of 147 ; and, of course, after selling 37 out of 74 to th e last, guard, she had still three dozen rem aining.

3 --4 -----6 ----0

-----

7 -----8 ----9 ------10 -----

11 ----12 ----13 14 15 10 17 18 19

---------- ------— --------------- -

20

-----

Odd or even.

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iW E B Y odd n um ber m ultiplied by an J odd num ber produces an odd n u m ­ b er ; every odd n u m b er m ultiplied by an even num ber produces an even nu m b er ; and every even num ber m ultiplied by an even num ber also produces an even n u m ­ ber. So, again, an even num ber added to an even num ber, and an odd num ber ad ­ ded to an odd num ber, produce an even n u m b e r; wliile an odd an oven n um ber added to g eth er produce an odd n u m ­ ber. If any one holds an odd num ber of counters in one hand, and an even n u m ­ b e r in th e other, it is n o t difficult to d is­ cover in w hich hand the odd o r even n u m ­ b er is. D esire th e p a rty to m ultip ly th e n um ber in th e rig h t hand by an even n u m ­ ber, and th a t in th e left h an d by an odd n um ber, th e n to add th e tw o sum s to ­ gether, and te ll you th e la st figure of tho product, if it is even, th e odd n u m b er will be in th e rig h t hand ; and if odd, in tho left h a n d ; th u s, supposing th ere are 5 counters in th e right hand, and 4 in th e left hand, m u ltiply 5 by 2, and 4 b y 3, th u s : 5 x 2 = 1 0 , 4 x 3 = 1 2 , and th e n adding 10 to 12, you have 1 0 + 1 2 = 2 2 , th e last figure of w hich, 2, is even, and th o odd num ber will consequently be in th e right hand.

How to rub out twenty chalks at five times, rubbing out every time an odd one. O do th is trick, you m ust m ake tw enty The old woman and her eggs. chalks, or long strokes, upon a board as in the m argin : T a tim e w hen eggs w ere scarce, an T hen begin and count back1 -----old w om an who possessed some re ­ w ards, as 20, 19, 18, 17, rub 3 ------ m arkably good-laying hens, wishing to

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THE MAGIC OF NUMBERS-, OR,

oblige h er neighbours, sent h er daughter round w ith a basket o£ eggs to three of th e m ; a t the first house, w hich was the squire’s, she left half the num ber of eggs she had and half a one o v e r; a t the second she left half of w hat rem ained and half an egg o v e r; and a t the th ird she again left half of the rem ainder, and half a one over ; she retu rn ed w ith one egg in h er basket, n o t having broken any. Required—th e num ber she set out with. A iis. 15 eggs.

Suppose the num ber of points of the first die which comes up to be 2, and th a t of th e other 3 ; then, if to four, th e double of th e points of th e first, th ere be added 5, and th e sum produced, 9, be m ultiplied by 5, the product will be 40; to which, if 3 ,'the num ber of points on the oth er die, be added, 48 will be produced, from which, if 25 be subtracted, 23 will re m a in ; the first figure of w hich is 2, th e num ber of points on th e first die, and th e second figure 3, th e pum ber on th e other.

The figures, up"to 100, arranged so The Sovereign and the Sage. as to make 505 in each column, when sovereign being desirous to confer a counted in ten columns perpendicularly liberal rew ard on one of his courtiers and the same when counted in ten who had perform ed some very im p o rtan t files horizontally. service, desired him. to ask w hatever he th ought proper, assuring him it should be granted. The courtier, who w as well tr I acquainted w ith th e science of num bers, only requested th a t th e m onarch would h-I 50 cp —T ca cn Oto K) O to to o O to to to o give h im a q u an tity of w heat equal to th a t which would arise from one grain -n 05 ** CTl rra hO t-l o05 05 05 05 05 U0 U0 00 w dsubled sixty-threo tim es successively. fcb The value of th e rew ard w as im m ense ; for it will be found by calculation th a t the sixty-fourth te rm of th e double progres­ sion divided by 1, 2, 4, 8, 10, 32, &c., is 9223372030854775808. B u t th e sum of all the term s of a double progression, beginning w ith 1, m ay bo obtained by doubling th e last term , and subtracting from it 1. The num ber of the grains of CO "“'I 00 ^ Cl O to M CD CO 05 05 00 CO 03 CP 05 03 CD w heat, therefore, in th e presen t ease, will bo 1844G744073709551015. Now, if a p in t contain 9210 grains of w heat, a gallon will contain 73728 ; and, as eight gallons m ake one bushel, if we divide the above resu lt by eight tim es 73728 we shall have 31274997411295 for th e num ber of The dice guessed unseen. th e bushels of w heat equal to th e above num ber of grains, a q u an tity g reater th an pair of dice being throw n, to find the the whole surface of the e arth could num ber of points on each die w ith ­ produce in several years, and w hich in out seeing them . T ell th e person who value would exceed all th e riches, perhaps casts th e die to double the num ber of on th e globe. points upon one of them , and add 5 to it th en to m ultiply th e sum produced by 5, December and May. and to add to th e product th e num ber of points upon th e other die. This being N old m an m arried a young w o m an ; done, desire him to tell you th e am ount, th eir united ages am ounted to O. and, having throw n out 25, th e rem ainder The m an ’s age m ultiplied by 4 and divi­ will be a num ber consisting of tw o figures ded by 9, gives th e w om an’s age. W hat th e first of which, to th e left, is th e n u m ­ w ere th eir respective ages ? b er of points on th e first die, and the A nsweb .— The m a n ’s age, 00 years 12 second figure, to the right, the num ber on w eek s; th e w om an’s age, 30 years 40 the other. T h u s : weeks.

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COMIC AND CURIOUS PROBLEMS IN ARITHMETIC.

The Mathematical Fortune Teller. I t is required to tell th e num ber thought RO C U RE six cards, and having ruled them th e sam e as the following dia­ of by any person, th e num bers being co n ­ gram s, w rite in th e figures neatly and tained in th e cards, and such num bers no t I to exceed GO. H ow is th is done ? legibly.

P

3

5

7

9

11

1

5

0

7

13

12

4

13

15

17

19

21

23

14

15

20

21

22

23

23

27

29

31

33

35

28

29

30

31

30

37

37

39

41

43

45

47

52

38

39

44

45

40

49

51

53

55

57

59

47

53

54

55

CO

13

9

10

11

12

13

8

3

G

7

10

11

2

14

15

24

25

20

27

14

15

18

19

22

23

28

29

30

.31

40

‘41

20

27

30

31

34

35

42

43

44

45

40

47

38

39

42

43

40

47

50

57

58

59

GO

13

50

51

54

55

58

59

17

18

19

20

21

1G

33

34

35

3G

37

32

22

23

24

25

2G

27

38

39

40

41

42

43

28

29

30

31

48

49

44

45

4G

47

48

49

50

51

52

53

54

55

50

51

52

53

54

55

5G

57

58

59

30

GO

50

57

58

59

GO

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R equest th e person to give you th e cards containing th e num ber, and th e n add th e rig h t hand upper corner figures together, w hich will give th e correct answ er. F o r exam ple: suppose 10 is th e

num ber th ought of, th e cards w ith 2 and 8 in th e corners will be given, w hich m akes th e answ er 10, and so on w ith the others.

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' t h e m a g ic o f n d m b e e s ; o r ,

