Chapter I: II. III. IV (2)

The least number that will answer this question is twelve; for if we suppose that each Grace gave one to each Muse, the latter would each have three, and there would remain three for each Grace. (Any multiple of 12 will answer the conditions of the question.)

THE JESUITICAL TEACHER.

A teacher, having fifteen young ladies under her care, wished them to take a walk each day of the week. They were to walk in five divisions of three ladies each, but no two ladies were to be allowed to walk together twice during the week. How could they be arranged to suit the above conditions?

SUN. MON. TUES. WEDN. THURS. FRID. SAT.
a b c|a d g|a k n|a e l|a h o|a f p|a i m
d e f|b e h|b l o|b f m|b i p|b d n|b g k
g h i|c m p|c f i|c g n|c d k|c h l|c e o
k l m|f k o|d h m|d i o|e m n|e i k|d l p
n o p|i l n|e g p|h k p|f g l|g m o|h f n

QUAINT QUESTIONS.

What is the difference between twenty four quart bottles, and four and twenty quart bottles?

_Ans._--56 quarts difference.

What three figures, multiplied by 4, will make precisely 5?

_Ans._--1¼, or 1·25.

What is the difference between six dozen dozen, and half-a-dozen dozen?

_Ans._--792: Six dozen dozen being 864, and half-a-dozen dozen, 72.

Place three sixes together, so as to make seven.

_Ans._--6⁶/₆.

Add one to nine and make it twenty.

_Ans._ IX--cross the _1_, it makes XX.

Place four fives so as to make six and a half. _Ans._--5⁵/₅ ·5.

A room with eight corners had a cat in each corner, seven cats before each cat, and a cat on every cat's tail. What was the total number of cats?

_Ans._--Eight cats.

Prove that seven is the half of twelve. _Ans._--Place the Roman figures on a piece of paper, and draw a line through the middle of it, the upper will be VII.

THE FOX, GOOSE AND CORN.

A countryman having a Fox, a Goose, and a peck of Corn, came to a river, where it so happened that he could carry but one over at a time. Now as no two were to be left together that might destroy each other, he was at his wit's end, for says he "Though the corn can't eat the goose, nor the goose eat the fox; yet the fox can eat the goose, and the goose eat the corn." How shall he carry them over, that they shall not destroy each other?

Let him first take over the Goose, leaving the Fox and Corn; then let him take over the Fox and bring the Goose back; then take over the Corn; and lastly take over the Goose again.

MULTIPLYING MONEY BY MONEY.

Amongst the various questions that are given for the purpose of puzzling the unwary arithmetician, the multiplication of money by money is one of the most curious: take for instance the following problems:

Multiply £99 19_s._ 11-3/4_d._ by £99 19_s._ 11-3/4_d._
Multiply £11 11_s._ 11_d._ by £11 11_s._ 11_d._

To the uninitiated they usually appear easy of solution but the various modes of working them out, and the different results obtained, prove that there is something absurd and wrong in the questions themselves. Some reduce all to farthings, and after multiplying one term by the other, return the product into pounds, shillings, and pence. Others convert them into decimals; whilst some work the problem in the style of duodecimals.

Having sufficiently puzzled the tyros, the querist remarks: "The problem itself is absurd, it is incapable of solution; for what is the nature of the product of pounds, shillings, and pence multiplied by pounds, shillings and pence? We know that a yard multiplied by a yard is a square yard, but who can tell what is a penny multiplied by a penny, or a penny by a pound?"

Now all this is quite correct, provided the question is limited, as above to the product of pounds, shillings, and pence, into pounds, shillings, and pence; but suppose the problem were put in this form--If a capital of £1 produces by compound interest, in a certain time, £99 19_s._ 11¾_d._, how much would be produced by a capital of £99 19_s._ 11¾_d.?_ It is evident that, to answer this, we must multiply £99 19_s._ 11¾_d._ by £99 19_s._ 11¾_d._: these are in fact the second and, third terms of an ordinary "rule of three;" and though one of the terms is a "concrete" quantity of pounds, shillings, and pence, the other must be regarded as an "abstract" mathematical quantity, being 99 and a fraction, of which the number of farthings in a pound is the denominator 960, and the number of farthings in the third term is the numerator, 959; or, instead of this, the shillings and pence might be converted into decimals of a pound, or into aliquot parts. The product of multiplying £99 19_s._ 11¾_d._ by 99-959/960 is £9,999 15_s._ 10 1/3840_d._; the quickest way of doing this, is to multiply by 100, and to subtract from the product the 960th part of the multiplicand.

In the other question proposed, the product of £11 11_s._ 11_d._ into £11 11_s._ 11_d._, or 11 143/240, is £134 9_s._ 3 49/240_d._

Number and value are distinct abstract ideas, and cannot, without committing a logical absurdity, be confused. To multiply is to repeat a certain number of _times_, and it is obviously impossible to bring _value_ into the question. Value is arbitrary; number is fixed. Put it in this way, and the absurdity is evident: One pound is equivalent to 20 shillings, or 240 pence, or 960 farthings. In value there is no difference whatever; but what an enormous difference between multiplying by 1, 20, 240, or 960!

