Part 22
........................................... . . . . . . . . . . . . . . . . 18 . 25 . 2 . 9 . . . . . . . . . +----------------------------------+....... | | | | | | . | | | | | | . | 17 | 24 | 1 | 8 | 15 | 17 . | | | | | | . |------+------+------+------+------+....... | | | | | | . | | | | | | . | 23 | 5 | 7 | 14 | 16 | 23 . | | | | | | . |------+------+------+------+------+....... | | | | | | . | | | | | | . | 4 | 6 | 13 | 20 | 22 | 4 . | | | | | | . |------+------+------+------+------+....... | | | | | | . | | | | | | . | 10 | 12 | 19 | 21 | 3 | 10 . | | | | | | . |------+------+------+------+------+....... | | | | | | . | | | | | | . | 11 | 18 | 25 | 2 | 9 | . | | | | | | . +------+------+------+------+------+....... ]
Magic squares of this class, however large in the number of compartments, can be easily filled up by attending to these two rules.
We give opposite, a seven-placed square.
There are various other kinds of magic squares; but explanations of them would be too lengthy for our work.
The invention of these contrivances has been traced back to the early ages of science, and talismanic properties were attributed to them. Modern philosophers have amused themselves in bringing them to perfection, and none has contributed so much as "the model of practical wisdom," Dr. Franklin.
THE SQUARE OF GOTHAM.
The wise men of Gotham, famous for their eccentric blunders, once undertook the management of a school; they arranged their establishment in the form of a square divided into 9 rooms. The playground occupied the center, and 24 scholars the rooms around it, 3 being in each. In spite of the strictness of discipline, it was suspected that the boys were in the habit of playing truant, and it was determined to set a strict watch. To assure themselves that all the boys were on the premises, they visited the rooms, and found three in each, or 9 in each row. Four boys then went out, and the wise men soon after visited the rooms, and finding 9 in each row, thought all was right. The four boys then came back, accompanied by four strangers; and the Gothamites, on their third round, finding still 9 in each row, entertained no suspicion of what had taken place. Then 4 more "chums" were admitted; but the clever men, on examining the establishment a fourth time, still found 9 in each row, and so came to an opinion that their previous suspicions had been unfounded. How was all this possible?
The following figures represent the contents of each room at the four different visits; the first, at the commencement of the watch; the second, when four had gone out; the third, when these 4, accompanied by another 4, had returned; and the fourth, when 4 more had joined them.
I. II. III. IV. ___________ ___________ ___________ ___________ | | | | | | | | | | | | | | | | | 3 | 3 | 3 | | 4 | 1 | 4 | | 2 | 5 | 2 | | 1 | 7 | 1 | |___|___|___| | __|___|___| | __|___|___| | __|___|___| | | | | | | | | | | | | | | | | | 3 | | 3 | | 1 | | 1 | | 5 | | 5 | | 7 | | 7 | |___|___|___| | __|___|___| | __|___|___| | __|___|___| | | | | | | | | | | | | | | | | | 3 | 2 | 3 | | 4 | 1 | 4 | | 2 | 5 | 2 | | 1 | 7 | 1 | |___|___|___| | __|___|___| | __|___|___| | __|___|___|
On each change the boys arranged themselves in the rooms in such a manner that, when the corner rooms were counted as a part of two rows, each entire row of three rooms contained the same number of boys. The illusion of the wise men was due to their mistake in counting each corner room twice.
THE MATHEMATICAL BLACKSMITH.
A blacksmith had a stone weighing 40 lbs. A mason coming into the shop, hammer in hand, struck it and broke it into four pieces. "There," says the smith, "you have ruined my weight." "No," says the mason, "I have made it better, for whereas you could before weigh but 40 lbs. with it, now you can weigh every pound from 1 to 40." Required size of the pieces?
_Ans._ 1, 3, 9, 27; for in any geometrical series proceeding in a triple ratio, each term is 1 more than twice the sum of all the preceding, and the above series might proceed to any extent. In using the weights, they must be put in one or both scales as may be necessary: as to weigh 2, put 1 in one scale, and 3 in the other.
CURIOUS PROPERTIES OF SOME FIGURES.
Select any two numbers you please, and you will find that one of the two, their amount when added together, or their difference, is always 3, or a number divisible by 3.
Thus, if the numbers are 3 and 8, the first number is 3; let the numbers be 1 and 2, their sum is 3; let them be 4 and 7, the difference is 3. Again, 15 and 22, the first number is divisible by 3: 17 and 26, their difference is divisible by 3, &c.