Now, if another person and yourself I have fifty counters a-piece, and agree never to stake m ore th a n te n a t a tim e, ■ you m ay tell him th a t if he p erm it you to stake first, you alw ays com plete the even century before him . I n order to succeed, you m u st first stake 1, and rem em bering th e order of the above series, constantly add to w hat he stakes as m any as will m ake one move h e? H e had 7 sheep : as m any m ore 7 ; half th a n th e num bers 11, 22, 33, &o., of as m any m ore, 3J ; and 2 i ; m aking in all w hich it is composed, till you come to 89, after which your opponent cannot possibly 20. reach tho even century him self, o r p re ­ The certain game. vent you from reaching it. If your opponent has no know ledge of WO persons agree to take, altern ately num bers less th an a given num ber, num bers, you m ay stake any o th er n u m ­ for example, 11, and to add th em to g eth erber first, under 10, provided you subse­ till one of them has reached a certain sum quently take care to secure ooie of th e last such as 100. B y w hat m eans can one of term s, 5G, 07, 78, &c.; or you m ay even th em infalliably a ttain to th a t num ber le t him stake first, if you take care a fter­ w ards to secure one of these num bers. before the other ? This exercise m ay be perform ed w ith The whole artifice in this consists in im m ediately m aking choice of th e n u m ­ o ther num bers, b u t, in order to succeed, bers, 1, 12, 23, 34, and so on, or of a you m ust divide th e num ber to be attain ed series w hich continually increases by 11, by a num ber w hich is a u n it g reater th a n up to; ‘100. L e t us suppose th a t tho first w hat you can stake each tim e, and tho person,' who kuows the game, m akes rem ainder will th e n be th e num ber you choice of 1 ; it is evident th a t his adver­ m u st first stake. Suppose, for example sary, as he m ust count less th a n 11, can th e num ber to be attain ed be 52 (malting a t m ost reach-11, by adding 10 to it. The use of a pack of cards instead of counters), first will th en take 1, w hich will m ake 12 and th a t you are never to add m ore th a n and w hatever num ber th e second m ay add 0 ; th e n , dividing 52 by 7, th e rem ainder, th e first will certainly win, provided he w hich is 3, will be th e num ber w hich you continually add th e num ber w hich forms m u st first s ta k e ; and w hatever your tho com plem ent of th a t of his adversary opponent stakes, you m u st add as m uch to 11; th a t is to say, if the la tte r tak e 8, to it as will m ake it equal to 7, th e n u m ­ lie m ust take 3 ; if 9 he m ust take 2 ; and ber by w hich you divided, and so in con­ so on. B y following th is m ethod he will tinuation. infalliably a tta in to 89, and it will th e n The unlucky hatter. be im possible for the second to prevent PE R SO N w ent into a h a tte r’s shop him from getting first to 100 ; for w h at­ and bought a h a t for a sovereign and ever num ber the second takes he can a ttain only to 99; after w hich th e first gave in p aym ent a five-pound note. The m ay say— “ and 1 m ake 100.” Betw een h a tte r called on a friend n ear by, who tw o persons who are equally acquainted changed th e note for him , and th e m an w ith the. game, he who begins m u st n e­ having received his change w ent his way. Shortly aiterw ards th o ta ilo r’s friend dis­ cessarily win. covered th e n ote to be a counterfeit, and The magical century. called upon th e h a tte r, who was com pel­ F th e num ber 11 be m ultiplied by any led forthw ith to borrow five-pounds of one of th e nine digits, th e tw o figures an o th er friend to redeem it w ith ; so th e of th e product will always be alike, as forged note was left on th e h a tte r’s appears in the following example ;— hands. The question is, w hat did he lose 11 11 11 11 11 11 11 11 11 —was it five-pounds beside th e h a t or 1 2 3 4 5 G 7 8 9 was it five-pounds including th e h a t ? This question is often given w ith nam es 11 22 33 44 55 00 77 88 99 and circum stances as a real transaction, tjndrif th e com pany knows suchpersonsso

The knowing Shepherd. S H E P H E R D was going to m arket w ith some -sheep, w hen he m et a m an who said to him , “ Good m orning, friend, w ith your score.” “ N o,” said tho shepherd. “ I have n o t a score ; b u t if I had as m any m ore, half as m any m ore, and tw o sheep and a half, I should have . ju st a score.” H ow m any sheep had

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COMIC AND CURIOUS PROBLEMS IN ARITHMETIC.

11

Place th ree sixes to g eth er, so as to m uch th e better, as it serves to w ithdraw atten tio n from th e question ; and in al­ m ake seven. A n s.—Cg. m ost every ease th e first im pressio n is,th at th e h a tte r lost five-pounds besides the Add one to nine and m ake it tw enty. hat, though it is evident he was paid for A n s.—IX —cross th e I , it m akes XX. th e hat, and had he kep t the sovereign he Place four fives so as to m ake six and a needed only to have borrowed four-pound half. A ns. 5J'5 additional to redeem the note. A room w ith eight corners h ad a c a t in each corner, seven cats before each cat, and a eat on every c a t’s tail. W h a t was The basket of nuts. th e to tal nninber of cats ? A n s. E ig h t PER SO N rem arked th a t when he cats. counted over his basket of nuts, two Prove th a t seven is th e half of tw elve. by two, three by three, four by four, five A n s .—Place th e H om an figures on ap iece by five, or six by six, th ere w as one r e ­ of paper, and draw a line th ro u g h th e m aining ; b u t w hen he counted th em by middle of it, th e u p per will be YI1. sevens, th ere was no rem ainder. How m any had lie ? The least comm on m ultiple of 2, 3, 4, 5, The council of ten. and C being GO, it is evident, th a t if Cl were divisible by 7, it would answ er the E N cards or counters, num bered from ••conditions of the questions. T his not one to ten, or tho first te n playing being the case,however let G 0X 2+1, G0x3 cards of any su it disposed in a circular + 1 , 0 0 x 4 + 1 , &c., be tried successively, form m ay be em ployed w ith g reat con­ and it will be found th a t 301 = 0 0 + 5 + 1 , is venience for perform ing th is feat. Tho divisible by 7 ;an d consequently th is n u m ­ accom panying figure shows th e cards ber answ ers th e conditions of the question. thus arranged, n um ber one, or th e ace, If to this we add 420, th e least common designated by A. and th e ten by K. m ultiple of 2,3, 4, 5, Gand 7, th e sum 721 t) wHl be another a n sw e r; and by adding 0 perpetually 420, we m ay find as m any 2B D4 answ ers as we please. 1A 10 5 10 K FG The united digits. 91 G7 II RRA N G E th e figures 1 to 9 in such 8 order th a t, by adding them together H aving placed the cards in th e above th ey am ount to 100. order, desi e a bystander to th in k of a card 15 or num ber, and when he has done so, to touch any other card or num ber. R equest him then to add to tho num ber of th e card touched the num ber of th e cards "'employed, w hich in th is ease is ten. T hen desire him to count th e sum in an o rd er co n trary to th a t of th e n atu ral num bers, 100 beginning a t tho card lie - touched, and assigning it th e num ber of Quaint Questions. the card he th o u g h t of. B y co u n t­ H A T is the difference betw een tw enty ing in this m anner, he w ill end a t four q u art bottles, and four and th e numb&r or card ho th o u g h t of, and tw enty q u art bottles ? consequently you will im m ediately know A n s.— 50 quarts difference. it. AVhat three figures, m ultiplied by 4, T hus, for exam ple, suppose tho person will m ake precisely 5 ? had th ought of 3 C, and touched G F ; A n s.— 1£, or l -25. th en , if 10 be added to G, th e sum will ba W hat.j is th e difference betw een six 1G; and if th a t num ber be counted from dozen dozen, and half-a-dozen dozen ? F , th e n u m b er touched, tow ards E D 15 A n s.— 792: Six dozen dozen being 804 C A, and so o n , in th e retrograde order, and half-a-dozen dozens, 72. counting F th ree, th e num ber thought

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THE MAGIC OF NUMBERS; OR,

of, E five, D six, and so round to six­ teen, th a t num ber will term inate a t 0 , showing th a t the person thought of 3, the num ber w hich corresponds to C. A greater or less num ber of cards or counters m ay be employed at pleasure ;

b u t in every instance th e whole num ber of cards m u st bo added to th e num ber of th e card touched. This trick done on th e dial of a w atch, using th e figures thereon, is even m ore surprising.

The two Travellers. Two travellers trudged along th e road together, Talking, as travellers do, about th e w e a th e r; W hen, lo 1 beside th e ir p a th th e forem ost spies I Three casks, and loud exclaims “ A prize, a prize I ” One large, tw o sm all, b u t all of various size. This way and th a t th ey gazed, and all around. E ach wondering if an ow ner m ight be found : B ut not a soul was th ere—th e coast was clear, So to th e barrels th ey a t once drew near, A nd both agree w hatever m ay be there I n friendly partn ersh ip th e y ’ll fairly share. Two they find em pty, b u t th e oth er full, And straightw ay from his pocket one doth pull A large clasp knife. A heavy stone lay handy, And th u s in tim e th ey found th e ir prize was brandy. ’Tis tasted and ap p ro v ed ; th e ir lips th ey smack, And each pronounces ’tis th e fam ed Cognac. “ W on’t we have m any a jolly night, m y boyl May no ill luck our presen t hopes d e stro y ! ” ’Twas fortunate one knew th e m athem atics, And had a sm atterin g of h y d ro sta tic s; T hen m easured ho th e casks, and said, “ I sea This is eight gallons, those are five and three. ” The question th e n was how they m ight divide The brandy, so th a t each should be supplied W itli ju s t four gallons, n either less nor m ore. W ith eight, and five, and three th e y puzzle sore, Filled up tlie five—filled up th e three, in v ain; A t length a happy th o u g h t came o ’er th e brain Of one ; ’tw as done, and each w ent liome content, And th eir good dames declared t ’was excellent. W ith those three casks they m ade division tru e ; I found th e puzzle out, say, friend, can you ? The five-gallon barrel was filled first, and from th a t th e three-gallon b arrel, thus leaving tw o gallons in the five-gallon b a rr e l; th e three-gallon barrel was th e n em ptied into th e eight-gallon barrel, and th e two gallons poured from th e five-gal­ lon barrel into th e em pty three-gallon barrel ; the five-gallon barrel was th e n filled,?and one gallon poured into the three-gallon barrel, therefore leaving four gallons in the five-gallon barrel, one gallon in th e eight-gallon barrel, and th ree gallons in the three-gallon barrel, w hich w as th e n em ptied into th e eightgallon barrel. Thus each person had

four gallons of brandy in th e eight and five-gallon barrels respectively. The Fox, Goose and Corn. countrym an having a Fox, a Goose, and a peek of Corn, cam e to a river, A w here it so happened th a t he could carry b u t one over a t a tim e. Now as no two w ere to be left to g eth er th a t m ight des­ tro y each other, he was a t his w it’s end, for says he “ Though th e corn can’t eat th e goose, n o r th e goose e at th e fo x ; y et th e fox can eat th e goose, and th e goose eat

oowtc a n d cuiiiotra p r o b l e m s

in a e i t h m e t i c .