THE UNFAIR DIVISION.

A gentleman rented a farm, and contracted to give to his landlord 2/5 of the produce; but prior to the time of dividing the corn, the tenant used 45 bushels. When the general division was made, it was proposed to give to the landlord 18 bushels from the heap, in lieu of his share of the 45 bushels which the tenant had used, and then to begin and divide the remainder as though none had been used. Would this method have been correct?

The landlord would lose 7⅕ bushels by such an arrangement, as the rent would entitle him to ⅖ of the 18. The tenant should give him 18 bushels from his own share after the division is completed, otherwise the landlord would receive but 2/7 of the first 63 bushels.

A POPULAR FALLACY.

It is often suggested from the pulpit and elsewhere, that enough persons have lived and died in the world to cover its whole surface with bodies; and even two or three strata deep. Is this probable?

Say the earth has existed 6000 years, the population always having been 800,000,000, and the average life of man 30 years; this being the utmost that could be claimed. Allow then the State of Virginia to contain 70,000 square miles, and each grave to occupy a space of 6 feet by 2; the territory of the State would contain 162,624,000,000; while the mighty army of the dead would number only 160,000,000,000; leaving 2,624,000,000 graves yet unoccupied. How wide of truth then is the position often set forth so positively!

FOOTNOTES:

[Footnote 10: Ancestor of the fighting and writing Napiers of our day.]

[Footnote 11: When an _exact half_ cannot be taken without a fraction, he must take the _larger half_--you must tell him this before he commences. Here it is the _larger half_.]

[Footnote 12: From Parkes' Philosophy of Arithmetic, a capital work published by Moss & Bro. Philadelphia.]

TRICKS IN GEOMETRY.

"Let young beginners come and try
Their hands at our geometry."

The word Geometry is derived from the Greek, and signifies the art of measuring land. The invention of it is ascribed by some to the Chaldeans and Babylonians, by others to the Egyptians, who were obliged to determine the boundaries of their fields after the inundation of the Nile, by geometrical measurements. According to Cassiodorus, the Egyptians either derived the art from the Babylonians, or invented it after it was known to them. Thales, a Phœnician, who died 548 years B.C., and Pythagoras of Samos, who flourished about 520 B.C., introduced it from Egypt into Greece. In elementary geometry, Euclid of Alexandria, as everybody knows, is particularly distinguished. Archimedes measured the sphere, and after him other philosophers prosecuted the science with the utmost assiduity. In Italy, where the sciences first revived after the dark ages, several mathematicians were distinguished in the 16th century. The French, and after them the Germans, followed; while in England, Hook, Newton, and others, carried the science to the highest pitch of usefulness, and through its aid made the most prodigious discoveries. It is not, however, our province to enter into a long disquisition on the subject, but simply to set before the young reader some of the more curious properties of the science, that he may be excited to study it for himself; and we will promise him that should he devote his mind to its study, he will be amply repaid for any amount of labor he may bestow upon it.

GEOMETRICAL DEFINITIONS.

In geometry a _point_ is said to have neither breadth, length, nor thickness. A _line_ is the distance between two points; parallel lines always keep at the same distance from each other. A _right_ line is what is commonly called a straight line. A _curve_ is a line which continually changes its direction. An _angle_ is the inclination or opening of two lines meeting in a point. A _figure_ is a bounded space, and is either a superficies or a solid. A _triangle_ is a figure with three sides and three angles. A _square_ has four equal sides, and four right angles. A _circle_ is a plane figure bounded by a curved line running into itself. Its diameter is a straight line drawn from one extremity of its circumference to the other, and its center is equally distant from every part of the circumference. A _solid_ is any body which has length, breadth, and thickness; and a sphere is a solid, terminated by a convex surface, every part of which is at an equal distance from a point within, called its center.

THE FIVE GEOMETRICAL SOLIDS.

The following figures will show how the five geometrical solids may be cut out of a piece of cardboard. Where the lines are drawn the board is to be partly cut through with a penknife, so as to render the angles of the models as sharp and as straight as possible. The edges which require joining are to be fastened together with a slip of thin paper and gum dissolved in just sufficient water to bring it to the consistence of treacle. Fig. 1 will form a tetrahedron, a figure with four sides, each shaped like an equilateral triangle. Fig. 2 forms a cube or hexahedron. Fig. 3 an octohedron, with eight triangular sides. Fig. 4, a dodecahedron, with twelve sides shaped like pentagons, with five equal sides. Fig. 5, an isocahedron, with twenty sides, formed of equilateral triangles.

HOW TO MAKE FIVE SQUARES INTO A LARGE ONE WITHOUT ANY WASTE OF STUFF.

Suppose you have five squares of cloth, or anything else, as in Fig. 7; find the center of one side of four of these squares, and cut them from that point to the opposite corner, then place the perfect square in the centre, and the other pieces round, as seen in Fig. 8.

DECEPTIVE VISION.