All the odd numbers above 3, that can only be divided by 1, can be divided by 6, by the addition or subtraction of a unit. For instance, 13 can only be divided by 1; but after deducting 1, the remainder can be divided by 6; for example, 5 + 1 = 6 ; 7 - 1 = 6; 17 + 1 = 18; 19 - 1 = 18; 25 - 1 = 24, and so on.
If you multiply 5 by itself, and the quotient again by itself, and the second quotient by itself, the last figure of each quotient will always be 5. Thus 5 × 5 = 25; 25 × 25 = 125; 125 × 125 = 625, &c. Again, if you proceed in the same manner with the figure 6, the last figure will constantly be 6; thus, 6 × 6 = 36; 36 × 36 = 216; 216 × 216 = 1,296, and so on.
To multiply by 2 is the same as to multiply by 10 and divide by 5.
Any number of figures you may wish to multiply by 5, will give the same result if divided by 2--a much quicker operation than the former; but you must remember to annex a cipher to the answer where there is no remainder, and where there is a remainder, annex a 5 to the answer. Thus, multiply 464 by 5, the answer will be 2320; divide the same number by 2, and you have 232, and as there is no remainder you add a cipher. Now, take 357, and multiply by 5--the answer is 1785. On dividing 357 by 2, there is 178, and a remainder; you therefore place 5 at the right of the line, and the result is again 1785.
There is something more curious in the properties of the number 9. Any number multiplied by 9 produces a sum of figures which, added together, continually makes 9. For example, all the first multiples of 9, as 18, 27, 36, 45, 54, 63, 72, 81, sum up 9 each. Each of them multiplied by any number whatever produces a similar result; as 8 times 81 are 648, these added together make 18, 1 and 8 are 9. Multiply 648 by itself, the product is 419,904--the sum of these digits is 27, 2 and 7 are 9. The rule is invariable. Take any number whatever and multiply it by 9; or any multiple of 9, and the sum will consist of figures which, added together, continually number 9. As 17 × 18 = 306, 6 and 3 are 9; 117 × 27 = 3,159, the figures sum up 18, 8 and 1 are 9; 4591 × 72 = 330,552, the figures sum up 18, 8 and 1 are 9. Again, 87,363 × 54 = 4,717,422; added together the product is 27, or 2 and 7 are 9, and so always. If any row of two or more figures be reversed and subtracted from itself, the figures composing the remainder, will, when added horizontally, be a multiple of nine:
42 886 326 24 688 1623 -- --- ---- 18 - 9 × 2. 198 - 9 × 2. 1638 - 9 × 2.
If a multiplicand be formed of the digits in their regular order, omitting the 8, a multiplier may be found by a rule, which will give a product, each figure of which shall be the same. Thus if 12345679 be given, and it be required to find a multiplier which shall give the product all in 2, that multiplier will be 18: if in 3, the multiplier will be 27: if all 4, it will be 36--and so forth.
12345679 12345679 12345679 18 27 36 -------- -------- ------- 98765432 86419753 74074074 12345679 24691358 37037037 --------- --------- --------- 222222222 333333333 444444444
The rule by which the multiplier is discovered (but which we do not attempt to explain) is this: Multiply the last figure (the 9) of the multiplicand by the figure of which you wish the product to be composed, and that number will be the required multiplier. Thus, when it was required to have the product composed of 2, the 2 multiplied by 9 gives 18, the multiplier: 3 multiplied by 9 gives 27, the multiplier to give the product in 3; &c.
If a figure, with a number of ciphers attached to it, be divided by 9, the quotient will be composed of one figure only, namely, the first figure of the dividend, as--
9)600,000 9)40,000 ---------- --------- 66,666-6 4,444-4
{ 9)549 If any sum of figures can be divided by 9 as, {------ { 61
the amount of these figures, when added together, can be divided by 9:--thus, 5, 4, 9, added together, make 18, which is divisible by 9. If the sum 549 is multiplied by any figure, the product can also be divided by 9, as--
} { 3 549 } { 2 6 } { 9 ------ } And the amount of the figures of { 4 9)3294 } the product can also be divided by { ---- ----- } 9, thus, { 2)18 366 } { --- } { 9
To multiply by 9, add a cipher, and deduct the sum that is to be multiplied: thus,
43,260 } { 4,326 4,326 } Produces the same result as { 9 ----- } { ---- 38,934 } { 38,934
In the same manner, to multiply by 99, add two ciphers; by 999, three ciphers, &c. These properties of the figure 9 will enable the young arithmetician to perform an amusing trick, quite sufficient to excite the wonder of the uninitiated.