10

This peculiar series of num bers is thus form ed : W rite down th e num bers 1, 2, 3, &c., as far as you please, in a vertical row. On th e rig h t hand of 2 place 1, add th em together, and place 3 u n d er the 1 ; th en 3 added to 3 = 6, w hich place under th e 3 ; 4 and 6 arc 10, w hich place Tinder the G, and so on as far. as you w ish. This is th e second vertical row, and tho th ird The visitors to the Crystal Palace. is form ed from th e second in a sim ilar N a fam ily consisting of 8 young people, way. This triangle has th e p ro p erty of it was agreed th a t 3 a t a tim e should inform ing us, w ith o u t th e trouble of cal­ visit th e C rystal Palace, and th a t the culation, how m any com binations can be v isit should be repeated each day as long m ade, tak in g any n u m b er a t a tim e out as a different trio could be selected. In of a larger num ber. liow m any days were the possible com ­ Suppose th e question w ere th a t ju st binations of 3 out of 8 com pleted ? given ; how m any selections can be m ade Wo m ust m ultiply 8 X 7 X 0 , and also of 3 at a tim e out of 8 ? On th e h o ri­ 3 x 2 x 2 , and divide th e product of the zontal row com m encing w ith 8,- look for former, 330, by th e product of th e latter, tho th ird num ber ; this is 06, w hich is 6 ; tho result is 5G, the num ber of visits, th e answ er. a different three going each tim e. So m uch gratified were they w ith th e results How many different deals can Da of th e ir agreem ent, th a t they w ished to be allowed another series of visits, to be made with 13 cards out of 52. continued as m any daj's as th ey could M10 discover this we m ust m ake a congroup 3 together in different order when JL tinued m ultiplication of 52 X 51 X 50 starting. If Paterfam ilias had granted X 4 9 x 4 8 X47 X 4GX 4 5 x 4 4 X 4 3 X 4 2 X 41 such perm ission he would have had to X40, being 13 term s for th e 13 cards, w ait 50 m ultiplied by 3 x 2 x 1 , ’r 330 also a continued m ultiplication of 13-f-12 days, before th is “ now series” of visits X11X10X9X 8 X7 X 6 x 5 x 4 x 3 x 2 x l , would have come to &fin is. and, having found th e tw o products, we m ust divide one by th e other, :i jits pi? I'ls

Tho 8-pint can full, and th e ae a i ea — ia ei ie others em pty - 8 0 0 Brave, dashing sea, like a g ian t revives 1. F illed th e ii-pint can - 3 5 0 itself. 2. Filled th e 3-pint can from the 5-pint - 3 2 3 The United Digits. j 3. P o u r tho contents of 3-pint N page eleven we showed how to w ith th e 8-pint - 6 2 0 place th e figures, 1 to 9, so th a t th e y m ight by adding them to g eth er4. Transfer th e 2-pints from th e 5-pints to th e 3-pint - 6 0 2 am o u nt to 100. U ntil now i t has been ! believed th a t th ere was only one way to ■ 5. Fill th e 5-pint from tho 8-pint - 1 5 2 do this w ithout using fractions. W e give 0. F ill up th e 3-pint from tho a n o th e r: 5-pint - 1 4 3 32 7. P o u r th e 3-pints into th e 57 8-pint, com pleting - 4 4 0 This was a dexterous expedient feat of th e w orthy sapper, th o only objections to i t being th e tim e tho th irty men had to wait, and th e resu ltin g flat condition of th e beer. The Difficult Case of Wine. 100. GENTLEM AN h ad a b o ttle contain­ To Tell what Figure a Pei’son ing 12 p in ts of wine, 6 of which he has struck out of the Sum of was desirous of giving to a friend, b u t lie C Two given Numbers. h ad n othing to m easure it w ith, except SSUME those num bers only th a t arc 2 o th er bottles, one of 7 pints and th e divisible by 9 ; such, for instance, oth er of 5. Plow did ho contrive to p u t as 18, 36, 63, 81, 117, 126, 162, 261, 315, 6 p in ts in to th o 7-pint b ottle ?

O

A

A

1ft

THE MAGIC OS’ iroMBEES ; OR) 12-pt. 7-pt. 5-pt.

Before lie comm enced, th e con­ te n ts of the bottles were - 12 0 1. H e filled the 5-pint - 7 0 2. E m ptied the [5-pint into the 7-pint - 75 \ Filled again the 5-pint from the 12-pint - 2 5 4. Filled up the 7-pint from the 5 - 27 5. E m ptied the 7-pint into the 12-pint - 90 0. Poured the 3 pints from the 5 into the 7 - 9 3 7. Filled the 5-pint from the 12-pint - 43 8. Filled up the 7-pint from th e 5-pint - 47 9. E m ptied th e 7-pint into the 12-pint - 11 0 10. Poured 1 p in t from the 5pint into the 7-pint - 11 1 11. Filled th e 5-pint from the 12-pint - 01 12. Poured the contents of the 5-pint into the 7-pint - 6 6

0 5

1 .2 2 .2 3 .3

H alf-full E m pty.

3 3 1

2 2 3

1 .2 2 . 3 3 . 4 1 .1 2 4 3 '. 4

3 3 0 5 1 1 0 5 0

5 3 1

2 3 4

Table F u ll H alf-full Em pi

5

CERTAIN hotel-keeper was dexterous in contrivances to produce a large' appearance w ith sm all m eans. In th e dining-room were three tables, betw een which he could divide 21 bottles, of which 7 only were full, 7 half full, and 7 apparently just em ptied, and in such a m anner th a t each table had tho sam e num ber of bottles, and the sam e quantity of wine. H e did this in two ways. T ab le F u ll

Tablo F u ll H alf-fu ll E m pty.

0

The wine and the tables.

A

Also w ith 27 bottles, 9 full, 9 hali* ful, and 9 e m p ty :

7 1 1

1 4 4 i

The three Travellers.

1

THREE m en m et a t a caravansary or . inn, in P e rs ia ;. and tw o of them brought th eir provisions along w it'i them according to the custom of the co u itr y ; but th e th ird n o t having provided any, proposed to th e others th a t th ey should e at together, and he would pay th e value of his proportion. This being agreed to, A produced 5 loaves, and B 3 loaves, all of which th e travellers ate together, and C paid 8 pieces of m oney as th e value of his share, w ith w hich th e others were satisfied, b u t quarrelled about th e divi­ sion ot it. U pon th is th e m a tte r was re ­ ferred to the judge, who decided im p a r­ tia lly .'"'W h at was his decision ? A t first sight it would seem th a t the money should be divided according to the bread furnished ; b u t we m u st con­ sider th a t, as th e 3 ate 8 loaves, each one ate 23 loaves of th e bread he furnished. T his from 5 would leav e 2J loaves fur­ nished th e stran g er by A ; and 3—2§ = | furnished by B, hence 2 | to | = 7 to 1, is th e ratio n in which th e m oney is to be di­ vided. If you im agine A and B to fu r­ nish, and C to consum e all, th en tho di­ vision will be according to am ounts fu r­ nished.

T ab le F u ll H alf-full E m p ty .

1 .3 1 3 Which counter has been thought of 2 .3 1 3 out of sixteen. 3 .1 5 1 A KE sixteen pieces of card, and n u m ­ H o also perform ed a sim ilar exploit ber th em 1 to 16. Arrange them in w ith 24 bottles, 8 full, 8 half-full, and 8 tw o row s, as a t A B. em pty. A B C B D M E B F N G B H T ab le F u ll H alf-full E m p ty. 1 9 19 2 2 2 9 4 2 2 9 6 1 .3 2 3 2 10 3 1 0 4 4 610 8 6 110 5 2 .3 2 2 3 11 511 6 G 111 3 1 411 8 3 .2 4 3 4 12 712 8 8 512 7 5 312 7 Table F u ll H alf-full E m pty. 5 13 13 1 13 4 13 1 . 2 4 2 6 14 14 3 14 8 14 2 .2 4 2 7 15 15 5 15 3 15 I . 4 0 4 8 16 16 7 16 7 16

T

C„41ic ANt> CtjiUOttS fr.O flL E llS IN A filTH M EflC .