The following sleight shows how easily the eye may be deceived. Take a piece of pasteboard, an inch and a half in width, and five inches in length, and divide it by inked lines into thirty squares, then cut it from corner to corner, so as to form two triangles. After this cut off the top of these triangles at C and D,[13] and arrange the pieces in this manner:--

On counting the squares in the first figure, there appear to be thirty, but the other arrangement of the same card seems to contain thirty-two. It does so, however, only in appearance, but it is only a very correct eye that can detect the imperfection.

THE CARPENTER PUZZLED.

A carpenter having a piece of mahogany of a triangular form, (see Fig.) wished to know how he could make it up to the best advantage. His first idea was to make an oblong square table of it, but he found that if he did so the waste of the wood would be very great. After consideration he discovered that the most economical method of using the wood would be to form it into an oval. To make this oval contain as much wood as possible, he proceeded in the following manner: Let B G D be the triangular piece of wood; take G H one half of the base, and divide the triangle by drawing a line from H to B. Take G H in the compasses, and set it off on one of the sides from G to E, draw the line E F, and the point I will be the center of the oval; draw K L parallel to E F, and at the same distance from the center as the base G. The points A and C are found by dividing the line from E to K and drawing A C, or by drawing the dotted lines D A and G C through the center at I. These points being found, the oval must be completed by the eye of the draughtsman.

THE BRICKLAYER PUZZLED.

A bricklayer had to construct a wall, whose length in the direction A B C was twenty-four feet. The one half of this wall, namely from B to C, had to be built over a piece of rising ground, so that the base of this part of the wall would necessarily be more than twelve feet. In making out his account he charged more for this half of the wall than for that which was built on level ground from A to B. A geometrician assured him that the square contents of both portions of the wall were exactly alike; which may be proved in the following manner:--

Cut two pieces of cardboard, in the form shown in Figs. 2 and 3, to represent the two parts of the wall; lay the piece representing the straight wall on the curved piece, and it will be found that the angles which project at A and B will exactly fill up the spaces at E and F. The piece of board representing the straight wall may thus be found to be exactly sufficient to form a piece equal to that representing the curved wall. You may then lay the curved piece upon the straight one, and reversing the experiment prove that the curved piece is capable of forming a rectangular piece equal to the other.

TRIANGULAR PROBLEM.

Take four square pieces of pasteboard of the same dimensions, and divide them diagonally, that is, by drawing a line from two opposite angles, as in the figures, into eight triangles. Paint seven of these triangles with the prismatic colors, red, orange, yellow, green, blue, indigo, violet, and let the eighth be white. To find how many chequers or regular four-sided figures, different either in form or color, may be made out of these eight triangles.

First, by combining two of these triangles there may be formed, either the triangular square A, or the inclined square B, called a Rhomb. Secondly, by combining four of the triangles the large square C may be formed, or the long square D, called a parallelogram. Now the first two squares, consisting of two parts out of eight, may each of them by the eighth rank of the triangle be taken twenty-eight different ways, which makes fifty-six. And the last two squares, consisting of four parts, may each be taken by the same rank of the triangle seventy times, which makes 140.

TO FORM A SQUARE.

Take a piece of card of the shape and size or proportions of the subjoined, and cut it into three parts, and with these three form a perfect square.

To do this, cut it in the direction of the dotted lines, and it will then be easy to lay down the pieces to form a perfect square.

SQUARING THE CIRCLE.

"Squaring the circle," as it is called, is the puzzle of puzzles, and there are many persons who fancy this can be accomplished, as there are also many who believe that they can discover "perpetual motion."

The meaning of this phrase _squaring_ is scientifically expressed by the term finding the quadrature of the circle; that is, the act of producing a square equal to a given circle; and many persons but slightly acquainted with mathematics have puzzled their brains to effect this object. The Cardinal de Cusa rolled a cylinder over a plane, till the point which was first in contact with the plane touched it again; and then, by a train of reasoning very unmathematical, he endeavored to determine the length of the line thus described. Oliver de Serras worked a circle, and also a triangle equal to an equilateral triangle, inscribed within the circle, and imagined that the former was exactly equal to two of the latter, forgetting that the double of this triangle is equal to the hexagon inscribed within the circle, and therefore smaller than the circle itself. A Frenchman challenged the world, and deposited 10,000 livres as a stake, that he could accomplish the feat. He reduced the problem to the mechanical process of dividing a circle into four quarters, and then turning these with their angles outwards, so as to form a square, which he asserted to be equal to the circle; this however was soon proved to be ridiculous.

Some persons have taken a piece of pasteboard, and cutting it out into a circular form, and by cutting that circular disc into pieces of a square form and definite dimensions, and fitting the same turned pieces one into the other, have come _near_ to a notion of the superficial area of a circle. But this kind of demonstration is purely mechanical, and is neither geometrical nor scientific, and, is in fact, no demonstration according to mathematics. For if we take the pieces of card, however exactly they may appear to be formed, and examine them with a microscope, we shall soon find that none of them are geometrically true, nor of the same length or breadth, and therefore the conclusion arrived at is a false one.