Any series of numbers that can be divided by 9, as 365, 472,821,754, &c., being shown, a person may be requested to multiply secretly either of these series by any figure he pleases, to strike out one number of the quotient, and to let you know the figures which remain, in any order he likes; you will then, by the assistance of the knowledge of the above properties of 9, easily declare the number which has been erased. Thus, suppose 365,472 are the numbers chosen, and the multiplier is six; if then, 8 is stuck out, the numbers returned to you will be
} 2 } 1 } 9 365472 } 2 6 } 3 ------ } 2 219232 } -- } 19
The amount of these numbers is 19; but 19, divided by 9, leaves a remainder of 1; you, therefore, want 8 to complete another 9: 8, then, is the number erased.
The component figures of the product made by the multiplication of every digit into the number 9, when added together, make NINE.
The order of these component figures is reversed after the said number has been multiplied by 5.
The component figures of the amount of the multipliers (viz. 45,) when added together, make NINE.
The amount by the several products, or multiples of 9 (viz. 405,) when divided by 9, gives for a quotient, 45; that is, 4 + 5 = NINE.
The amount of the first product (viz. 9,) when added to the other product, whose respective component figures make 9, is 81; which is the square of NINE.
The said number 81, when added to the above mentioned amount of the several products, or multiples of 9 (viz. 405) makes 486, which, if divided by 9, gives for a quotient 54: that is, 5 + 4 = NINE.
It is also observable, that the number of changes that may be rung on nine bells is 362,880; which figures, added together, make 27; that is, 2 + 7 = NINE.
And the quotient of 362,880, divided by 9, will be 40,320; that is 4 + 0 + 3 + 2 + 0 = NINE.
If number 37 be multiplied by any of the progressive numbers arising from the multiplication of 3 with any of the units, the figures in the quotient will be similar, and the result may be known beforehand by merely inspecting the progressive numbers, thus, 3, 6, 9, 12, 15, 18, 21, 24, 27, &c., are the progressive numbers formed by 3 multiplied by the units 1 to 9; and the result of the multiplication of any of these numbers with 37 may be seen in the following examples:--37 × 3 = 111; 37 × 6 = 222; 37 × 12 = 444; 37 × 24 = 888; by which it appears that the numbers of which the quotient is formed are the same as the units by which number 3 was multiplied to obtain the respective progressive numbers. Thus--3 multiplied by 2 is equal to 6, and 37 multiplied by 9 is equal to 222; so, again, 4 multiplied by 3 produces 12, and 37 multiplied by 12 is equal to 444, and so on.
THE INDUSTRIOUS FROG.
There was a well 30 feet deep, and at the bottom a frog anxious to get out. He got up 3 feet per day, but regularly fell back 2 feet at night. Required the number of days necessary to enable him to get out?
The frog appears to have cleared one foot per day, and at the end of 27 days, he would be 27 feet up, or within 3 feet of the top, and the next day he would get out. He would therefore be 28 days getting out.
THE COUNCIL OF TEN.
Ten cards or counters, numbered from one to ten, or the first ten playing cards of any suit disposed in a circular form may be employed with great convenience for performing this feat. The accompanying figure shows the cards thus arranged, number one, or the ace, designated by A, and the ten by K.
3 C 2 B D 4 1 A E 5 10 K F 6 9 I G 7 H 8
Having placed the cards in the above order, desire a bystander to think of a card or number, and when he has done so, to touch any other card or number. Request him then to add to the number of the card touched the number of the cards employed, which in this case is ten. Then desire him to count the sum in an order contrary to that of the natural numbers, beginning at the card he touched, and assigning it the number of the card he thought of. By counting in this manner, he will end at the number or card he thought of, and consequently you will immediately know it.
Thus, for example, suppose the person had thought of 3 C, and touched 6 F; then, if 10 be added to 6, the sum will be 16; and if that number be counted from F, the number touched, towards E D B C A, and so on, in the retrograde order, counting F three, the number thought of, E five, D six, and so round to sixteen, that number will terminate at C, showing that the person thought of 3, the number which corresponds to C.
A greater or less number of cards or counters may be employed at pleasure; but in every instance the whole number of cards must be added to the number of the card touched.
THE TWO TRAVELERS.