17

q uotient by itself, th e la s t figure of each q u o tien t w ill alw ays be 5. T hus 5 x 5 = 25; 2 5 X 2 5 = 1 2 5 ; 1 2 5x125=025, &c. Again, if you proceed in th e sam e m a n ­ lier w ith th e figure 0, th e la st figure will constantly be 0 ; th u s, 0 x 0 = 3 0 ; 3 0 x 3 0 = 210 ; 216x210=1,290, and so on. To "multiply by 2 is th e sam e as to m ultiply by 10 and divide by 5. Any num ber of figures you m ay w ish to m ultiply by 5, will give th e sam e resu lt if divided by 2—a m uch quicker operation th an th e form er ; b u t you m u st rem em ber to annex a cipher to th e answ er where there is no rem ainder, and w here th ere is a rem ainder, annex a 5 to the answ er. Thus, m ultiply 404 by 5, th e answ er will be 2320; divide th e sam e num ber by 2, and you have 232, and as th ere is no re ­ m ainder you add a cipher. Now, tak e 357 and m ultiply by 5—th e answ er is 1785. On dividing 357 by 2, th ere is 178 and a rem ainder ; you therefore place 5 a t th e rig h t of th e line, and th e resu lt is again 1785. T here is som ething m ore curious in th e properties of th e num ber 9. Any n u m ­ b er m ultiplied by 9 produces a sum of figures which, added together, continually m akes 9. F o r example all tho first m u l­ tiples of 9, as 18, 27, 30, 45, 54, 03, 72, 81 sum up 9 each. E ach of them m ultiplcd by any num ber w hatever produces a si­ m ilar re s u lt; as 8 tim es 81 aro 048, these added to g eth er m ake 18, 1 and 8 are 9. M ultiply 648 by itself, th e product is 419, 904—th e sum of those digits is 27, 2 and 7 Curious Properties of some figures. are 9. Tho rule is invariable. Take any E L E C T any tw o num bers you please, num ber w hatever and m ultiply it by 9 ; and you will find th a t one of th e two, or any m ultiple of 9, and th e sum will th eir am ount w hen added together, orconsist of figures w hich, added to g eth er th e ir difference, is alw ays th ree, or a continually num ber 9. As 1 7 X 18= 300, num ber divisible by 3. 0 and 3 are 9 ; 117 X 27=3,159, th e figures T hus, if th e num bers are 3 and 8, th e sum up to 18, 8 and 1 are 9 ; 4591X 7 2 = first num ber is 3 ; let th e num bers be 1 330,552, th e figures sum up ta 18, 8 and 1 and 2, th eir sum is 3 ; le t th em be 4 and are 9. Again, 87,803x54=4,717,422 ; 7, the difference is 3. Again 15 and 22, added together, th e product is 27, or 2 and th e first num ber is divisible by 3 ; 17 and 7 are 9, and so always. I f any row of 20, th e ir difference is divisible by 3, &c. tw o or m ore figures be reversed and sub­ All the odd num bers above 3, th a t can tra c te d from itself, th e figures composing only bo divided by 1, can be divided by th e rem ainder, will, w hen added horizon­ 0, by th e addition or subtractio n of a tally, be a m ultiple of n in e : un it. F o r instance, 13 can only be divi­ 42 880 320 ded by 1; but after deducting 1, th e re ­ 24 088 1623 m ainder can be divided by 0 ; for exam ple 5 + 1 = 6 ; 7—1 = 6 ; 1 7 + 1 = 1 8 ; 19—1 19—8 x 2 . 198—9 x 2 . 1038—9 x 2 = 1 8 ; 25— 1= 24, and so on. If a m ultiplicand be form ed of the dig­ If you m ultiply 5 by itself, and th e its in th eir regular order, o m itting the 8, quotient again by itself, and th e second ■ a m ultiplier m ay be found by a rule, which

D esire th e person to th in k of one of the um bers, and to tell you in w hich row it is. Suppose he fixes on C ; he will tell you th a t th e row A contains the n u m b er lie thought of. ‘"ake up th e row A, and arrange the num bers on each side of the row B, as shown at 0 P , so th a t the first num ber of the row A m ay be th e first of th e row C, th e second of A be th e first of D. the th ird of A be th e second of C, and 60 on. Ask in w hich of th e row s, C or D, is th e num ber th o u g h t o f; in th e case supposed it is in D. Take up th e rows C D, anrJ p u t one u n derneath .the other as a t M, taking care th a t the half-row in w hich is th e num ber thought of, shall be above the other. Divide it again into tw o row s, as a t E F , on each side of B, in th e sam e way as before. Ask again in w hich row it is; it is now in E . Place one row under th e other, as at N, and divide again into tw o rows, w hich w hich will now be as G H , You will be inform ed th a t th e num ber is in row H , and you m ay th en announce it to be th e top num ber of th a t row.The num ber thought of will alw ays be a t the top o f one o f the rows a fte r three transpositions. If there were 32 coun­ ters it would be a t th e top after four transpositions.

S

THE MAGIC OF NUMBERS ; Oh,

18

will give a product, cack figure of w hich shall bo tho same. Thus if 12345079 be given, and it be required to find a m u lti­ plier which shall give the product all in 2, th a t m ultiplier will bo 18; if in 8, the m ultiplier will be 27 ; if all 4, it will be 30—and so forth. 12345G79 12315079 12315079 18 27 3G 98705432 12345079

80419753 24091358

74071074 3703(037

222222222 333333333 414444444 The rule by which the m ultiplier is dis­ covered, (but w hich we do not a ttem p t to explain) is t h i s : M ultiply the last figure (the 9) of the m ultiplicand by th e figure of which you wish the product to be com ­ posed, and th a t num ber will bo tho re ­ quired m ultiplier. T hus, when it was required to have tho product composed of 2, the 2, m ultipied by 9 gives 18, the m u l­ tiplier ; 3 m ultiplied by 9 gives 27, the m u ltiplier gives the product in 3 ; &c. If a figure w ith a num ber of ciphers a t ­ tached to it, bo divided by 9, th e quotient will be composed of ono figure only, nam ely, the first figure of tho dividend, as— 9)000,000 0)40,000 GG,GG0G—0 4,444—4 If any sum of figures can be f 9)549 divided by 9 as, 1 ^ tho am ount of these figures, w hen added together, can be divided by 9 :—th u s, 5, 4 , ‘9, added together, m ake 18, which is divisible by 9. If the sum 549 is m u lti­ plied by any figure, the product can also be divided by 9, as— 2 And the am ountof th e figures of tho product can also bo divided by 9; thus,

9

4 2)18

\

9 To m ultiply by 9, add a cipher, and dcduct the sum th a t is to bo m ultiplied : thus, 43,200) f 4,820 4 ,3 2 0 1Produces the sam e resu lt J 9 38,934;

1 38,934 In the sam e m anner, to m ultiply by 99

add tw o ciphers ; by 999, threo ciphers,&c. These properties cf th e figure 9 will enable th e young arith m etician to perform an am using trick, q u ite sufficient to excite th e w onder of th e u ninitated. Any series of num bers th a t can be divided by 9, as 805,4.72,821,754, &c., being shown, a person m ay be requested to m ultiply secretly eith er of these series by any figure he pleases, to strike out one num ber of tho quotient, ifad to le t you know th e figures w hich rem ain, in any order he lilces ; you will then, by th e assistance of th e knowledge of th e above properties of 9, easily declare / 2 th e num ber w hich has been / 1 805472 erased. Thus, suppose 805,472 I 9 0 are th e num bers clioson, and / 2 th e m ultip lier is six ; if then, 1 \ 3 219232 is struck out, th e num ber r e ­ tu rn ed to you will bo '19 Tho am ount of these num bers is 19; b u t 19, divided by 9, loaves a rem ainder of 1 ; you, therefore, w ant 1 to com plete another 9 : 8, then, is the num ber erased. The com ponent figures of th e product made by tho m ultiplication of every digit in to th e num ber 9, w hen added together, m ake N i n e . Tho order of theso com ponent figures is reversed after th e said num ber has been m ultiplied by 5. The com ponent figures of th e am ount of th e m ultipliers (viz. 45,) w hen added together, m ake N i n e . The am ount by the several products, or m ultiplies of 9, gives for a quotient, 4 5 ; th a t is, 4 + 5 = N i n e . The am ount of tho first product (viz. 9) w hen added to th e o th er product, whose respective com ponent figures m ake 9, is, 81 ; which is th e square of N i n e . The said num ber 81, when added to the above m entioned am ount of th e several products, or m ultiples of 9 (viz. 405) make 48G, which, if divided by 9, gives for a quotient 54 : th a t is, 5 - |- 4 = N in e . I t is also observable, th a t th e num ber of changes th a t m ay bo rung on nine bells is 362,880; w hich figures, added together m ake 2 7 ; th a t is, 2 + 7 = N in e . And th e quotient of 302,880, divided by 9, wiU be 4 0 ,3 2 0 ; th a t is 4 + 0 + 3 + 2 + 0 = N in e .

