• > » IN-DO0R FUN & ENTERTAINMENT ^
No.
l ] I l i L T J S T E A T E D . [ i d .London : ALDINE PUBLISHING Co., 9, Red Lion Court, Fleet S treet, B.C.
THE
BOYS’
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ONE PEfiHY EACH, or Post Free lor l j d . In Stamps. Beautiful Coloured Covers.
1, 2
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3. 4, - 5. 6. 7. 8. 9. 0 11.
12. 1 3. '1 4. 1 5. 1 6 . 1 7. 18. 1 9. 20. 2 1.
2 2. 23. 24. 25. 2 6 . 27. 2 8 . 29. 30. 31. 32. 33. 34. 35. 36. 37. 3 8. 39. ‘4 0. 41. 42. 4 3. 44. 45. 4 6. 4 7. 4 8 . 4 9 . 50. 51. 52. 53. 54. 55. 5 6. 57. 58. 5 9 . 60. 61. 62. 6 3. 64. 65. 66. 67, 68.. 6 9. 70. 71. 72. 73. V O L U M E S N O W R E A D Y . 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 O U R S C H O O L , A N D A L L A B O U T T H E B O Y S i / 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 . 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 t s * 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 - I ■ T H E C A V A ! lcr,:> -T'ADTAIN Artv^ntures-on th e H ig h w a ys o f O ld L o n d o n F O U G H T F< G E N E R A L * L I F E F O R I M O R G A N , ‘ T R A C K E D ’ R E C K L E S S N I C E B O Y S T H E W JL D B I L L Y BR IC S A M S L A B T H E S L A S t D O C T O R C T H E K I N G ' T H E W IL D IN S T E E L / T H E M E R C T H E B A N K T H E C A N A IR O N C L A D T H E S O L D T H E D IA M i T H E O C A E T H E Y O U N T H E H A L F T H E IR O N T H E F R O N T H E H U N T D E A D W O O D O U B L E D B U F F A L O W IL D IV A N T H E P H A N D IC K IN D D E A D W O O C A L A M IT Y D IC K D A R I C O R D U R O T H E S E A < R O V E R S 0 Mile D E A D W O O D E A D W O O D E A D W O Ol^ w v w — _______________ _______ B L O N D B IL L , o r D ead w o o d D ick's D isco very 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 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 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 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 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 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 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 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 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 D E A D W O O D D I C 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 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 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 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 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 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 . D E A D W O O D D IC K IN D E A D C IT Y 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 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 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 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 N IC K E L - P L A T E D N E D S U N F L O W E R S A M O F S H A S T A F L U S H F A N T H E F E R R E T 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.WILL ALMA
M.I.M.C. (LONDON)The State L ib ra ry of V ictoria
Cocker and P ilu o r th , 'W alkingam e and Yyse, I n th e ir own sphere, by B idder were outshone. T h ey , w ith pen or pencil, problems s o l\c d — H e, w ith no aid, b u t w ondrous mem ory ; They, w hen of years m a tu re ac quired th e ir fam r, l i e “ lisped in num bers for th e num bers cam e.”
T
H E delightful an d valuable science of num bers first arrived a t any degree of perfection in E urope am ong th e G reeks, who m ade use of tho letters of th e alphabet to express th o ir num bers.A sim ilar mode was followed by tho Homans, who, besides characters for each ra n k of classes, in tro d u ced others for five, fifty, an d five hundred, w hich are still used for chapters of books, an d some o th er u n im p o rtan t purposes.
The common arithm etic, in w hich th e te n Arabic figures, 1, '2,3, 4, 5, G, 7, 8, 9, 0, are used, was unknow n to th e Greeks and R om ans. They came in to E urope by w ay of Spain from th e A rabians, who are believed to have received th em from th e an c ie n t philosophers of India.
The A rabic system is supposed to have ta k e n its origin from th e ton fingers of th e hand, which were used in m aking calculations before arith m etic was b rought a n a rt, a n d i t is to th is a r t t h a t we
in te n d to introduce our readers.
B u t as th e principal o b ject of th is volume is to enable th em to learn som ething in th e ir sports, an d to u n d er sta n d w hat th ey are doing, wo shall, before proceeding to tho curious trick s an d feats cenucctod w ith th o scienco of num bers, presen t th em w ith some arithm etical aphorisms, upon w hich m ost of th e following exam ples arc founded.
A p h o ris m s o f N u m b er. 1. If tw o even num bers bo a d d ed to g e th e r, or su b stracted from each other, th e ir sum or difference will be a n even num ber.
2. If tw o uneven num bers bo added or su b tracte d th e ir sum or differcnce'w ill bo. an even number.
3. The sum or difference of a n even an d an uneven liutnber added or sub tra c te d will be an uneven num ber.
4. The product of tw o even num bers will bo an even num ber, an d th e p ro d u ct1 of tw o uneven num bers will be an uneven number.
G. The p ro d u ct of an even and an uneven num ber will be an even num ber.
THE MAGIC OF NUMBERS; OK,
G. If tw o different num bers be divisible by auy one num ber, th e ir sum and th e ir difference will also be divisible by th a t num ber.
7. If several different num bers, divided by 13, b:i added or m ultiplied together, th e ir sum and th eir product will also be divisible by 3.
8. If two num bers, divisible by 9, be added together, th e sum of th e figures in the am ount will be eith er 9, or a num ber divisible by 9.
! 9. If auy num ber be m ultiplied by 9, or by any other num ber divisible by 9, the am ount of th e figures of th e product w ill be eith er 9, or a num ber divisible by 9.
10. In every arith m etical progression, if th e first and last te rm ba each m u lti plied by the num ber of term s, and th e sum of th e two products be divided by 2, the quotient will be th e sum of th e series.
11. In every geom etric progression, if auy tw o term s be m ultiplied together, their product will bo equal to th a t term , w hich answ ers to the sum of these tw o indices. Thus, in th e series—
1 2 3 4 5 2 4 8 10 32
If th e th ird and fourth term s 8 and 10 be m ultiplied together, th e p roduct 128 will be the seventh term of th e series. Tn j like m anner, if th e fifth te rm be m ultiplied into itself ,the product will be the tw entieth term , and if th a t sum be m ultiplied into itself, th e product will be the tw en tieth term . Thci'cfore, to find the last, or any other te rm of a geom etric series it is not necessary to continue th e series beyond a few of th e first term s.
Previous to th e num erical recreations, wo shall here describe certain m echanical m ethods of perform ing arithm etical cal culations, such as are n o t only in th e m selves en tertain in g , b u t will be found m ore or loss useful to th e young reader.
Arithmetic.
1
1HE
blind m ath em atician , Dr. Saunder- son, adopted a very ingenious device for perform ing arith m etical operations by I th e sense of touch.Small cubes of wood were provided, and in one face of each, nine holes wore pierced, th u s :
1 2 3 o o o
4 5 I! o o o
7 8 9 o o o
These holes represented the nine
digits, as in the figure, and to denote any figure, a sm all peg w as in s e rte d into th e hole corresponding to it. If th e num ber consisted of several figures, m ore cubes were used one for each. A cipher w as r e p resented by a peg of different shape from th a t of th e others, and inserted in the central hole.
To perform any arith m etical process, a square board w as provided, jdivided by ridges into recesses of the sam e w idth as t h 3 cubes, and by th is the cubes were retained in th e required horizontal and p erpendicular lines. Suppose it was necessary to add to g eth er th e num bers 763, 124, 859, th e cubes and pegs w ould be arranged th u s : 0 0 0 0 0 0 0 0 r 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 r. 0 0 0 c 0 0 0 0 0 0 0 0 0 0 8 -0 0 0 0 0 o' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 £ 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The Abacus.