The early mathematicians, in their attempts to solve this problem, generally proceeded on the following plan. If we draw a square exterior to a circle, that is, touching the square in four points, each side of the square being equal to the diameter of the circle, we can soon convince ourselves that the boundary of the square will be greater than the circumference of the circle, and the area of the former greater than that of the latter. But if the square be drawn within the circle, so that only the four corners touch it, then it is equally evident that the circle is larger, both in boundary and area, than the square. By this proceeding, we arrive at the conclusion that a circle is _smaller_ than a square _external_ to it, and _larger_ than one _internal_ to it. Let us next suppose that we draw a regular pentagon, that is, a figure of five equal sides, exterior to the circle, and touching it on five points; then it is evident that as the circle is wholly contained within the pentagon, it must be smaller than that which contains it. But if the pentagon be described within the circle, touching it at the five angular points, then of course the circle is larger than the pentagon which it contains.

Now, in geometry, my young readers must bear in mind, the exact periphery or circumference, and the exact area of any figure bounded by straight lines, may be determined with rigorous accuracy; and if we draw two polygons--say of one hundred sides, one within and one without the circle--we can ascertain the exact area of those polygons, and affirm that the area of a circle is greater than a certain amount, and less than another certain amount. These two amounts, if the number of the sides of the polygon be so large as we here suppose, may be so very nearly alike, that either one will give the area of the circle with great closeness.

By some such means as these Archimedes found that if the diameter of a circle be called 7, then the circumference will be nearly 22; and that if the square of the diameter be 14, then the area of the circle will be equal to about 11; but this computation was slightly in error, and gave to the area of the circle too great a measure by about one three-thousandth part of the whole. At a later period, however, a European mathematician, named Metius, discovered a method which makes an extraordinary approach to accuracy, and is at the same time easily remembered. He found that if the diameter be considered equal to 113, then the circumference would equal 355; or if we multiply the square of the radius by 355, and divide it by 113, the area will be given. Now this method is so very nearly correct, that the area of a circle one foot in diameter is given within the fifty-thousandth part of a square inch.

Other mathematicians have carried the approximation still further. Ludolph Van Ceulen worked it out to 36 places of figures, showing that if the diameter be 1, the circumference will be

3.14,159,265,358,979,323,846,264,338,327,950,288.

or that if the last figure be 8, the result will be a little below the truth, and if 9, a little above it.

Since this, Mr. Sharp, an English mathematician, carried the approximation to 72 places of figures; Mr. John Machin to 100 figures, and eclipsed all others. M. de Lagny worked it out to 128 places of figures, and of the degree of _nearness_ to which this computation brings the proportion, Montucla says, "If we suppose a circle, the _diameter_ of which is a thousand million times greater than the distance between the sun and the earth, the error in the proportion of the circumference would be a thousand million times less than the thickness of a hair."

But after all, none of these computations are quite correct; they all deviate from the truth, and bring us to the conclusion that there are no numbers or collection of numbers which will give the exact ratio of the circumference, or of the area of a circle to its diameter. We offer this explanation on the subject to our young friends that they may not be puzzled by the question; and that should they be asked to square the circle, or hear any one assert that he can do so, they may be able to show that they are "awake" to the question, and know how to explain it.

FOOTNOTES:

[Footnote 13: _I. e._, at the fourth square from the right angle.]

PRACTICAL PARADOXES AND PUZZLES.

A puzzle is not solved, impatient sirs,
By peeping at its answer, in a trice--
When Gordius, the plow-boy, king of Phrygia,
Tied up his implements of husbandry
In the far-famed knot, rash Alexander
Did not undo, by cutting it in twain.

Paradoxes and Puzzles, although by many persons looked upon as mere trifles, have, in numerous instances, cost their inventors considerable time, and exhibit a great degree of ingenuity. We can readily imagine that some of the complicated puzzles in the ensuing pages may have been originally constructed by captives, to pass away the hours of a long and dreary imprisonment; thus does the misery of a few frequently conduce to the amusement of many. We look upon a Paradox as a sort of superior riddle, and a tolerable Puzzle, in our opinion, takes precedence of a first-rate rebus. There is often considerable thought, calculation, patience, and management, required to solve some of these strange enigmas; and we have, ere now, followed the mazes of a Puzzle so ardently, as to be entirely absorbed in devising means to extricate ourself from its bewildering difficulties; and felt almost as much pleasure in eventually achieving victory over it, as we have in conquering an adversary at some superior game of skill. It is "in good sooth, a right dainty and pleasant pastime," to watch the stray wanderings of another person attempting to elucidate a Paradox, or perform a Puzzle, with which one is previously acquainted. It is laughable to see him elated with hope at the apparent speedy end of his troubles, when you know that, at that moment, he is actually farther from his object than he was when he began; and it is no less amusing to watch his increasing despair, as he conceives himself to be getting more and more involved, when you are well aware that he is within a single turn of a happy termination of his toils; but what a mirthful moment is that, when there being only two ways to turn, the one right and the other wrong, as is usually the case, he takes the latter, and becomes more than ever

"Pozed, puzzled, and perplexed."