If th e num ber 37 be m ultiplied by any of tho progressive num bers arising from the m ultiplication of 3 'w ith any of the

1-9

COMIC AND CURIOUS MOBLEMS IN AIUTIIMKTK,.

Units, the figures in th e quotient will be sim ilar, and the resu lt m ay be know n beforehand by m erely inspecting tho p ro ­ gressive num bers, thus, 8,0,9,12,15,18,21, 24,27, &c., are th e progressive num bers form ed by 3 m ultiplied by th e u n its 1 to 9 ; and the result of th e m ultiplication of any of these num bers w ith 37 m ay bo seen infollow ingexam ples :—87X 3 = 111; U7X 0 = 2 2 2 X 3 7 X 1 2 = 4 4 4 ; 8 7 x 2 4 = 8 8 8 ; by whieli in appears th a t th e num bers of w hich th e quotient is formed are th e sam e as th e units, by w hich num ber 8 was m u lti­ plied to obtain th e respective progressive num bers. T hus—3 m ultiplied by 2 is equal to G, and 37 m ultiplied by 9 is equal to 222 ; so, again, 4 m ultiplied by 3 produces 12, and 87 m ultiplied by 12 is equal to 444, and so on.

8

3

oO ------- ---8

3

ii 8

------3

So th a t there were nine boys on each side of tho establishm ent, while tho w orthy doctor occupied th e centre cham ­ ber him self. Strange to say, however, despite tho old g entlem an's eagle eye, it was suspectcd th e boys were in th e h a b it of slipping out after dark four a t a tim e. One night th e doctor resolved to find out all about it. So, after he im agined tho four h ad departed, he m ade a round of The industrious Frog. th e room s. Much to his astonishm ent, H E R E was a well 30 feet deep, and, at however, th ere were still nine on each side th e bottom , a,irog anxious to get out. in th is way :— H e got up 3 feet per day, b u t regularly fell back 2 feet a t night. R equired the num ber of days necessary to enable him 4 4 1 1 to g et out ? The frog appears to have cleared one foot per day, and a t th e end of 27 days, he 1 1 would be 27 feet up, or w ithin 3 feet of tho top, and the n ex t day he would g et out. H o would therefore be 28 days getting 4 4 1 out.

T

The Mathematical Blacksmith; B LA C K SM ITH h ad a stone weighing 40 lbs. A m ason com ing in to th e shop, ham m er in hand, struck it and broke it into four pieces. “ T h ere,” says th e sm ith, “ you have ruined m y w eight.” “ N o,” says th e m ason, “ I have m ade it b e tter, for w hereas you could before w eigh b u t 40 lbs. w ith it, now you can weigh every pound from 1 to 40.” R e­ quired—size of the pieces ? A?is. 1, 3 ,0 ,27 ; for in any geom etrical scries proceeding in a trip le ratio n , each te rm is 1 m ore th a n tw ice th e sum of all th e preceding, and th e above series m ight proceed to any extent. In using th e w eights, they m ust be p u t in one or both scales, as m ay bo necessary ; as for ex­ am ple, to weigh 2, p u t 1 in one scale, and 3 in the other.

A

O

The four boys who w ont out brought back four eliums from a neighbouring academ y who, against all rules, rem ained for th e night. N evertheless, th e doctor on his n ext round could only find nine boys to a side. This is how th o young rascals m ar aged i t :—

By and by four m ore chum s a rm e d , and stole up to th e dorm itories on tiptoe ; The Doctor and his Pupils. b u t when th e w atchful dom inie again paid LD D r. B razenose took a school, and tho corridors a visit, th ere rscrc still but had four and tw enty boys as boarders. nine to a side (though eight extra boya These he placed in dorm itories th u s :— were present) th u s :—

so

THE MAGIC O f NUMBERS ;

F o u r m ore lads, m issing th e ir com ­ panions, now w ent to th e opposite academ y in search of tlicm , and they, too, entered unperceived. Twelve strange youths were now the guests of D r. B razcnose's four and tw enty pupils, yet when the doctor, hearing an unw onted talking and laughing, stole out to discover th e cause of tho strange voices, tho n u m ­ ber to a side was still tho same, in th is fashion:— 9

The Fanner’s Sons. W ELL-TO -D O farm er died, and left his property to his three sons in shares. The eldest son was to have one half as his share, th e second a th ird , and tho th ird a nin th . All w ent well till th ey began to divide^ th e live-stock, when it w as found th ere were seventeen cows upon the farm , and over these th e brothers began to squabble. I t would hardly do to slaughter tho anim als for they were of an exceedingly fine breed ; still th e y could n o t see how else the division w as to be accom plished. In th eir quandary th ey applied to on old friend of th eir father. This geutlem an th ought for a tim e, and th en an idea struck him . “ Take one of m y cow s,” said ho, “ it originally belonged to your own herd, and I fancy we can m anage it.” The cow was brought over to tho farm and now as th ere were eighteen, th e division was accom plished. The eldest got 9 cows The second C The th ird 2

A

9

9

OR,

eighty, th e fifth a hunded, th e sixth a hundred and tw enty, th e seventh a h u n ­ dred and fort}7. All sold th eir apples at the sam e rate, and w hen th eir stocks were disposed of, every one had taken exactly th e sam e sum . "What were the ra te at w hich tho apples were sold ? Answer.— Seven a penny and th re e ­ pence ap iece for all th a t were over.. 2 0 = tw o p e n n y w o rth + 6 threepences =20d. j 4 0 = five p e n n y w o rth + 5 threepences =20d. 0 0 = eig lit pcnnyw ortli-|-4 threepences =20d. S 0= elevcn pennyw orth-|-3 threepences = 20d. 100 = fourteen pennyw orth -f 2 threepen ces= 2 0 d . 120= soventeen pennyw orth-)-1 th re e ­ pence = 2 0 d . 1 4 0 = tw e n ty p en n y w o rtli= 2 0 d .

9

N ext niglitD r. Brazcnoso’s boys slipped out in a body of six, leaving th e ir feilowpupils behind disposed in th e ir room s th u s :—

B u tth a tn ig h t th e doctor discovered the secret, and henceforward th e nocturnal games were p u t a stop to. I t will be seen th a t’th e old boy's m istake lay in counting each com er room twice.

17 So, you see, th eir old friend got back his cow after all.

The Seven Apple Women.

S

E V E N women sat a t th eir apple-stalls. The first had tw enty apples, the second forty, th e th ird sixty, th e fourth

The Shepherds.

T

WO shepherds wore feeding th eir flocks on th e m ountain-side. Said

003IIC AXD CURIOUS PROBLEMS I.V ARITHMETIC.

one to th e other, “ Jack, give m e one of your sheep, and I shall have as m auy as you.” “ Nay, “ replied the other greedily,

21

“ Give m e one of yours and I shall have as m a n y again as you, H ow m any sheep had each ? The first had five,the second had seven

The Ten Tens. AKE ten pieces of card, and upon each on each card, you can determ ine its n u m ­ w rite any ten words ; there is no r e ­ ber. striction as to the initial le tte r of nine of H ere are te n cards, (call th ese th e th e words, but the la s t w ord on each card Selecting Cards), w hich we give by way m u st com m ence w ith certain letters of example, though our readers will p e r­ w hich you m ust in your own m ind asso­ haps prefer having w ords of th e ir own ciate w ith the num bers 1 to 10, so th a t by selection. know ing th e initial le tte r of th e la s t word E llen. George. Jane. Jam es Newton. F anny Mary. W illiam Clem ent. D avy. Caroline. Frederick. M atilda. E dw ard. Morse. Sarah. Isabel. B obert. R alph. F ulton. Flora. E dm und. Bosa. Francis. F raukliu. L aura. John. E lizabeth. Edwin. Arago. M aria. Alfred. W alter. H arriet. Spurzheim. Ann. F rances. A lbert. Charles. L aplacc. E m ily. E dith. H enry. Samuel. Steers. Isaac. E m m a. D orothea. T heodore. H erschel.