1
1 IIIS in stru m e n t is used for teaching _ n um eration, and th e first principles of arith m etic.U pon a fram e are placed, wires parallel to onean- o th er an d ate q u al d is ta n c e s . Ten sm all balls are stru n g upon each w i r e , b e i n g placed as in the m a r g i n . T h e r i g h t w ire de notes units, the n ex t tons, and so on, 7th wire being the place of m illions. In Using the abacus, all th e balls arc first ranged a t one end, and a n um ber of th e m are th en m oved to th e o th er end of each w ire, to correspond to th e figures req u ired . The exam ple given in th e m argin is 15,781, .th e height of M ount W ane.
COMIC AND CURIOUS fcBOBLEMS iN ARITHMETIC. t
Progression.
I
F a hund red stones be placed in n straig h t line, a t the distance of a yard from each o th er, tho first being a t tho panic distance from a basket, how m any y ards m u st tho person w alk who engages to pick th em up, ono by one, and p u t th em in to th e basket ? I t is evident th a t to pick up th e first stone, and p u t it into th e b asket, th e person m u s t w alk tw o yards ; for th e second, he m u st w alk four ;for th e th ird , six ; and so on, increasing by tw o, to th e hu n d red th .
The n um ber of y ards, th erefo re, w hich tho p erson m u st w alk w ill be equal to th e sum of th e progression, 2, 4, 0, &c,* th e la s t te rm of w hich is 200 (22). B u t tho sum of th e progression is equal to 202, th e sum of th e two extrem es, m ultiplied by DO, or half th e n um ber of term s ; th a t is to say, 10,100 yards, w hich m akes m ore th a n 5J m iles.
H
OW can num ber 45 be divided in to four sn c h p a rts th a t, if to th e first p a rt yon add 2, from th e second p a rt yon su b tract 2. tho th ird p a rt you m u ltip ly b y 2, and tho fourth p a rt you divide by 2, th e sum The 1st is 8 ; to w hich The 2nd is 1 2 ; su b tract The 3rdTho 4tll
The famous forty-five.
of th o addition, tho rem ain d er of tho su b tractio n , th e p ro d u ct of th e m u ltip li cation, and tho quo tien t of tho division m u st be all equal ? add is 0 ; m ultiplied by is 2 0 ; divided by 2, th e sum is 10 2, th e rem ainder is 10 2, th e p ro d u ct is 10 2, th e quo tien t is 10 45
R equired to su b tra c t 45 from 45, and leave 45 as a rem ain d er 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 = 4 5 V / 8 + 0 + 4 + 1 + 9 + 7 + 5 + 3 + 2 = 4 5
Subtraction.
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,'ROM 1 m ile su b tra c t 7 furlongs, 39 rods, 5 yards, 1 foot, 5 inches.miJes, fu rio n g s, r o d s , y a rd s , feet, in ch es.
F ro m 1 0 0 0 0 0
T ake 0 7 ' 89 5 1 5
0 0 0 0 0 1
I n th is problem , in stead of borrow ing 1 foot, we borrow j a foot= G inches, from w hich we tak e 5 inches, an d 1 rem ains ; we th e n carry J to 3, and borrow ing J a y a r d = l j feet, wo have 1J from 1 £ = 0 , and afterw ards proceed as usual.
The expunged figure.
I
N th e first place desire a person to w rite dow n secretly, in a line any n u m b er' of figures he m ay choose, and add th em to g eth er as u n its ; having done th is, te ll him to su b tra c t th a t sum from th e line of figures originally se t d o w n ; th e n desire him to strike out any figure lie pleases, and add th e rem aining figures in th e lino to g eth er as u n its, (as in th e first instance,) and inform you of th e result, w hen you w ill tell him th e figure he has stru ck out.7G542-24 Suppose for exam ple, the 24 figures p u t dow n arc 70542 ; --- these, added to g eth er, as nn-7G518 its, m ake a to ta l of 24 ; deduct 24 from th e first line, and 7G518 re m a in ; if 5, tho centre figure be stru ck o ut, th e to ta l will be 22. I f 8, th e first figure be stru ck out, 19 w ill be th e to tal.
I n order to ascertain w hich figure has b een stru c k o ut, you m ake a m e n ta l sum one m ultiple of 9 higher th a n th e to tal given. I f 22 bo given as th e to tal, th en 3 tim es 9 are 27, an d 22 from 27 show th a t 5 w as stru c k out. If 19 bo given, th a t sum deducted from 27 show s 8
.
Should th e to ta l bo equal m ultiples of 9, as 18, 27. 30, th e n 9 has been ex punged.
W ith very little p ractice any person m ay perform this w ith rap id ity , it is therefore needless to give any furth er exam ples. The only w ay in w hich a person can fail in solving th is riddle is w hen eith er the n um ber 9 or a cipher is stru ck out, as it th e n becom es im possible to tell w hich of th e tw o it is, th e sum of
4
THE MAGIC OF ND3IEEES J OR,th e figure iii th e line being an even n u m ber of nines in both cases.
The mysterious addition.
I
T is required to nam e th e quotient of five or throe lines of figures—each line consisting of five o r m ore figures — only seeing the first line before th e other lines are even p u t down. Any person m ay w rite down th e first line of figures for you. H ow do you find th e quo tie n t ?Ex a m p l e.—AVhen th e first lino of fig ures is set down, su b tra c t 2 from th e la st right-hand figure, and place it before the first figure of th e line, and th a t is the quotient for five lines. F o r example, suppose th e figures given are 8G,214, the quotient will be 280,212. You m ay allow any person to p u t dow n th e tw o first and th e fourth lines, bu t you m u st always set down th e th ird an d fifth lines, and in doing so, alw ays m ake lip 9 w ith the line above, as in th e follow ing example,
8G,214 Therefore in th e annexed 42,GS0 diagram y o uwill see th a t y o u
57,819 have m ade 9 in th e th ird and 62,85-1 fifth lines w ith th e lines above iJ7,145 them . If th e person desire --- to p u t down th e figures should Qt. 208,212 set down a 1 or 0 for the last
figure, you m u st say we will have a n o th er figure, and a n other, and so on u n til he sets down som ething above 1 or 2.
I n solving th e puzzle w ith G7,85G three lines, you su b tract 1 47,218 from th e last figure, and place 52,781 it before th e first figure, and ---m ake up th e th ird line your-Qt 167,855 self to 9. F o r exam ple :—
67,856 is given, and th e quo tie n t will be 167,855, as shown in th e above diagram .
To tell at what hour a person intends to rise.
L
E T th e person set th e h an d of the dial . of a w atch a t any hour he pleases and tell j’oii w hat hour th a t is ; and to the nu m b er of th a t hour you add in your m ind 12 ; th e n tell him to count privately th e n u m b er of th a t am ount upon th e dial, begin ning w ith th e n ex t hour to th a t on w hich ho proposes to rise, and counting back w ards, first reckoning th e num ber of theh our a t w hich he h as placed th e hand. F o r e x a m p le :—
Suppose th e h our a t w hich he intends to rise be 8, and th a t he has placed the hand a t 5 ; you will add 12 to 5, and tell him to count 17 on th e dial, first reckon ing 5, th e h our a t w hich th e index stands, and counting backw ards from th e hour a t w hich he intends to rise ; and th e num ber 17 will necessarily end a t eight, w hich shows th a t to be th e h our h^ chose.