Puzzles are by no means of modern origin; the Sphynx puzzled the brains of some of the heroes of antiquity, and even Alexander the Great, as it is written, made several essays to untie the knot with which Gordius, the Phrygian king, who had been raised from the plow to the throne, tied up his implements of husbandry in the temple, in so intricate a manner, that universal monarchy was promised to the man who could undo it: after having been repeatedly baffled, he, at length, drew his sword, considering that he was entitled to the fulfillment of the promise, by cutting the Gordian knot.

1. THE CHINESE CROSS.

Have six pieces of wood, bone, or metal, made of the same length as No. 6, in the above figures, and each piece of the same size as No. 7. It is required to construct a cross, with six arms, from these pieces, and in such a manner that it shall not be displaced when thrown upon the floor.

The shaded parts of each figure represent the parts that are cut _out_ of the wood, and each piece marked _a_ is supposed to be facing the reader, while the pieces marked _b_ are the _right_ side of each piece turned over _towards_ the left, so as to face the reader. No. 7 represents the end of each piece of wood, &c., and is given to show the dimensions.

2. THE PARALLELOGRAM.

A parallelogram, as in the illustration, fig. 1, may be cut into two pieces, so that by shifting the position of the pieces, two other figures may be formed, as shown by figs. 2 and 3.

3. THE DIVIDED GARDEN.

A person let his house to several inmates, who occupied different floors, and having a garden attached to the house, he was desirous of dividing it among them. There were ten trees in the garden, and he was desirous of dividing it so that each of the five inmates should have an equal share of garden and two trees. How did he do it?

4. THE ENDLESS STRING.

Now, sir, your coat is off!
And see--
Your right-hand pocketed!
So let it be:
While o'er your arm
An endless string--
Some three yards round--
Hangs like a sling.
Take the string off--
But, just for fun,
It must be done
Keeping your right-hand in its place,
And not a smile must stir your face.
Until you find this puzzle out,
No coat shall wrap your back about.

5. CHINESE MAZE. THE WILLOW-PATTERN PLATE.

Ye fair ones who, in continent or isle,
Long for delights which love alone can bring;
Whilst ruby lips display affection's smile,
Haste through the maze, and reach the "wedding ring"
The sweet Koong-see, whose spirit hovers near,
Shall watch thee wand'ring through the doubtful way;
And when thou showest aught of hope or fear,
Shall whisper to thee, as thy footsteps stray!

6. THE VERTICAL LINE PUZZLE.

Draw six vertical lines, as below, and, by adding five other lines to them, let the whole form nine.

| | | | | |
| | | | | |
| | | | | |
]

7. THE THREE RABBITS.

Draw three rabbits, so that each shall appear to have two ears, while, in fact, they have only three ears between them.

8. THE ACCOMMODATING SQUARE.

Make eight squares of card, then divide four of them from corner to corner, so that you will now have twelve pieces. Form a square with them.

9. THE CIRCLE PUZZLE.

Draw a circle upon a piece of paper, and thrust a pin through it without crossing the circle, or thrusting it downward through the center.

10. THE CARDBOARD PUZZLE.

Take a piece of cardboard or leather, of the shape and measurement indicated by the diagram, cut it in such a manner that you yourself may pass through it, still keeping it in one piece.

+-----------+
| 3 Inches. |
| |
|6 |
| |
|I |
|n |
|c |
|h |
|e |
|s |
|. |
| |
+-----------+
]

11. THE BUTTON PUZZLE.

In the center of a piece of leather make two parallel cuts with a penknife, and just below a small hole of the same width; then pass a piece of string under the slit and through the hole, as in the figure, and tie two buttons much larger than the hole to the ends of the string. The puzzle is, to get the string out again without taking off the buttons.

12. THE QUARTO PUZZLE.

|\ /|
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \/ |
| |
| |
| |
+--------------+
]

Divide this figure into four equal parts, each of the same figure.

13. THE PUZZLE OF FOURTEEN.

Cut out fourteen pieces of paper, card, or wood, of the same size and shape as those shown in the diagram, and then form an oblong with them.

14. THE SQUARE AND CIRCLE PUZZLE.

+---------------+
|o o o|
| |
| |
|o oo o|
| oo |
| |
| |
|o o o|
+---------------+
]

Get a piece of cardboard, the size and shape of the diagram, and punch in it twelve circles or holes in the position shown. The puzzle is, to cut the cardboard into four pieces of equal size, each piece to be of the same shape, and to contain three circles, without cutting into any of them.