T

Friendship. Bose. Sister. P ntnan. Clay. B rother. Violet. H appiness. L afayette. W ebster. Lupin. ' In d u stry . Steuben. Uncle. Calhoun. Daisy. Am bition. Scott. B enton. Aunt. Energy. G randm other. Tr.lip. Taylor. Jefferson. Fidelity. Green. G randfather. l’eony. Adams. H yacinth. Affection. Nephew. H arrison. M adison. Pink. H ope. Niece. H am ilton, Jackson. Cousin. Justice. Snowdrop. W ayne. Monroe. O rd er. W ashington F a th e r. N apoleon. L ily F o r these the key w ords are, “ E d ith F low n,” so th a t th e letters E D I T II F L O W N Stand for 1 2 3 4 5 C 7 8 9 10 F o r the success of th e game, th e key N ew ton. F ulton. Sister. w ords and the num bers denoted by th eir Aunt. letters, m ust be carefully concealed. Bose. Daisy. Take ten o th er cards, w hich call th e F riendship. A m bition. “ grouped cards " and up on one w rite P stm an . Scott. Clay. dow n th e first w ord from each of th e B enton, The object of the game is to guess selecting cards, being careful to w rite th em in th e sam e order. L e t an o th er w hich of the words from any of th e card contain all th e w ords w hich are selecting cards any person m ay have fixed second from th e top, and so on till all the upon. w ords have been grouped together. As L e t any one choose a card out of th e an example, we give th e 1st and 4tli selecting cards, and after he h as fixed upon a wo»d, give it back to y o u ; when grouped cards. 1st. 4th. receiving it, carefully note th e last word Jan e. Sarah. upon it, w hich w ill give you, by the aid E llen. Isabel. of th e key w ord, th e num ber of the card ; George. R obert. th is you m u st keep secret, and you then Jam es, R alph, give him all th e grouped, cards. th sum s, 2 2 + 3 4 = 5 6 , S ubtract you know in w hich four halvings he was having one from the other, leaving 12; tho 2d obliged to take a “ larger half num ber will be 6, th e half of th is ; take ascertained th is point, you discover the th e 2d from th e sum of th e 1st and 2d an exact h a lf ca n n o t be ta k e n w ith o u t a you will get th e 1 st; take the 2d from the fractio ( *'"When n , he m u s t ta k e tb e la rg e r h alf—you m u st tell sum of t i e 2d and 3d, and you will have h] im th is before he com m ence?. Ife re it is th e lurger th e 3d, and so on. ih a lf

A

O

25

COMIC AND CURIOUS PEOBI.F.JIS IN A R IT H M E T IC .

n um ber fixed upon in tho following m a n ­ ner. Carry in jo u r m ind, or on a slip of paper, the following list of nam es in w hich the le tte r A occurs in one or m ore of th e throe syllables of all except the last. The three syllables are intended to rep resent th e 1st, 2d, and 3d halvings, and the occurrence of th e le tte r A corres­ ponds to the occurrence of a “ larger half ” in one or m ore of these th ree h alv­ ings. H aving been inform ed w here th e larger h a lf w as tak en , refer to th e w ord w hich has A in th e corresponding syllable and against it stan d tw o num bers, one of w hich was the num ber though t o f ; and of these two, tho rig h t h and num ber is th e correct ono i f a larger h a lf was taken in the 4 th stage, and tho left hand one i f the 4 th halving w as exact. I n th e exam ple given, a larger h a lf occurred in th e 1st and 3d s ta g e ; th is points us to Car-ro-way, and th e halving in th e 4th stage being exact, shows us th a t 8 was the num ber fixed upon. If the 4th halving If a la r g e r h a l f occurs is e x a c t . in the 4th halving. W Ash-ing-ton 4 • 12 IiA-fAy-ette 2 10 CAr-row-wAy 8 0 MAn-liAt-tAu G ■ . It Q c r-m -A n y 13 • .'j Tel-e-grAph 3 • • 11 Bo-nA-pArte 1 ■ 9 Long-fel-low 15 7

I t will be observed th a t there is always a difference of 8 betw een th e num bers of th e colum ns, so th a t it is necessary to recollect only one of them . Perhaps some of our readers who w ish to be adepts in th is game, would prefer recollecting th e above table if p u t in th is form : 2-3 1-2 3 1-2-3 1-3 2 none ‘1 2 3 4 8 13 15 w here the upper line denotes th e cases in w hich th e “ larger h a lf ” was tak en , and th e low er line th e num bers of the left hand colum n above given, A nother Method. The person having chosen any n um ber from ono to fifteen, he is to ad d ono to th a t num ber, an d trip le th e am ount. Then, 1st. H e is to take half of th a t triple, and trip le th a t half. 2nd. To take tho half of th e last triple and triple th a t half. 3rd. To take the half of the last triple.

4 th. To tak e tho half of th e la st half. In th is operation th ere are four distinct cases or stages w here th e half is to ba tak en . The th ree first are denoted by one of th e eight following L a tin words, each word being composed of th ree syll­ ables, and th e syllables containing the le tte r i corresponding in num erical order w ith th e cases w here th e half cannot bo taken w ithout a fraction ; consequently, in those cases th e person who m akes th e deduction is to add one to th e n u m b er to be divided. Tho fo urth case show s w hich of th e tw o num bers corresponding to each w ord has been chosen. F o r if th e fourth half can be tak e n w ithout adding ono, th e n u m b er chosen is in th e first, or left-hand c o lu m n ; b u t if n o t, it is in tho second colum n to th e right. The words. M i-ser-is O b-tin-git Ni-m i-um N o-tar-i In-fer-nos Or-di-nes Ti-mi-di Te-ne-ant

Tho num bers denoted. 8

0

1

9

2

10

3 4 13 0 15

11 -1 2 C> 14 7

E x a m p le.— Suppose th e num ber chosen to be nine, to w hich is to be added one, m aking ten . and w hich last, being tripled give3 th irty . T h e n : 1st case. Tho half of th e trip le is 15 which tripled, m akes 45 2nd case The half of th a t triple, 1 being added to m ake an even num ber, is 23 and th a t tripled m akes C9 3rd case. The half of th e last triple, 1 being added, is 35 4 th case. The half of the la st half, 1 being again added, is 18 H ere we see, th a t in th e second and th ird case, one had to be added and, look­ ing a t th e table, we find th a t th e only corresponding word having an i in its second and th ird syllables is O b-tin-git, w hich represents th e figures one and n ine. Then, as one had to be added in th e fourth case, we know by th e rule, th a t th e figure in th e second colum n, 9, is th e one required. Observe, th a t if no addition be required at any of the four stages, th e num ber th o u g h t of will be fifteen ; and if one addition only be re ­ quired a t th e fo urth stage, th e num ber will be seven.

26

THE MAGIC OF NUMBERS J OR,

Odd Magic Squares. QQTJARES ol th is kind are formed O thus. Im agine an exterior line of squares above the magic square you wish to form, and another exterior line on th e rig h t hand of it. These two im aginary lines are shown in th e figure. T hen attend to th e tw o following ru le s : 1st. In placing the num bers in the squares we m ust go in th e ascending oblique direction from left to r i g h t ; any num ber which, by pursuing this direction, w ould fall into th e exterior line, m u st be carried along th a t line of squares, w hether vertical or horizontal, to the la st square. Thus, 1 having been placed in tho centre

: 17

18

:

25

13 12 18

: 9 15

14

4

11

2

24

23

10

:

19

20

21

10

of th e to p line, (see th e first table, 2 woul'd fall into th e exterior square above th e fourth vertical line ; it m u st be therefore carried down to th e low est square of th a t line ; then, ascending obliquely, 3 falls in th e square, but four falls out of it, to th e end of a h orizantal line, and it m ust be carried along th a t line to the extrem e left, and th ere placed. R esum ing our oblique ascension to th e right, we jjlace 5, where the reader sees it, and would place Gin tho m iddle of th e to p band, b u t finding it oc­ cupied b y l, w e lo o k fo rth e d ire c tio n to th e

30

39

48

1

10

19

28

38

47

7

9

18

27

29

40

6

8

17

20

35

37

5

14

10

25

34

30

45

13

15

24

33

42

44

'4

i

21

23

32

41

43

3

12

•

22

31

40

49

2

11

20

:

! 17

1

23

22

4 10

25

2d Rule, w hich prescribes th a t, when in ascending obliquely we come to a square already occupied, we m u st place th e num ber, w hich according to th e first rule should go into th a t occupied square, directly under th e last num ber placed. Thus, in ascending w ith 4, 5, G, th e G

m u st be placed directly u n d er th e 5, because th e square n ext to 5 in oblique direction is “ engaged.” Magic squares of th is class, how ever large in th e num ber of com partm ents, can be easily filled up b y atten d in g to these two rules.

The Arithmetical Boomerang. H E boom erang is an in stru m en t of tical processes by w hich you can divine a peculiar form , used by th e natives num ber thought of by another. You of New South W ales, for th e purpose ofthrow forw ards th e num ber by m eans of killing wild fowl and sm all anim als. If addition and m ultiplication, and then, by projected forwards, it at first proceeds in m eans of sub tractio n and division, you a straight line, b u t afterw ards rises in the bring it bacfc to th e original startin g point air, and after perform ing sundry peculiar m aking it proceed in a track so circu­ gyrations, retu rn s in the direction of th e itous as to evade the superficial notice of place whence it was throw n. the tyro. The term is applied to those arith m e­

T

COMIC AND CURIOUS PROBLEMS IN ARITHMETIC.