To find the difference between two numbers the greater of which is un known.
rilA K E as m any nines as th ere arc figures
JL
in th e sm allest num ber, and su b tract th a t sum from th e n um ber of nines. Lot an o th er person add th e difference to th e larg est num ber, and tak in g aw ay th e first figure of th e am ount add it to th e last figure, and th a t sum will be th e difference of th e tw o num bers.F o r example : Jo h n , who is 22, tells Thom as, who is older, th a t he can dis cover th e difference of th e ir ages ; he therefore privately deducts 22 from 99 (his age consisting of tw o figures, he of course tak es tw o nines) ; th e difference, w hich is 77, he tells T hom as to add to his age, and to tak e aw ay th e first figure from th e am ount, and add it to th e la st figure and th a t will bo th e difference of th e ir ages ; th u s,—
The difference betw een J o h n ’s age and 99 i s ... 77
To w hich T hom as adding his ag e . . 85 The sum i s ...112 Then by taking away the first
figure 1, and adding it to the figure 2, th e sum i s ... 13 W hich add to J o h n ’s a g e ...22 Gives th e age of T h o m as... 85
The Remainder.
A
very pleasing w ay to arrive a t an a rith m etical sum , w ithout th e use of e ith er slate or pencil, is to ask a person to th in k of a figure, th e n double it, th en add a certain figure to it, now halve the whole . sum , and finally to su b tract from th a t th e figure first th ought of. You are th en to tell th e th in k er w hat is th e rem ainder.The key to th is lock of figures is, th a t
COillC AND CURIOUS PROBLEMS IN ARITHMETIC.
added during th e w orking of th e sum is
t h e r e m a i n d e r. I n the exam ple given, five is th e half of te n , th e num ber r e quested to he added. A ny am ount m ay he added, h u t th e operation is simplified by giving only even num bers, as th ey will divide w ithout fractions.
E xa m p le. Think o f ... 7 . Double i t ... 14 Add 10 to i t ...10 H alve i t ... 2)24 "Which will le a v e ... 12 S u b tract th e num ber thought o f . . . . 7 The Bemainder w ill b e ...5
A person having an equal number of counters or pieces of money in ea§h hand, to find how many he has alto gether.
T ) equest th e person to convey any num-
JLij ber, as 4, for exam ple, from th e one hand to th e other, and th e n ask how m any tim es th e less n um ber is contained in th e greater. L e t us suppose th a t ho says th e one is th e trip le of th e o th e r ; and in,.this case, m ultiply 4, th e num ber of th e coun te rs conveyed, by 3, and add to th e p ro d u ct th e sam e num ber, w hich will m ake 1C. L astly , tak e 1 from 8, and if 10, be divided by th e rem ain d er 2, the quo tien t w ill he th e n um ber contained in each hand, and consequently th e w hole n u m ber is 1G.
T his curious problem deserves a n o th er exam ple. L e t us again suppose th a t 4 counters are passed from one h an d to th e oth er, and th e less n um ber is contained in the g re a te r 2J tim es. I n th is case we m u st, as before, m u ltip ly 4 by 2J, w hich will give 9 J ; to w hich if 4 be added, we shall have 13J, or : if 1, th en , be ta k e n from 2J, th e rem ain d er will be 1$, or by w hich, if 43° be divided, th e quo tie n t 10 will be th e n u m b er of counters in each hand.
■The three jealous Husbands.
/T1H BEE jealous husbands, A, B, 0 , w ith
JL th eir wives, being ready to pass by
nig h t over a river, find a t th e w a te r side a b o at w hich can carry b u t tw o a t a tim e, and for w ant of a w aterm an tliey are com pelled to row them selves over th e river a t several tim es. The question is how those six persona shall pass, tw o a t a tim e, so
th a t none of th e th re e w ives m ay be found in th e com pany of one or tw o m en unless h e r husband bo p re s e n t?
T his m ay be effected in tw o o r th ree w a y s; the following m ay be as good as any :—L e t A an d w ife go over—le t A r e tu rn —le t B ’s and C’s wives go over—A’s wife re tu rn s—B and 0 go over—B and wife re tu rn , A and B go over—C’s wife re tu rn ’s, an d A ’s and B ’s wives go over— th e n O com es back for h is wife. Sim ple as th is question m ay appear, i t is found in th e w orks of A lcuin, who flourished a th ousand years ago, h u ndreds of y ears b e fore th e a rt of p rin tin g w as invented.
The false scales.
A
C H E E S E being p u t in to one of th e scales of a false balance, w as found to w eigh 10 lbs., and w hen p u t in to th e o th er only 9 lbs. W h at is th e tru e w eight ?The tru e w eight is a m ean proportional betw een th e tw o false ones, and is found b y extracting th e square ro o t of th e ir p ro duct. T h u s lG x 9 = 1 4 4 ; and square ro o t 1 4 4 = 1 2 lbs., th e w eight required.
The apple woman.
A
rO O H w om an, carrying a b ask et of apples, was m e t by th ree boys, th e first of w hom bought half of w h at she had, and th en gave h e r back 10; th e second boy bought a th ird of w h a t r e m ained, and gave h er back 2 ; and th e th ird bought half of w hat she h a d now left and retu rn ed h er 1 ; after w hich she found she had 12 apples rem aining, AVhat n u m b e r h a d she a t first ?F ro m th e tw elve rem aining, deduct 1, and 11 is th e ntim ber she sold th e la s t boy, w hich w as half she had ; h e r n u m b er a t th a t tim e, therefore, w as 22. F ro m 22 deduct 2, and th e rem aining 20 w as § of h e r p rio r stock, w hich w as therefore 30. F ro m 30 deduct 10, an d th e rem ainder 20 is half h er original sto ck ; consequently she h ad a t first 40 apples.
The Graces and Muses.
T
H E th ree G races, carrying each an equal num ber of oranges, w ere m et b y th e nine M uses, w ho asked for some of th em : and each G race having given to each Muse th e sam e num ber, it was th e n found th a t th e y h ad all equal shares, H ow m any h a d th e G races a t first ?0
THE MAGIC OP NUMBERS ; OB,Tho least num ber thas ,ttll answ er this question is tw e lv e ; for if we suppose th a t each Grace gave one to each Muse, th e la tte r would each have three, and there
would rem ain th ree for each G race. (Any m ultiple of 12 will answ er th e conditions of th e question.)
The Jesuitical Teacher.
A
T E A C H E R , having fifteen young ladies under h er care, w ished th em to take a w alk each day of th e week.th ree ladies each, b u t no tw o ladies were to bo allowed to walk to g eth er tw ice d u r ing th e w eek. H ow could tlicy be ar. ranged to su it th e above conditions ?
SUN. MON. TUB. WED. THU. FRT. sKt.
a b c a d L k n a e ] a h 0 a f r a i m a e f b e h b 1 0 b f m b i p b d n b g k g h i 0 m P c f i c g n 0 d k c h 1 c 0 o k 1 m f k o d h m d i 0 e m n 0 i k d 1 p n o & 1 n 0 g h h k P f S 1 S m 0 h f n Atithmetical Puzzle.
I
f from C you take 9, and from 9 you take 10 ; and if 50 from 40 be taken, th ere will ju st half a dozen rem ain.ANSW ER.
P ro m S IX I F ro m IX I F rom X L Take IX I Take X | Take L
S I X R em s.
The' money game.