15. THE SCALE AND RING PUZZLE.

Provide a thin piece of wood of about two inches and a half square; make a round hole at each corner, sufficiently large to admit three or four times the thickness of the cord you will afterwards use, and in the middle of the board make four smaller round holes in the form of a square, and about half an inch between each. Then take four pieces of thin silken cord, each about six inches long, pass one through each of the four corner holes, tying a knot underneath at the end, or affixing a little ball or bead to prevent its drawing through; take another cord, which, when doubled, will be about seven inches long, and pass the two ends through the middle holes _a a_, from the front to the back of the board, (one cord through each hole,) and again from back to front through the other holes _b b_; tie the six ends together in a knot, so as to form a small scale, and proportioning the length of the cords, so that when you hold the scale suspended, the middle cord, besides passing through the four center holes, will admit of being drawn up into a loop of about half an inch from the surface of the scale; provide a ring of metal or bone, of about three quarters of an inch in diameter, and place it on the scale, bringing the loop through its middle; then, drawing the loop a little through the scale toward you, pass it, double as it is, through the hole at the corner A, over the knot underneath, and draw it back; then pass it in the same way through the hole at corner B, over the knot, and draw it back; then, drawing up the loop a little more, pass it over the knot at top, and afterward through the holes C and D in succession, like the others, and the ring will be fixed.

16. THE HEART PUZZLE.

Cut a piece of thin wood the shape indicated by the diagram, and having perforated it as above, draw a piece of string, with a smaller heart attached at the end, through No. 1, pass it behind, and bring it through 2 before, and through 3, and so on to 6, when a loop must be made so as to enclose that part of the string which runs from 2 to 3. The puzzle is to remove the string from the large heart altogether, without unfastening the loop.

Care should be taken to avoid twisting or entangling the string. The length of the string should be proportioned to the size of the heart; if you make the heart two inches and a half high, the string when doubled should be about nine inches long.

17. THE CROSS PUZZLE.

Cut three pieces of paper to the shape of No. 1, one to the shape of No. 2, and one to that of No. 3. Let them be of proportional sizes. Then place the pieces together so as to form a cross.

18. THE YANKEE SQUARE.

Cut as many pieces of each figure in cardboard as they have numbers marked on each; then form the pride of the American army.

19. THE CARD PUZZLE.

One of the best puzzles hitherto made is represented in the annexed cut. A is a piece of card; _b b_ a narrow slip divided from its bottom edge, the whole breadth of the card, except just sufficient to hold it on at each side; _c c_ is another small slip of card with two large square ends, _e e_; _d_ is a bit of tobacco-pipe, through which _c c_ is passed, and which is kept on by the two ends _e e_. The puzzle consists in getting the pipe off without breaking it or injuring any other part of the puzzle. This, which appears to be impossible, is done in the most simple manner. On a moment's consideration it will appear plainly that there must be as much difficulty in getting the pipe in its present situation, as there can be in taking it away. The way to put the puzzle together is as follows: The slip _c c e e_ is cut out of a piece of card in the shape delineated in Fig. 3. The card in the first figure must then be gently bent at A, so as to allow of the slip at the bottom of it being also bent sufficiently to pass double through the pipe, as in Fig. 2. The detached slip with the square ends (Fig. 3) is then to be passed half way through the loop _f_ at the bottom of the pipe; it is next to be doubled in the center at _a_, and pulled through the pipe, double; by means of the loop of the slip to the card. Upon unbending the card the puzzle will be complete, and appear as represented in Fig. 1.

20. THREE SQUARE PUZZLE.

Cut seventeen slips of cardboard of equal lengths, and place them on a table to form six squares, as in the diagram. It is now required to take away five of the pieces, yet to leave but three perfect squares.

21. THE CYLINDER PUZZLE.

Cut a piece of cardboard about four inches long, of the shape of the diagram, and make three holes in it, as represented. The puzzle is, to make one piece of wood to pass through, and also exactly to fill, each of the three holes.

22. PUZZLE OF THE FOUR TENANTS.

I have a square plot of ground, in one quarter of which I have built a house, which I have let to four tenants. I tell them that if they can divide the remaining ground into four equal plots, alike in shape, and each containing one of the four apple trees I have planted, they shall have it without any increase of rent. How may they succeed?

23. THE PUZZLE WALL.

Suppose there was a pond, around which four poor men built their houses, thus:

Suppose four evil-disposed rich men afterwards built houses around the poor people, thus:

and wished to have all the water of the pond to themselves. How could they build a high wall, so as to shut out the poor people from the pond?

24. THE NUNS.

Twenty-four nuns were arranged in a convent by night, by a sister, to count nine each way, as in the diagram. Four of them went out for a walk by moonlight. How were the remainder placed in the square so as still to count nine each way? The four who went out returned, bringing with them four friends; how were they all placed still to count nine each way, and thus to deceive the sister, as to whether there were 20, 24, 28, or 82, in the square?

25. THE HORSE SHOE PUZZLE.

Cut a piece of apple or turnip into the shape of a horse shoe, stick six pins in it for nails, and then, by two cuts, divide it into six parts, each to contain one pin.

26. THE CARD SQUARE.

With eight pieces of card or paper, of the shape of Fig. _a_, four of Fig. _b_, and four of Fig. _c_, and of proportionate sizes, form a perfect square.