27

To find a number thought of. F irs t M ethod. fjU IIS is an arithm etical trick which, to JL those who are unacquainted w ith it, seems very s u rp ris in g ; but, w hen ex­ plained it is very si-mple. F o r instance, ask a person to th in k of any num ber under 10. W hen he says he has done so, desire him to treble th a t num ber. T hen ask h im w hether the sum of tho num ber lie has thought of (now m ultiplied by 3) be odd or even; if odd, tell him to add 1 to make the sum .even. H e is next to halve th e sum, and then treble th a t half. Again ask w hether the am ount be odd or even If odd, add 1, (as before) to m ake it even and th e n halve it. Now ask how m any nines are contained in the rem ainder. The secret is, to b ear in m ind w hether the first sum be odd or even ; if odd, reta in 1 in the m em ory ; if odd a second tim e, re ­ tain 2 m ore (making in all 0 to be retained in the m e m o ry ;) to w hich add 4 for every nine contained in th e rem ain­ der. 3 3

F o r exam ple, No. 7 is odd th e first and also th e second tim e ; and th e rem ainder (17) contains one nine : so th a t 1, added to 2, m ake 3, and 3, added to 4, m ake 7, the num ber th o u g h t of. No. 1 is odd the first tim e (retain 1), and even th e second (of w hich no notice is taken), b u t tho r e ­ m ainder is n o t equal to nine. No. 2 is even th e first and odd th e second tim e (retain 2), b u t th e rem ainder contains no nine. No. 3 is odd th e first and tho sec­ ond tim e, still there is no nine in th e re ­ m ainder. No. 4 is even both tim es, and contains one nine. No. 5 is odd th e first tim e and the rem ainder contains one nine. No. C is odd th e second tim e, and contains one nine in th e rem ainder. No. 8 is even both tim es, and th e rem ainder contains tw o nines. No notice need be tak en of any overplus of a rem ainder, a fter being divided by nine. The following are illustrations of tho result w ith each n u m b e r; G 3

7 3

3 2)0 9 2)12 13 2)18 21 2)24 27 Add 1—A d d l —Add 1 —Add 1 —A d d l — 8 — 0 — 9 — 12 — 2)4 3 2)10 3 2)10 3 2)22 3 2)28 2 9 SA ddl

5 3 15 2)0 2)10 — — Add 1 3 5 — 2)10 8

Second M ethod. ■ EX A M PLE.

2)18 8 27 2)30 11 14 — 3A ddl 3 — 3 9)9 — — — 9)18 — — 2)24 2)28 33 — 2)42 1— —A d d l 2 — 9)12 9)14 — 9)21 — — 2)34 1 1 — S 9)17

T h ir d M ethod, EXAM PLE.

L e t a person think of a num ber, say 6 Suppose th e num ber th ought of to be 0 1. L e t him m ultiply it by 3 - 18 1. L et him double it - 12 2. Add 1 ; . 19 2. Add 4 ........................................1G 3. M ultiply by 8 - 57 3. M ultiply by 5 • . -8 0 4. Add to this the num ber thought of 03 4. Add 1 2 ....................................... 02 L et him inform you w hat is th e num ber 5. M ultiply by 10 . 920 p roduced; it will alw ays end w ith 3. L et him inform you w hat is the num ber Strike off the 3, and W o rm him th a t lie produced. You m ust in every case sub­ th ought of 0. tra c t 320; th e rem ainder is, in this ex­

28

THE 3IAGIC OF NUMBERS, OR;

ample, GOO; strike Oi7 th e two ciphers, and announce 0 as the num ber thought of. F o u rth Method. Desire a person to think of a num ber, aay 0. IIo m ust then proceed— EXAMPLE.

1. To m ultiply this num ber by itself DO 2. So take 1 from the num ber thought of . . . . . . 5 S. To m ultiply this by itself - 25 4. To tell you tho difference betw een this product and the form er - 11 You m ust then add 1 to it - 12 And halve this num ber . . . 6 W hich will be the num ber thought of. F if th M ethod. Desire a person to think of a num ber, say 0. H e m ust th en proceed as fol­ lows : EXAMPLE.

1. Add 1 to it • • 7 2 . M ultiply by 3 • • .2 1 3. Add 1 again " - 22 4. Add the num ber thought of - 28 L et him tell you the figures produced (28 ) :

5. You then substraet 4 from it - 24 0. And divide by 4 G W hich you can say is the num ber thought of. S ix th M ethod.

4

EXAMPLE.

Suppose the num ber thought of • 6 1. L et him double it - 12 2. Desire him to add to this any num ber you tell him , say 4 - 1G 3. To halve it . . . . 8 You can th en tell him th a t if he will subtract from th is the num ber he thought of, th e rem ainder will be, in th e case supposed, 2. N ote.—The rem ainder is always half of the num ber you toll him to add. Who wears the ring. H IS is an elegant application of the principles involved in discovering a num ber fixed upon. Tho num ber of p e r­ sons participating in th e game should not exceed nine. One of them p u ts a ring on one of his fingers, and it is your object to discover—1st, The w earer of the ring. 2d. The hand. 3d. The finger. 4th. The joint. The company being seated in order the persons m ust be num bered 1, 2, 3, &c. ; th e thum b m ust be term ed the first finger,

T

th e fore finger being th e second ; th e joint nearest th e extrem ity m ust be called th e first joint ; th e right hand is one, and th e left hand two. These prelim inaries having been arran ­ ged, leave the room in order th a t th e ring m ay be placed unobserved by you. W e will suppose th a t the th ird person lias th e ring on the right hand, third finger, and first j o i n t ; your oblect is to discover th e figures 3131. Desire one of th e com pany to perform secretly th e following arithm etical opera­ tions ; 1, Donble th e num ber of th e person who has th e ring ; in the case supposed, th is will pro d u ce........................................ G 2. Add 5 ................................................11 3. M ultiply by 5 ............................... 55 4. Add 10........................................... G5 5. Add the num ber denoting the h a n d ............................................................. GG G. M ultiply by 10...........................GC0 7. Add the num ber of the finger. . 663 8. M ultiply by 10...........................6630 9. Add the num ber of the jo i n t.. G631 10. Add 3 5 ....................................... 66GC H e m ust apprise you of th e figure! now produced, 666G ; you will th en in all caSes substraet from it 3535 ; in the present instance there will rem ain 3181, denoting the person No. 3, tho hand No, 1, th e finger No. 3, and th e joint No. 1.

The Astonished Farmer. and B took each 30 pigs to m a rk e t; A sold his a t th ree for a pound, 15 a t tw o for a pound, and to g eth er they received 25 pounds. A afterw ards took 00 alone, which ho sold as before, a t five for two pounds, an d received b u t 24 pounds ; w hat became of th e other pound ?

A

This is rath e r a catch question, tho insinuation th a t th e first lo t were sold a t th e ra te of five for two pounds being only tru e in p art. They commence selling a t th a t rate, but, a fte r making ten sales, A’s pigs are exhausted, an d they have received 20 p o u n d s; B still has ten, which he sells a t “ two for a pound,” and of course receives five pounds; whereas h ad he sold them a t th e ra te of five for tw o pounds, he would have received b u t four pounds. H ence th e difficulty is easily settled.

COMIC AKD OUKIO'

'BLEM3 IN ARITHMETIC.

29

HOW t o T E L L A PE R S O N ’S AGE.

th e la tte r so rt in th e above m anner, she OUNG ladies of a m arriageable rfge w ould only have sold eighty of th e former, do n o t like to tell how old th ey for th ere are as m any threes in one are, h u t you can find o u t hy following h undred and tw en ty as tw os in e ig h ty ; tho subjoined instructions. L ot tho th en tho rem aining fo rty m u st be sold person whoso age is to bo discovered do a t five for a penny, which were b ought a t th o figuring. Suppose, for example, th a t tho ra te of four for a penny, v iz :— h er age is fifteen, an d th a t she was born A : D : : A : D in A ugust. L e t her pu t down th e num ber ost of 40 of the of tho m onth in which she was born, and I f 4 :1 : : 40 : lo } rrime £ first sort. toll her to proceed as follows :— 5 : 1 I : 40 : 8 Selling price of ditto. N um ber of m onth 8 M ultiply by 2 10 J Tcncc Loss, A dd 5 21 M ultiply by GO - 10-50 T h e D ro v e r’s P roblem . Then add th e ago (15) - 1065 Su b tract 3G5, leaving - 700 One m orning11 chanced w ith a d ro \c r to m eet, W ho was driving some sheep up to tow n, A dd 115 . . . 815 W hich seemed very near ready to drop from the heat, She th e n announces th e resu lt 815, W hereupon I exclaim ed wi;ii a. frow n: w hereupon you inform her th a t h er age “ D on’t you th in k it is w ro n j lo tre a t anim als so, is fifteen, and A ugust, or th o eighth W hy n o t take t /U e r car* nf j our flock '!” ould do so,” said lie, •* c u t I ’ve some miles to go m onth, is th e m onth of her birth, for th e “ I wBetween th is and eleven o’clock.” tw o figures to th e rig h t in th e re su lt will “ W ell, supposing you h a \e ,” I rep lied ,“ y o u sh o u ld let always indicate th e age, an d tho rem ain­ Them have re s t now and then by tho w ay.” ing figure or figures th e m onth th e “ So I w ill, iny good friend, it' jo u th in k I can get T h e re in tim e for the m ark et to-day. b irth d ay falls in. This rule never fails o t v , as you seem to know such a lo t ab o u t sheep, for all ages up to one hundred. In ages “ N Perhaps you’ll tell us how m any I ’ve got under ten a cypher will appeal’ in th e “ No, a casual glance, as they stan d in a heap, W on’t p erm it of it, so 1 can n o t.” result, b u t no account is tak en of this.