A
PE R SO N having in one h an d a piece of gold, and in th e o th er a piece of silver, you m ay tell m w hich hand he has th e gold, and in w hich 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 u st bo assigned to th e gold ; and a value represented by an odd num ber, such as three, m u st be assigned to th e silver ; afte r which, desii'o the person to m ultiply the num ber in th e rig h t 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 th e tw o p ro ducts, and if th e whole sum bo odd, tho gold will be in th e rig h t hand, and the silver in tho left ; if th e sum be even, the contrary will be tho case,To conceal th e artifice b e tte r, it will be sufficient to ask w h eth er th e sum of th e tw o 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 th e contrary easo odd.
I t m ay be readily seen, th a t th e pieces, iustead of being in th e two h an d s of tho sam e person, m ay be supposed to be in the hands of tw o persons, one of w hom has the even num ber, or piece of gold, and
th e o th er th e odd num ber, or piece of silver. Tho sam e operations m ay th e n be perform ed in regard to these tw o persons, as are perform ed in regard to th e tw o hands of th e sam e person, calling th e one privately tho right, and th e other tho left.
The philosopher’s pupils.
T
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 am ong th o ancient G recian arith m etician s, 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 h alf,” replied th e philosopher, “ study m ath em atics, one fourth n atu ral philosophy, one seventh observe silence, and th ere are th ree fem ales besides.”
Tho answ er is, 28 : 1 4 + 7 + 4 + 3 = 2 8 .
To discover a square number.
A
SQUA RE num ber is a n um ber p ro duced by th e m ultiplication of any 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 tak es som e 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 know ing th a t every square n um ber ends eith er 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 ro p erty of a square num ber is, th a t if it be divided by 4, th e rem ainder, if any, w ill be 1—th u s, th e square of five
COMIC AND CURIOUS PROBLEMS IN ARITHMETIC.
is 25, and 25 divided by 4 leaves a re m ain d er of 1 •, and again, 16, being a square num ber, can be divided by 4 w ithout leaving a rem ainder.
The Sheepfold.
A
F A R M E R had a pen m ade of CO hurdles, capable of holding 100 sheep only : supposing he w anted to m ake it sufficiently large to hold double th a t num ber, how m any addional hurdles would he have occasion for ?A nsw er.—Two. T here were 24 hurdles on each side of th e p en ; a h u rd le a t th e top. and an o th er a t th e b o tto m ; so th a t, by m oving one of th e sides a little back, and placing an additional hurdle a t the to p and b ottom , th e size of th e p en w ould be exactly doubled.
Countrywoman and eggs.
COUNTRYW OMAN carried eggs to a garrison, whore she h ad th ree guards to pass. She sold to th e first guard half th e n u m b er she had, and half a n egg m ore ; to th e second, th e half of w hat rem ained, and an half egg besides ; and to th e th ird guard she sold th e h alf of th e rem ainder, an d half a n o th er egg. W hen she arrived a t th e m arket-place, she had th ree dozen still to s e ll; how w as thisjpos- sible, w ithout breaking an y 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 b e sold w ith o u t breaking an y of th e eggs ? The possibility of all th is seem ing im possibility w ill be evident, w hen it is considered, th a ^ b y tak in g th e g re a te r half of an odd num ber, we tak e th e exact half + J. W hen th e countryw om an passed the first guard she had 295 eggs ; by sel ling to t h a t guard 148, w hich is th e half + J, she h ad 147 re m a in in g ; to th e second guard she disposed of 74, w hich is th e m ajo r half of 147 ; and, of course, a fte r selling 37 out of 74 to th e last, guard, she had still th ree dozen rem aining.
How to rub out twenty chalks at five times, rubbing out every time an odd one.
T
O do th is trick, you m u st m ake tw en ty chalks, or long strokes, upon a board as in th e m argin :T hen begin and count back- 1 ---w ards, as 20, 19, 18, 17, ru b 3
---o u t th ese f---our ; th e n pr---oceed saying, 16, 15, 14, 13, rub out th ese lour ; an d begin agaiu, 12, 11, 10, 9, and ru b out th e s e ; and proceed again, 8, 7, 6, 5, th e n ru b o u t th ese ; and lastly say, 4, 3, 2, 1, w hen th ese fou>r are rubbed out. The whole tw e n ty are rubbed out a t five tim es, and every tim e an odd one, th a t is, 17th, 13tli, 9th, 5th, and 1st.
This is a tric k which, if once seen, m ay be easily r e tu r n e d ; and th e puzzle a t first is, it n o t occurring im m ediately to th e m ind to begin to rub th em out backw ards. I t is as sim ple as any th in g possibly can be.
Odd or even.
I
iW E B Y odd n u m b er m u ltip lied by an J odd n um ber produces a n odd n u m b er ; every odd n u m b er m ultiplied by an even n um ber produces a n even n u m b er ; and every even n u m b er m ultiplied by an even n um ber also produces a n even n u m ber. So, again, a n even n u m b er added to an even num ber, an d a n odd n u m b er a d ded to a n odd n um ber, produce a n even n u m b e r; wliile a n odd an oven n u m b er added to g eth er produce an odd n u m ber.I f any one holds an odd n u m b er of counters in one hand, an d a n even n u m b e r in th e other, it is n o t difficult to d is cover in w hich han d the odd o r even n u m b er is. D esire th e p a rty to m u ltip ly th e n u m b er in th e rig h t hand by a n even n u m b er, and th a t in th e left h a n d by an odd n u m b er, th e n to add th e tw o sum s t o gether, an d te ll you th e la s t figure of tho p ro d u ct, if i t is even, th e odd n u m b er w ill be in th e rig h t hand ; an d if odd, in tho left h a n d ; th u s, supposing th e re are 5 counters in th e rig h t hand, and 4 in th e left hand, m u ltip ly 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 la st figure of w hich, 2, is even, an d th o odd n um ber w ill consequently be in th e rig h t hand.
The old woman and her eggs.
A
T a tim e w hen eggs w ere scarce, an old w om an who possessed som e r e m arkably good-laying hens, w ishing to 3 ---4 ---6 ---0 ---7 ---8 ---9 ---10 11 12 ---13 14 -15 ---10 - — 17 18 19 -20 ---O8 THE MAGIC OF NUMBERS-, OR,
oblige h e r neighbours, sen t h e r daughter round w ith a basket o£ eggs to th ree of th e m ; a t the first house, w hich was the squire’s, she left half th e 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 th e th ird she again left half of the rem ainder, and half a one over ; she retu rn e d w ith one egg in h er basket, n o t having broken any. R equired—th e num ber she set out w ith. A iis. 15 eggs.
The figures, up"to 100, arranged so as to make 505 in each column, when counted in ten columns perpendicularly and the same when counted in ten files horizontally. t r I fcb cp—T ca cnO K) h-I 50 to o O to to to to O o to <r>-n05** CTlrrahO t-l o 00w 05 05 05 05U0U0 05 CO "“'I 00 ^ Cl O to M CD CO 05 05 00 CO 03 CP 05 03 CD
The dice guessed unseen.
A
p air of dice being throw n, to find the num ber of p oints on each die w ith out seeing them . T ell th e person who casts th e die to double th e num ber of p oints upon one of th em , and add 5 to it th e n to m ultiply th e sum produced by 5, and to add to th e product th e n um ber of points upon th e o th er die. T his being done, desire him to tell you th e am ount, and, having throw n out 25, th e rem ainder w ill be a num ber consisting of tw o figures th e first of w hich, to th e left, is th e n u m b er of points on th e first die, and th e second figure, to th e right, th e num ber on the other. T h u s :Suppose th e n um ber of points of the first die w hich com es up to be 2, and th a t of th e oth er 3 ; th en , 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 p roduct will be 40; to which, if 3 ,'the num ber of points on the o th er die, be added, 48 will be produced, from which, if 25 be su b tracte d , 23 will re m a in ; the first figure of w hich is 2, th e n um ber of p oints on th e first die, and th e second figure 3, th e p um ber on th e other.