27. THE DOG PUZZLE.

The dogs are, by placing two lines upon them, to be suddenly aroused to life and made to run. Query, How and where should these lines be placed, and what should be the forms of them?

28. PUZZLE OF THE TWO FATHERS.

Two fathers have each a square of land. One father divides his so as to reserve to himself one fourth; thus--

+----------------+
| |
| |
| |
+------+ |
| | |
| | |
| | |
+------+---------+
]

The other father divides his so as to reserve to himself one fourth in the form of a triangle; thus--

+--------+
| |
| |
| /\ |
| / \ |
| / \ |
|/ \|
+--------+
]

They each have four sons, and each divides the remainder among his sons in such a way that each son will share equally with his brother, and in similar shape. How were the two farms divided?

29. THE TRIANGLE PUZZLE.

Cut twenty triangles out of ten square pieces of wood; mix them together, and request a person to make an exact square with them.

30. CUTTING OUT A CROSS.

How can be cut out of a single piece of paper, and with one cut of the scissors, a perfect cross, and all the other forms as shown in the cuts?

31. ANOTHER CROSS PUZZLE.

With three pieces of cardboard of the shape and size of No. 1, and one each of No. 2 and 3, to form a cross.

32. THE FOUNTAIN PUZZLE.

A is a wall, B C D three houses, and E F G three fountains or canals.

It is required to bring the water from E to D, from G to B, and from F to C, without one crossing the other, or passing outside of the wall A.

33. THE PUZZLE OF THE STARS.

Friends one and all, I pray you show
How you _nine stars_ would so bestow,
_Ten_ rows to form--in each row _three_--
Tell me, ye wits, how this can be?

34. THE COUNTER PUZZLE.

Place eight counters or coins, as in the diagram; it is then required to lay them in four couples, removing only one at a time, and in each removal passing the one in the hand over _two_ on the table.

35. THE JAPAN SQUARE PUZZLE.

Cut out ten pieces of card or wood of the same sizes and shapes as in the diagram, and then form a square with them.

36. THE CABINET MAKER'S PUZZLE.

A cabinet-maker has a circular piece of veneering with which he has to veneer the tops of two oval stools; but it so happens that the area of the stools, exclusive of the hand-holes in the center, and that of the circular piece, are the same. How must he cut his stuff so as to be exactly sufficient for his purpose?

37. THE STRING AND BALLS PUZZLE.

Get an oblong strip of wood or ivory, and bore three holes in it, as shown in the cut. Then take a piece of twine, passing the two ends through the holes at the extremities, fastening them with a knot, and thread upon it two beads or rings, as depicted above. The puzzle is to get both beads on the same side, without removing the string from the holes, or untying the knots.

38. THE DOUBLE-HEADED PUZZLE.

Cut a circular piece of wood as in the cut No. 1, and four others, like No. 2. The puzzle consists in getting them all into the cross-shaped slit, until they look like Fig. 3.

39. ARITHMETICAL PUZZLE.

The sum of four figures in value will be.
Above seven thousand nine hundred and three;
But when they are halved, you'll find very fair
The sum will be nothing, in truth I declare.

40. GRAMMATICAL PUZZLE.

Let the rich, great, and noble, banquet in the festal halls,
And pass the hours away, as the most thoughtless revel;
Then seek the poor man's dreary home, whose very dingy walls
Proclaim full well to all how low his rank and level.

Take away one letter from a word in the above stanza, and substitute another, leaving the word so metamorphosed still a word of the English language; and, by that change, totally after the syntactical construction of the whole sentence, changing the moods and tenses of verbs, turning verbs into nouns, nouns into adjectives, and adjectives into adverbs, &c., and so make the entire stanza bear quite a different meaning from that which it has as it stands above.

41. THE TREE PUZZLE.

Plant an orchard of twenty-one trees, so that there shall be nine straight rows, with five trees in each row, the _outline_ a regular geometrical figure, and the trees all at unequal distances from each other.

42. AN EPITAPH ON ELLINOR BACHELLOR, AN OLD PYE WOMAN.

Bene A. Thin Thed Ustt HEMO. Uld yo
L.D.C. RUSTO! Fnel L.B.
Ach El Lor. Lat. ELY,
Wa. S. shove N. W. How--Ass! kill'd I. N. T. H.
Ear T. Sofp, I, Escu Star.
D. San D T Art. San D K. N E. W. E
Ver----Yus E.--Oft He ove N W. Hens He
'Dli V'DL. on geno
Ug H S hem A.D.E. he R. la Stp. Uf----fap
Uf. F. B Y he. R hu
S. Ban D. M.
Uch pra is 'D. No. Wheres Hedot
HL. i. e. Tom. A kead I.R.T.P. Yein hop Esthathe
R. C. RUSTWI,
L L B. Era is '----D!

43. A CURIOUS LETTER.

Friends Sir, friends,
stand your disposition;
I bearing
a man the world
is whilst the
contempt,
ridicule.
are
ambitious.

44. A PUZZLING INSCRIPTION.

P R S V R Y P R F C T M N
V R K P T H S P R C P T S T N.