Two travelers trudged along the road together, Talking, as Yankees do, about the weather; When, lo! beside their path the foremost spies Three casks, and loud exclaims "A prize, a prize!" One large, two small, but all of various size. This way and that they gazed, and all around, Each wondering if an owner might be found: But not a soul was there--the coast was clear, So to the barrels they at once drew near, And both agree whatever may be there In friendly partnership they'll fairly share. Two they find empty, but the other full, And straightway from his pocket one doth pull A large clasp-knife. A heavy stone lay handy, And thus in time they found their prize was brandy. 'Tis tasted and approved: their lips they smack, And each pronounces 'tis the famed Cognac. "Won't we have many a jolly night, my boy! May no ill luck our present hopes destroy!" 'Twas fortunate one knew the mathematics, And had a smattering of hydrostatics; Then measured he the casks, and said, "I see This is eight gallons, those are five and three." The question then was how they might divide The brandy, so that each should be supplied With just four gallons, neither less nor more. With eight, and five, and three they puzzle sore, Filled up the five--filled up the three, in vain; At length a happy thought came o'er the brain Of one: 'twas done, and each went home content, And their good dames declared 'twas excellent. With those three casks they made division true; I found the puzzle out, say, friend, can you?
The five-gallon barrel was filled first, and from that the three-gallon barrel, thus leaving two gallons in the five-gallon barrel; the three-gallon barrel was then emptied into the eight-gallon barrel, and the two gallons poured from the five-gallon barrel into the empty three-gallon barrel; the five-gallon barrel was then filled, and one gallon poured into the three-gallon barrel, therefore leaving four gallons in the five-gallon barrel, one gallon in the eight-gallon barrel, and three gallons in the three-gallon barrel, which was then emptied into the eight-gallon barrel. Thus each person had four gallons of brandy in the eight and five-gallon barrels respectively.
ARITHMETICAL PUZZLE.
If from 6 you take 9, and from 9 you take 10; and if 50 from 40 be taken, there will just half a dozen remain.
ANSWER.
From SIX From IX From XL Take IX Take X Take L --- -- -- S I X Remains.
THE MONEY GAME.
A person having in one hand a piece of gold, and in the other a piece of silver, you may tell in which hand he has the gold, and in which the silver, by the following method: Some value, represented by an even number, such as 8, must be assigned to the gold; and a value represented by an odd number, such as three, must be assigned to the silver; after which, desire the person to multiply the number in the right hand by any even number whatever, such as 2, and that in the left by an odd number, as 3; then bid him add together the two products, and if the whole sum be odd, the gold will be in the right hand, and the silver in the left; if the sum be even, the contrary will be the case.
To conceal the artifice better, it will be sufficient to ask whether the sum of the two products can be halved without a remainder; for in that case the total will be even, and in the contrary case odd.
It may be readily seen, that the pieces, instead of being in the two hands of the same person, may be supposed to be in the hands of two persons, one of whom has the even number, or piece of gold, and the other the odd number, or piece of silver. The same operations may then be performed in regard to these two persons, as are performed in regard to the two hands of the same person, calling the one privately the right, and the other the left.
THE PHILOSOPHER'S PUPILS.
To find a number of which the half, fourth, and seventh added to three shall be equal to itself.
This was a favorite problem among the ancient Grecian arithmeticians, who stated the question in the following manner: "Tell us, illustrious Pythagoras, how many pupils frequent thy school?" "One half," replied the philosopher, "study mathematics, one fourth natural philosophy, one seventh observe silence, and there are three females besides."
The answer is, 28: 14 + 7 + 4 + 3 = 28.
TO DISCOVER A SQUARE NUMBER.
A square number is a number produced by the multiplication of any number into itself; thus, 4 multiplied by 4 is equal to 16, and 16 is consequently a square number, 4 being the square root from which it springs. The extraction of the square root of any number takes some time; and after all your labor you may perhaps find that the number is not a square number. To save this trouble, it is worth knowing that every square number ends either with a 1, 4, 5, 6, or 9, or with two cyphers, preceded by one of these numbers.
Another property of a square number is, that if it be divided by 4, the remainder, if any, will be 1--thus, the square of 5 is 25, and 25 divided by 4 leaves a remainder of 1; and again, 16, being a square number, can be divided by 4 without leaving a remainder.
THE SHEEP-FOLD.
A farmer had a pen made of 50 hurdles, capable of holding 100 sheep only; supposing he wanted to make it sufficiently large to hold double that number, how many additional hurdles would he have occasion for?
_Answer._--Two. There were 24 hurdles on each side of the pen; a hurdle at the top, and another at the bottom; so that, by moving one of the sides a little back, and placing an additional hurdle at the top and bottom, the size of the pen would be exactly doubled.
COUNTRYWOMAN AND EGGS.