The Sovereign and the Sage.
A
sovereign being desirous to confer a liberal rew ard on one of h is courtiers w ho had perform ed som e very im p o rtan t service, desired him. to ask w hatever he th o u g h t proper, assuring him it should be granted. The courtier, who w as well acquainted w ith th e science of num bers, only requested th a t th e m onarch would give h im a q u an tity of w heat equal to th a t w hich would arise from one grain d subled sixty-threo tim es successively. 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 th e term s of a double progression, beginning w ith 1, m ay bo obtained by doubling th e la st term , and sub tractin g from it 1. The n um ber of the grains of w heat, therefore, in th e p re se n 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 re su lt by eight tim es 73728 we shall have 31274997411295 for th e n u m b er of th e bushels of w heat equal to th e above n um ber of grains, a q u an tity g reater th an the whole surface of the e a rth could produce in several years, and w hich in value would exceed all th e riches, perhaps on th e globe.December and May.
A
N old m an m arried a young w o m a n ; th e ir united ages am ounted to O. The m a n ’s age m ultiplied by 4 an d divi ded by 9, gives th e w om an’s age. W h at w ere th e ir respective ages ?Answ eb.—The m a n ’s age, 00 years 12 w e e k s; th e w om an’s age, 30 y ears 40 weeks.
COMIC AND CURIOUS PROBLEMS IN ARITHMETIC. 9 The Mathematical Fortune Teller.
P
R O C U R E six cards, and having ruled th em th e sam e as th e following dia gram s, w rite in th e figures neatly and legibly. 3 5 7 9 11 1 13 15 17 19 21 23 23 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 9 10 11 12 13 8 14 15 24 25 20 27 28 29 30 .31 40 ‘41 42 43 44 45 40 47 50 57 58 59 GO 13 17 18 19 20 21 1G 22 23 24 25 2G 27 28 29 30 31 48 49 50 51 52 53 54 55 5G 57 58 59 30 GOR equest th e person to give y ou th e cards containing th e num ber, an d th e n add th e rig h t h an d u p p er corner figures together, w hich will give th e correct answ er. F o r e x a m p le : suppose 10 is th e
I t is req u ired to tell th e n um ber th ought of by any person, th e num bers being co n tain ed in th e cards, and such num bers no t I to exceed GO. H ow is th is done ?
5 0 7 13 12 4 14 15 20 21 22 23 28 29 30 31 30 37 52 38 39 44 45 40 47 53 54 55 CO 13 3 G 7 10 11 2 14 15 18 19 22 23 20 27 30 31 34 35 38 39 42 43 40 47 50 51 54 55 58 59 33 34 35 3G 37 32 38 39 40 41 42 43 44 45 4G 47 48 49 50 51 52 53 54 55 50 57 58 59 GO 41
n um ber th o u g h t of, th e cards w ith 2 and 8 in th e corners w ill be given, w hich m akes th e answ er 10, and so on w ith the others.
10 't h e m a g ic o f n d m b e e s; o r,
The knowing Shepherd.
A
S H E P H E R D was going to m ark et w ith some -sheep, w hen he m e t 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 hadh e?
H e had 7 sheep : as m any m ore 7 ; half as m any m ore, 3J ; and 2 i ; m aking in all 20
.
The certain game.
T
WO persons agree to take, a ltern ately num bers less th an a given num ber, for exam ple, 11, and to add th e m to g eth er till one of th em has reached a certain sum such as 100. B y w h at m eans can one of th em infalliably a tta in to th a t num ber before th e o th er ?The whole artifice in th is consists in im m ediately m aking choice of th e n u m bers, 1, 12, 23, 34, and so on, or of a series w hich continually increases by 11, up to; ‘100. L e t us suppose th a t tho first person,' who kuows th e gam e, m akes choice of 1 ; it is evident th a t his adver sary, as he m u st count less th a n 11, can a t m ost reach-11, by adding 10 to it. The first will th e n tak e 1, w hich w ill m ake 12 and w hatever n u m b er th e second m ay add th e first will certainly w in, provided he continually add th e n um ber w hich form s tho com plem ent of th a t of h is adversary to 11; th a t is to say, if th e la tte r ta k e 8, lie m u st tak e 3 ; if 9 he m u st tak e 2 ; and so on. B y following th is m ethod he will infalliably a tta in to 89, and it w ill th e n be im possible for th e second to prevent him from getting first to 100 ; for w h at ever num ber th e second tak es he can a tta in only to 99; a fter w hich th e first m ay say— “ and 1 m ake 100.” B etw een tw o persons who are equally acquainted w ith the. gam e, he who begins m u st n e cessarily win.
The magical century.
I
F th e num ber 11 be m ultiplied by any one of th e nine digits, th e tw o figures of th e product w ill alw ays be alike, as appears in th e following exam ple ;—11 11 11 11 11 11 11 11 11
1 2 3 4 5 G 7 8 9
11 22 33 44 55 00 77 88 99
Now, if a n o th er person and yourself I have fifty counters a-piece, an d 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 e rm 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 stak e 1, and rem em bering th e order of th e above series, co n stan tly add to w h at he stakes as m any as w ill m ake one move th a n th e num bers 11, 22, 33, &o., of w hich it is composed, till you come to 89, after w hich your opponent cannot possibly reach tho even cen tu ry him self, o r p re v en t you from reaching it.
If your opponent has no know ledge of num bers, you m ay stake any o th er n u m ber first, u n d er 10, provided you su b se quently tak e care to secure ooie of th e last term s, 5G, 07, 78, &c.; or you m ay even le t him stake first, if you tak e care a fte r w ards to secure one of th ese num bers.
This exercise m ay be perform ed w ith o th er num bers, b u t, in order to succeed, you m u st divide th e n um ber to be a tta in e d by a num ber w hich is a u n it g reater th a n w h at you can stake each tim e, and tho rem ainder will th e n be th e n um ber you m u s t first stake. Suppose, for example th e n um ber to be a ttain ed be 52 (malting use of a pack of cards in stead of counters), and th a t you are never to add m ore th a n 0 ; th e n , dividing 52 by 7, th e rem ainder, w hich is 3, will be th e num ber w hich you m u st first s t a k e ; and w hatever y our opponent stakes, you m u st add as m uch to i t as w ill m ake it equal to 7, th e n u m b er by w hich you divided, and so in con tin u atio n .
The unlucky hatter.
A
P E R S O N w ent into a h a tt e r ’s shop and bought a h a t for a sovereign and gave in p ay m en t a five-pound note. The h a tte r called on a friend n ear by, who changed th e n o te for him , an d th e m an having received his change w ent his way. S hortly aiterw ard s th o ta ilo r’s friend d is covered th e n o te to be a counterfeit, and called upon th e h a tte r, who w as com pel led fo rth w ith to borrow five-pounds of a n o th er friend to redeem it w i t h ; so th e forged n o te w as left on th e h a tt e r ’s hands. The question is, w h at did he lose —was it five-pounds beside th e h a t or w as it five-pounds including th e h a t ?T his question is often given w ith nam es and circum stances as a real tran sactio n , tjndr if th e com pany know s su ch p erso n sso
COMIC AND CURIOUS PROBLEMS IN ARITHMETIC. 11
m uch th e b etter, as it serves to w ithdraw a tte n tio n from th e question ; and in a l m ost every ease th e first im p ressio n is,th at th e h a tte r lo st five-pounds besides the hat, though it is evident he was paid for th e h a t, and had he k e p t th e sovereign he needed only to have borrow ed four-pound additional to redeem th e note.