Y

T h e M ark et W o m a n ’s P u zzle. M ARKET-W OM AN bo u g h t 120 apples a t tw o a halfpenny, a n d 120 more of an o th er so rt a t th ree for a halfpenny; b u t n o t liking her bargain, she m ixed them together, and sold them o u t again a t five for a penny, th in k in g she w ould g et th e sanio su m ; b u t on counting up her money, she found, to her surprise, th a t she h ad lost twopence. H ow did this happen ? On tho first view of th e question th ere docs n o t appear to be any lo s s; b u t if it bo supposed th a t in selling five apples for a penny she gave th ree of th e la tte r sort, v iz : those a t th ree for a halfpenny, and tw o of th e former, v iz : those a t tw o for a halfpenny, she w ould receive ju s t th e same money as she bought them fo r; b u t th is will n o t be th ro u g h o u t th e whole, for ad m ittin g th a t she sells them as above, it m u st be evident th a t th e la tte r stock w ould be exhausted first, an d conse­ quently she m ust sell as m any of th e form er as rem ained overplus a t five for a penny, which sho bought a t th o ra te of tw o for a halfpenny, or four for a penny, and would therefore lose. I t will be readily found th a t when she had sold all

A

“ W ell, supposing as how I ’d as many again, H a lf as m any, anil seven, as tru e A« vou’re there, i t would pay me to ride u p by tra in , Because I should have th irty -tw o .’’

There were ton sheep in th o flock ; ten , as many again ; five, half as m a n y ; and seven besides. T o ta l: thirty-tw o.

M ore Q u e e r Q u e s tio n s . F you c u t u p th irty yards of cloth into one-yard pieces, and c u t one yard off every day, how long will it ta k e ? A n s : Twenty-nino. days. W h at two num bers m ultiplied to g eth er will produce 7 ? A ns: 7 and 1. W hat is th e difference betw een twice 25 an d tw ice 5 and 20 ? ^ A n s : Twice 25 is 50. Twice 5, and 20 is th irty —difference 20. W h a t is th e tw o-thirds of three-fourths of elevenpence-halfpenny ? Ans : Five-pence three-farthings. The tw o-thirds of th e three-fourths of an y ­ th in g are ju s t one-lialf th e whole. H ow much is a th ird and half-a-third of five P A ns: Two and a half. There are exactly th ree-th ird s in five, therefore a th ird an d half-a-third make exactly half,

I

30

THE MAGIC Oi' UIJilJJELlS; OB,

Divide th e num ber 50 into two such ! To find six times thirteen in parts th a t, if tho greater p a rt be divided ; twelve. by seven, and th e lesser m ultiplied by LACE your figures th u s :— throe, tho sum of th e q u o tien t an d th e j 1 ,2 ,3 , 4 ,5 , 6 ,7 ,8 , !), 10, 11, 12, p roduct will make SO ? ! and tak in g always tho first and last A n s : 35 and 15. figure together, you say :— Jf a goose weighs 10 lbs. and hall its 1 and 12 m ake 13 \ its own weight, whi.it is th e w eight of th e j 2 „ n „ 13 goose ? | 3 „ 10 „ 13 ! A ns: 20 lb3. 10 lbs., an d 10 lbs. for 0 tim es. 4 „ 9 „ 13 ( half its own weight. 5 ,, 8 „ 13 \ A snail climbing up a post 20 feet high, 0 „ 7 „ 13 / ascends five feet every day and slips down four foot every night. How long will it Peculiar Properties of the Numbers tako to got to tho top of tho post ? 37 and 73. A ns; 10 days. I t is perhaps u n ­ H E num ber 37 being m ultiplied by necessary to p o in t out th a t tho snail each of tho num bers in tho arithm e­ would gain ono foot a day for 15 days, tical progression—3, 6, 9, 12, 15, 18, 21, and on tho 10th day reach th o to p of tho 21, 27, all products will bo composed of polo, and th ere remain. th ree similar figures, an d th e sum is A train sta rts daily from San Francisco always equal to th o num ber by which 37 to Now York, and ono daily from Now was m u ltip lie d : York to San Francisco, tho journey 37 37 37 37 37 37 37 37 37 lasting five days. IIow m any train s will ’3 0 9 12 15 18 21 21 27 a traveller moot in journeying from New York to San Francisco ? Ans: Ton. A bout ninety-nine persons 111 222 3334 I ! 555 (i0(i 777 888 909 o ut of a h undred would say five trains, Tho num ber 73 being m ultiplied by as a m atter of course. Tho fact is over­ each of th e aforc-givon progression, th e looked th a t every day during th o journey p roducts will term in ate by ono of th e a fresh train is startin g from th e other nino d ig its—1, 2. 3, 4, 5, 6, 7, 8, 9 in a end, while th ere are five train s on tho reverse. Again, if wo refer to th o sums way to begin w ith. C onsequently the produced by th o m ultiplication of 73 by ■traveller will moot n o t five train s, b u t 3, 6, 9, 12, and 15, it will be found- th a t ten. by reading th e tw o figures to th e left of each am ount backwards, it will givo 1, 2, Tho unfair Division. G ENTLEM AN ren ted a farm and 3, 4, 5, 0, 7, 8, 9, 0. contracted to give to his landlord The Basket of Eggs. two-fifths of th o produce; b u t prior to WOMAN carrying eggs to m arket tho timo of dividing th e corn th e te n a n t was asked how m any she had. SI so used 45 bushels. AVhcn tho general division was niado, it was proposed to replied th a t when she counted th em by give th e landlord 18 bushels' from th e tw os th ere was ono l e f t ! when by threes heap in lieu of his share of th e 45 bushels th ere was ono le f t; and when by fours which tho te n a n t h ad used, an d th e n to th e re was one lo f t; b u t when she counted begin and divide tho rem ainder as though them by fives th ere were none left. How none had boon used. W ould this m ethod many had sho ? have boon correct P The least num ber th a t can bo divided No. Tho landlord would lose seven by 2, 3, and 4 respectively, w ithout a and one-fifth bushels by such an arrange­ rem ainder, is tw elve; and th a tth c ro may m ent, as tho re n t would en title him to bo ono rem aining, th e num ber m ust bo two-fifths of th e 18. Tho te n a n t should 13; b u t th is is n o t divisible by 5 w ithout give him 18 bushels from his own share a rem ainder. The n ex t g reater num ber a fter tho division is com pleted, otherwise is 24, to which add 1, and it becomes 25; tho landlord would only receivo two- th is is divisible by 5 w ithout a rem ainder, sevenths of th e first 03 bushels. an d is therefore th e num ber required.

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COMIC AN1) CUltlOUS l'UOCLU.M.S IN AUITllMKTIC.

A Tell-Tale Table.

r n H E R J i is a good deal of am usem ent I in tlie following table of figures. I t will enable you to te ll how old th e young ladies are. J u s t hand th is table to a young lady and request her to tell you in which column or colum ns h er age is contained, add to g eth er th e figures a t th e to p of tho column in which h er age is found, and you havo tho groat secret. Thus, suppose h er ago to be seventeen, you will find th a t mfmber in tho first and fifth columns. A dd th e first figures of these columns and you havo h er age. H ere is th e magical table 2 1 4 8 32 16 o «*> O 5 9 33 17 5 0 6 10 34 18 7 7 7 11 35 19 9 10 12 12 36 20 11 11 13 13 21 37 14 13 14 14 22 38 15 15 15 15 23 39 24 17 18 20 24 40 19 19 25 41 21 25 oo 21 22 26 42 26 23 23 27 23 27 43 25 26 28 28 44 28 27 27 29 29 29 45 29 30 30 30 46 30 31 31 31 31 47 31 33 31 40 48 36 48 49 35 35 41 37 49 50 37 38 42 38 50 39 51 39 39. 43 51 42