The basket of nuts.
A
P E R SO N rem arked th a t w hen he counted over his basket of n u ts, tw o by tw o, th ree by th ree, four by four, five by five, or six by six, th e re w as one r e m aining ; b u t w hen he counted th em by sevens, th e re was no rem ainder. H ow m any h ad lie ?The least com m on m ultiple of 2, 3, 4, 5, and C being GO, i t is evident, th a t if Cl were divisible by 7, it w ould answ er th e ••conditions of th e questions. T his not being th e case,how ever le t G 0X 2+ 1, G 0x3 + 1 , 0 0 x 4 + 1 , &c., be trie d successively, and it will be found th a t 301 = 0 0 + 5 + 1 , is divisible by 7 ;a n d consequently th is n u m ber answ ers th e conditions of th e question. If to th is we add 420, th e least com m on m ultiple of 2,3 , 4, 5, G and 7, th e sum 721 wHl be a n o th er a n s w e r; and by adding perpetually 420, we m ay find as m any answ ers as we please.
The united digits.
A
RRA N G E th e figures 1 to 9 in such order th a t, by adding th em tog eth er th ey am ount to 100.15
100
Quaint Questions.
W
H A T is the difference betw een tw en ty four q u a rt bottles, an d four and tw en ty q u a rt b o ttles ?A n s .— 50 q uarts difference.
AVhat th ree figures, m ultiplied by 4, will m ake precisely 5 ?
A n s.— 1£, o r l -25.
W hat.j is th e difference betw een six dozen dozen, an d half-a-dozen dozen ?
A n s .— 792: Six dozen dozen being 804 and half-a-dozen dozens, 72.
Place th ree sixes to g eth er, so as to m ake seven.
A n s.—Cg.
Add one to nine and m ake it tw en ty . A n s.—IX —cross th e I , it m akes XX. Place four fives so as to m ake six an d a half. A n s. 5J'5
A room w ith eight corners h a d a c a t in each corner, seven cats before each cat, and a eat on every c a t’s ta il. W h a t was th e to ta l n ninber of cats ? A n s. E ig h t cats.
Prove th a t seven is th e half of tw elve. A n s .—Place th e H om an figures on a p ie c e of paper, and draw a line th ro u g h th e m iddle of it, th e u p p er will be Y I1.
The council of ten.
T
E N cards or counters, nu m b ered from one to ten, or tho first te n p laying cards of any su it disposed in a circu lar form m ay be em ployed w ith g reat co n venience for perform ing th is feat. Tho accom panying figure show s th e cards th u s arranged, n u m b er one, or th e ace, designated by A. and th e te n by K. t) 0 2 B D 4 1 A 10 5 10 K FG 9 1 G 7 II 8H aving placed th e cards in th e above order, desi e a bystander to th in k of a card or num ber, and w hen he h as done so, to touch any oth er card or num ber. R eq u est him th en to add to tho n um 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 a tu ra l num bers, beginning a t tho card lie - touched, and assigning it th e n u m b er of th e card he th o u g h t of. B y c o u n t ing in th is m anner, he w ill end a t th e numb&r o r card ho th o u g h t of, and consequently you w ill im m ediately know it.
T hus, for exam ple, suppose tho person h ad th o u g h t of 3 C, and touched G F ; th en , if 10 be added to G, th e sum w ill ba 1G; an d if th a t n u m b er be counted from F , th e n u m b er touched, tow ards E D 15 C A, and so o n , in th e retrograde order, counting F th ree, th e n um ber th ought
12 THE MAGIC OF NUMBERS; OR,
of, E five, D six, and so round to six teen, th a t num ber will term in ate a t 0 , show ing th a t the person th o u g h t of 3, the n um ber w hich corresponds to C.
A greater or less num ber of cards or counters m ay be em ployed a t pleasure ;
b u t in every in stan ce th e whole num ber of cards m u st bo added to th e n um ber of th e card touched.
This trick done on th e dial of a w atch, using th e figures th ereo n , 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 T hree casks, and loud exclaim s “ 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 a c h w ondering if an ow ner m ig h t be found : B u t no t a soul was th ere—th e coast w as clear, So to th e barrels th e y a t once drew near, A nd b o th agree w hatever m ay be th ere I n friendly p artn e rsh ip th e y ’ll fairly share. Two they find em pty, b u t th e o th 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 w as b randy. ’Tis ta ste d and ap p ro v e d ; th e ir lips th ey sm ack, And each pronounces ’tis th e fam ed Cognac. “ W on’t we have m any a jolly night, m y bo y l May no ill luck our pre se n t hopes d e s tro y ! ” ’Twas fo rtu n ate one knew th e m athem atics, And had a sm atterin g of h y d ro s ta tic s; T hen m easured ho th e casks, and said, “ I sea This is eight gallons, those are five and th ree. ” The question th e n was how th ey m ight divide The brandy, so th a t each should be supplied W itli ju s t four gallons, n eith er less nor m ore. W ith eight, and five, and th ree th e y puzzle sore, F illed up tlie five—filled up th e th ree, in v ain ; A t length a h appy th o u g h t cam e o ’er th e brain Of one ; ’tw as done, and each w ent liome content, And th e ir good dam es declared t ’was excellent. W ith those th ree casks th ey m ade division tru e ; I found th e puzzle out, say, friend, can you ? The five-gallon barrel w as filled first,
and from th a t th e three-gallon b arrel, thus leaving tw o gallons in th e five-gallon b a r r e l; th e three-gallon barrel w as th e n em ptied into th e eight-gallon b arrel, and th e two gallons poured from th e five-gal lon barrel into th e em pty three-gallon b arrel ; th e five-gallon barrel w as th e n filled,?and one gallon poured into the three-gallon b arrel, therefore leaving four gallons in th e five-gallon b arrel, one gallon in th e eight-gallon barrel, and th re e gallons in th e three-gallon barrel, w hich w as th e n em ptied in to th e eight- gallon barrel. Thus each person had
four gallons of b randy in th e eig h t and five-gallon barrels respectively.
The Fox, Goose and Corn.
A
countrym an having a Fox, a Goose, and a peek of Corn, cam e to a river, w here it so happened th a t he could carry b u t one over a t a tim e. Now as no tw o w ere to be left to g eth er th a t m ig h t des tro y each other, he w as 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 a t th e fo x ; y e t th e fox can e a t th e goose, and th e goose eatoow tc a n d cuiiiotra p r o b l e m s in a e i t h m e t i c . 10 th e c o m .” II<5W shall he carry th em over,
th a t th ey shall n o t destroy each o th er ? L e t h'im first tak e over th e Goose, leaving th e F ox and Corn ; th e n le t him tak e over th e Fox and b ring th e Goose b a c k ; th e n ta k e over the C o rn ; and lastly tak e over th e Goose again.
The visitors to the Crystal Palace.
I
N a fam ily consisting of 8 young people, it w as agreed th a t 3 a t a tim e should v isit th e C rystal Palace, and th a t th e v isit should be rep eated each day as long as a different trio could be selected. In liow m any days were th e possible com binations of 3 out of 8 com pleted ?Wo m u st m ultiply 8 X 7 X 0 , and also 3 x 2 x 2 , and divide th e product of the form er, 330, by th e p ro d u ct of th e la tte r, 6 ; tho resu lt is 5G, th e n um ber of visits, a different th ree going each tim e. So m uch gratified were th ey w ith th e results of th e ir agreem ent, th a t th ey w ished to be allowed a n o th er series of visits, to be continued as m any daj's as th ey could group 3 tog eth er in different order w hen starting. If P aterfam ilias h ad granted such perm ission he would have had to w ait 50 m ultiplied by 3 x 2 x 1 , ’<>r 330 days, before th is “ now series” of visits would have come to & fin is .
How many changes can be given to 7 notes of a piano ?
riAHAT is to say, in how m an y ways can JL 7 keys be stru ck in succession, so th a t th ere shall bo som e difference in th e order of the n otes each tim e ?
The resu lt of m ultiplying 7 X G X 5 X 4 X 3 X 2 X 1 is 5,040, th e n u m b er of changes.
The Arithmetical Triangle.
I
if llS nam e h as been given to a con triv an ce said to have originted w ith th e fam ous Pascal, or to have been p erfected by him . 1 2 1 3 3 1 4 6 4 1 . 5 10 10 5 1 6 15 20 15 6 1 7 21 35 35 21 7 1 8 28 56 70 5G 28 8 1 &c. &c.T his p eculiar series of num bers is th u s form ed : W rite dow n th e num bers 1, 2, 3, &c., as far as you please, in a vertical row. On th e rig h t h an d of 2 place 1, add th em to g eth er, and place 3 u n d er th e 1 ; th en 3 added to 3 = 6, w hich place u n d er 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 is form ed from th e second in a sim ilar way. This triangle has th e p ro p e rty of inform ing us, w ith o u t th e tro u b le of c a l culation, how m any com binations can be m ade, tak in g any n u m b er a t a tim e out of a larger num ber.
Suppose th e question w ere th a t ju st given ; how m an y selections can be m ade of 3 a t a tim e out of 8 ? On th e h o ri zontal row com m encing w ith 8,- look for tho th ird n u m b er ; th is is 06, w hich is
th e answ er.
How many different deals can Da made with 13 cards out of 52.
M10 discover th is we m u st m ake a con-
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tinued m ultiplication of 52 X 51 X 50X4 9 x 4 8 X47 X 4GX4 5 x 4 4 X 4 3 X 4 2 X 41 X40, being 13 term s for th e 13 cards,
also a continued m u ltip licatio n of 13-f-12
X11X10X9X 8 X7 X 6 x 5 x 4 x 3 x 2 x l ,
and, having found th e tw o p roducts, we m u st divide one by th e other, <inl the q u o tien t is th e n um ber of different deals out of 52 cards. This “ su m ,” th it looks so form idable w ith n a tu ra l figures, is a very sh o rt one by logarithm s.
The Three Graces.
T
H R E E articles, or th ree nam es in scri bed on cards, having been d istrib u ted betw een th re e persons, you are to tell w hich article or card each p erson lias.D esignate th ree persons in y our own m ind, as 1st, 2d, and 3d, and tho throe articles, A, e, i. Provide 24 counters, and give 1 to th e first person, 2 to th e 2d 3 to th e 3d. Place th e rem aining 18 on th e table. R equest th a t th e th re e p e r sons will d istrib u te am ong them selves th e th ree articles, and, th at, having done so, tho person who has th e one w hich you have secretly denoted by a,
will tak e as m a n y counters as he m ay have already ; th e holder of E m u s t take tw ice as m an y as lie m ay have ; and the holder of I m u st ta k e four tim es as m any. T h en leave th e room , in order th a t th e
ii
THE MAGlC o r NUMBURil J Oil,distribution of articles and of counters m ay bo m ade unobserved by you. W e will suppose th a t d ie th re e articles are th ree cards, on w hich are th e w ords Clara, Eosa, E m ily, w hich you w ill y o u r self secretly denote by th e le tte rs A, e, i.
Suppose also th a t in th e division th e first person has E m ily (i), th e second has Clara (a) , and th e th ird has E osa (e),
th e n th e 1st will tak e four tim es as m any counters as he lias (1), and will therefore talce 4 ; j th e 2d will ta k e as m any as he lias (2), and w ill therefore tak e 2 ; the 3d will tak e 0, being tw ice as m any as he lias (8). On the table will be left six counters. The d istribution having been m ade, you will re tu rn an d observe th e num ber of counters on th e table, from w hich you can find who is th e holder of each card by th e following m ethod.
I t is plain th a t if th e cards hold b y the 1st and 2d can be told, th a t held by the
3d will be know n. I t will bo found th a t only six num bers can rem ain, viz. 1. 2, 8, 5, 0. 7, never four, and n ever m ore th a n 7. Now th e G com binations of a, e, and i,h e rc given, rep resen t th e articles held by th e 1st and 2<1 persons.
1 2 3 4 5 0 7
ae ea ai — ci ia ie I n the case supposed, G counters being on th e table, th e combination-ia. indicates th a t th e first person lias th e card you have called i ( l i i t i l l y ) , th e 2d has a(Clara), so
th a t th e 3d has e (Eosa),
I n order to recollect th e com binations of A, e, an d I, it will be b est to keep ia m em ory some 7 w ords w hich form a se n tence, and w hteh contain these vowels in th e order ju s t given.
O uryoungfriends can am use them selves in form ing a sentence for them selves, b u t - as exam ples we supply th ree.
1 2 3 4 5 6 7
a e e a a i — e i i a i e
J a m e s easy adm ires now reigning w ith a bride Anger, fear, p ain m ay be hid w ith a sm ile G raceful E m m a, charm ing she reigns in all circles
Or, if they prefer L atin , th ey can use th e , 1 \ 2 3 5 6 7 p en ta m e te r m ade up by th e in v e n to r of Salve c erta anim ae sem ita vita quieg.
th is beautiful p astim e : j
Another Method.
r r i l E perform er m u st m en tally distin-
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guish th e articles by the le tte rs a , b , c ,and th e persons as 1st, 2d, and 3d. The p e r sons having m ade th e ir choice give 12 counters t o th e 1st, 24 to th e 2d, and 3G
t o th e 3d. T hen req u est th e 1st person to add to g eth er th e half of th e counters of th e person who has chosen a , th e 3d of th e person who has chosen b , an d th e 4 th of those of th e person w ho has chosen c, and th e n ask th e sum , w hich m u s t be cith er 23, 24, 25, 27, 28, or 29, as in th e following ta b l e ,
F irst. Second. Third.
12 24 36 A ii c 23 A 0 B 24 11 A C 25 C A Ti 27 Ji 0 A 28 <J B A . 20
This tab le show a-that if th e sum bo 25,
f o r exam ple, th e 1st person m u st have
c h o s e n b, t h e 2d a , and th e 3d c; o r i f i t
be 28 th e 1st m u s t have chosen b, the 2 d c, and th e 3 d a.
Another Method.
T
H IIE E th in g s h a v in g b een divided be tw een th re e persons, you arc to determ ine th e holder of cach.Call th e persons in y our m in d 1st, 2d, 3d.
Give to th e 1st a card on w hich you have w ritte n th e n u m b er 12 ; to the 2d th e n u m b er 24; to th e 3d 36.
T he th re e th in g s you denote as a,e j.
To sim plify it you m ay have th re e cards w ith a nam e upon each, of w hich th e in itial le tte rs are A, E , r, as A nna, E m m a, Isabel.
R equest your friends to divide betw een th em th e th ree articles, and th e n to add to g eth er certain p a rts of th e n u m b ers on th e ir cards, as follows :
W hoever has A m u s t supply one half of th e n um ber on h is c a rd ;
c o iu c A s a g u iu o ttf i'i;ocLJ;.Mti i s AiUt iim e ij.cj, W hoever lias e m u st supply o n e -th ird ;
whoever lias I m u st supply one-fourth. This half, th ird , an d fo u rth having been added to g eth er, th e sum m u st be announced to you on y o u r r e t u r n ; and from th is num ber you can te ll who has A,
who has e, an d who has r.
I f th e ^ o . is th e 1st has th e 2nd has th e 3rd has
23 A U I 24 A I V. & i i: A I 27 l A i: 28 Y. 1 A 29 I E A
The sum which will be given to you can bo one of six only. There are only six ways in which th e articles can be divided, an d th ere is a definite num ber for each of them .
Tho num ber 26 can n ever occur, an d to recollect tho six which do occur, and which you perceive are consecutive, you need ta k e n o te only of w h at th e 1st and 2nd persons have.
23 24 15 26 27 28 29
a e a i e a — i a e i ie
I f you m ake up a lino of good (or bad) English, having th e vowels in th e order hero given, you will find it will aid you in th e ir recollection. W e give one as a specim en:
a e a i ea — i a e i ie
B rave, dashing sea, like a g ia n t revives itself.
The United Digits. j
O
N page eleven we show ed how to place th e figures, 1 to 9, so t h a t th e y m ig h t by adding th em to g e th e r a m o u n t to 100. U n til now i t h as been ! believed t h a t th e re was only one way to ■ do th is w ithout using fractions. W e give a n o th e r :32 57
100.
To Tell what Figure a Pei’son
has struck out of the Sum of
Two given Numbers.
A
SSUM E those num bers only th a t arc divisible by 9 ; such, for instance, as 18, 36, 63, 81, 117, 126, 162, 261, 315,360, 891, &c. A sk a p erson to cliooso any tw o of th ese num bers, a n d a fte r a d d in g th em to g e th e r in h is m ind, strik e o u t from th e sum any one of th e figures h e pleases. Then desire him to te ll you tho sum of th e rem aining figures ; a n d it follows th a t th e n u m b er which you are obliged to a d d to th is am o u n t, to m ake i t 9 o r 18, is th e one he s tru c k o ut. T h u s :
Suppose ho choose th e num bers 117 > an d 360, m aking to g e th e r 477, an d th a t he strik e o u t th e centre figure, th e tw o o th er figures will, a d d ed to g e th e r, p ro duce 11, which, to m ake 18, requires 7, th e n um ber stru c k out.
Dividing the Beer,
D
u rin g th e siege of. Sebastopol, when th e troops were on ‘ sh o rt allowance,’ a can of eight p in ts of p o rte r was ordered to be equally divided betw een tw o messes ; b u t having only a five p in t can, a n d one t h a t held th re e p in ts, i t was found im possible to m ake th is division, till one of tho clever sappers suggested th e follow ing m e th o d ; and, to u n d e rsta n d it, we will p u t down th e co n te n ts of each of th ree cans a t each stage of th e process ; com m encing w ith :i s r > :i jits pi? I'ls
Tho 8-pint can full, an d th e
others em pty - - - 8 0 0 1. F illed th e ii-pint can - - 3 5 0 2. Filled th e 3-pint can from th e
5-pint - - - - 3 2 3
3. P o u r tho contents of 3-pint
w ith th e 8-pint - - - 6 2 0 4. T ransfer th e 2-pints from th e
5-pints to th e 3-pint - - 6 0 2 5. F ill th e 5-pint from tho
8-pint - - - - 1 5 2
0. F ill up th e 3-pint from tho
5-pint - - - - 1 4 3
7. P o u r th e 3-pints in to th e
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 m en had to w ait, and th e resu ltin g flat condition of th e beer.
The Difficult Case of Wine.
A
G EN TLEM A N h a d a b o ttle contain ing 12 p in ts of wine, 6 of which he was desirous of giving to a friend, b u t lie C h ad n o th in g to m easure it w ith, except 2 o th e r b o ttles, one of 7 pints and th e o th er of 5. Plow did ho contrive to p u t 6 p in ts in to th o 7-pint b o ttle ?1ft THE MAGIC OS’ iroMBEES ; OR)
12-pt. 7-pt. 5-pt.
Before lie com m enced, th e con te n ts of th e bottles w ere - 12 0 0 1. H e filled the 5-pint - - 7 0 5 2. E m p tied th e [5-pint into the
7-pint - - - - 7 5 0
\ Filled again the 5-pint from th e 12-pint - - - - 2 5 5 4. Filled up th e 7-pint from
th e 5 - - - 2 7 3 5. E m ptied the 7-pint into th e
12-pint - - - - 9 0 3
0. P oured th e 3 p in ts from the 5 into th e 7 - - - 9 3 0 7. F illed th e 5-pint from th e
12-pint - - - - 4 3 5
8. Filled up th e 7-pint from th e 5-pint - - - - 4 7 1 9. E m p tied th e 7-pint into th e
12-pint - - - - 11 0 1 10. Poured 1 p in t from th e
5-p in t into th e 7-5-pint - - 11 1 0 11. Filled th e 5-pint from the
12-pint - - - - 0 1 5
12. P oured th e contents of th e 5-pint into th e 7-pint - - 6 6 0
The wine and the tables.
A
C ER TA IN hotel-keeper was dexterous in contrivances to produce a large' appearance w ith sm all m eans. In th e dining-room w ere th ree tables, betw een w hich he could divide 21 bottles, of w hich 7 only were full, 7 half full, and 7 apparently ju st em ptied, and in such a m an n er th a t each tab le had tho sam e num ber of b o ttles, and th e sam e quan tity of wine. H e did th is in tw o w ays.T a b le F u ll H alf-fu ll E m p ty . 1 . 2 3 2 2 . 2 3 2 3 . 3 1 3 T a b le F u ll H alf-fu ll E m p ty . 1 . 3 1 3 2 . 3 1 3 3 . 1 5 1
H o also perform ed a sim ilar exploit w ith 24 bottles, 8 full, 8 half-full, and 8 em pty. T ab le F u ll H alf-full E m p ty . 1 . 3 2 3 2 . 3 2 2 3 . 2 4 3 T ab le F u ll H alf-fu ll E m p ty . 1 . 2 4 2 2 . 2 4 2 I . 4 0 4
Also w ith 27 bottles, 9 full, 9 hali* ful, and 9 e m p ty : T ablo F u ll H alf-fu ll E m p ty . 1 .2 5 2 2 . 3 3 3 3 . 4 1 4 T ab le F u ll H alf-full Em pi 1 .1 7 1 2 4 1 4 3 '. 4 1 4 i
The three Travellers.
1
TH REE m en m e t a t a caravansary or . inn, in P e r s ia ;. and tw o of th em brought th e ir provisions along w it'i th em according to the custom of the co u it r y ; but th e th ird n o t having provided any, proposed to th e others th a t th ey should e a t 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 w hich th e travellers a te 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 w as r e ferred to the judge, who decided im p a r tia lly .'"'W h a t w as his decision ?A t first sight it would seem th a t the m oney should be divided according to th e 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 w ould leav e 2 J loaves fu r nished th e stran g er by A ; and 3—2§ = | furnished by B, hence 2 | to | = 7 to 1, is th e ra tio n in w hich 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 e n tho di vision will be according to am ounts fu r nished.
Which counter has been thought of out of sixteen.