The two lines above were affixed to the communion table of a small church in Wales, and continued to puzzle the learned congregation for several centuries, but at length the inscription was deciphered. What was it?

45. THE PUZZLING RINGS.

This perplexing invention is of great antiquity, and was treated on by Cardan, the mathematician, at the beginning of the sixteenth century. It consists of a flat piece of thin metal or bone, with ten holes in it; in each hole a wire is loosely fixed, beaten out into a head at one end, to prevent its slipping through, and the other fastened to a ring, also loose. Each wire has been passed through the ring of the next wire, previously to its own ring being fastened on; and through the whole of the rings runs a wire loop or bow, which also contains, within its oblong space, all the wires to which the rings are fastened; the whole presenting so complicated an appearance, as to make the releasing the rings from the bow appear an impossibility. The construction of it would be found rather troublesome to the amateur, but it may be purchased at most of the toy shops very lightly and elegantly made. It also exists in various parts of the country, forged in iron, perhaps by some ingenious village mechanic, and aptly named "The Tiring Irons." The following instructions will show the principle on which the puzzle is constructed, and will prove a key to its solution.

Take the loop in your left hand, holding it at the end, B, and consider the rings as being numbered 1st to 10th. The 1st will be the extreme ring to the right, and the 10th the nearest your left hand.

It will be seen that the difficulty arises from each ring passing round the wire of its right-hand neighbor. The extreme ring at the right hand, of course, being unconnected with any other wire than its own, may at any time be drawn off the end of the bow at A, raised up, dropped through the bow, and finally released. After you have done this, try to pass the second ring in the same way, and you will not succeed, as it is obstructed by the wire of the first ring; but if you bring the first ring on again, by reversing the process by which you took it off, viz., by putting it up through the bow, and on to the end of it, you will then find that by taking the first and second rings together, they will both draw off, lift up, and drop through the bow. Having done this, try to pass the third ring off, and you will not be able; because it is fastened on one side to its own wire, which is within the bow, and on the other side to the second ring, which is without the bow. Therefore, leaving the third ring for the present, try the fourth ring, which is now at the end all but one, and both of the wires which affect it being within the bow, you will draw it off without obstruction; and, in doing this, you will have to slip the third ring off, which will not drop through for the reasons before given; so, having dropped the fourth ring through, you can only slip the third ring on again. You will now comprehend that (with the exception of the first ring) the only ring which can at any time be released is that which happens to be second on the bow, at the right-hand end; because both the wires which affect it being within the bow, there will be no impediment to its dropping through. You have now the first and second rings released, and the fourth also--the third still fixed; to release which we must make it last but one on the bow, and to effect which pass the first and second rings together through the bow, and on to it; then release the first ring again by slipping it off and dropping it through, and the third ring will stand as second on the bow, in its proper position for releasing, by drawing the second and third off together, dropping the third through, and slipping the second on again. Now to release the second, put the first up, through and on the bow; then slip the two together off, raise them up, and drop them through. The sixth will now stand second, consequently in its proper place for releasing; therefore draw it toward the end, A, slip the fifth off, then the sixth, and drop it through; after which replace the fifth, as you cannot release it until it stands in the position of a second ring; in order to effect this you must bring the first and second rings together, through and on to the bow; then in order to get the third on, slip the first off and down through the bow; then bring the third up, through and on to the bow; then bring the first ring up and on again, and, releasing the first and second together, bring the fourth through and on to the bow, replacing the third; then bring the first and second together on, drop the first off and through, then the third the same, replace the first on the bow, take off the first and second together, and the fifth will then stand second, as you desired; draw it toward the end, slip it off and through, replace the fourth, bring the first and second together up and on again, release the first, bring on the third, passing the second ring on to the bow again, replace the first, in order to release the first and second together; then bring the fourth toward the end, slipping it off and through, replace the third, bring the first and second together up and on again, release the first, then the third, replacing the second, bring the first up and on, in order to release the first and second together, which having done, your eighth ring will then stand second, consequently you can release it, slipping the seventh on again. Then to release the seventh, you must begin by putting the first and second up and on together, and going through the movements in the same succession as before, until you find you have only the tenth and ninth on the bow; then slip the tenth off and through the bow, and replace the ninth. This dropping of the tenth ring is the first effectual movement toward getting the rings off, as all the changes you have gone through were only to enable you to get at the tenth ring. You will then find that you have only the ninth left on the bow, and you must not be discouraged on learning, that in order to get that ring off, all the others to the right hand must be put on again, beginning by putting the first and second together, and working as before, until you find that the ninth stands as second on the bow, at which time you can release it. You will then have only the eighth left on the bow; you must again put on all the rings to the right hand, beginning by putting up the first and second together, till you find the eighth standing as second on the bow, or in its proper position for releasing; and so you proceed until you find all the rings finally released. As you commence your operations with all the rings ready fixed on the bow, you will release the tenth ring in one hundred and seventy moves; but as you then have only the ninth on, and as it is necessary to bring on again all the rings up to the ninth, in order to release the ninth, and which requires fifteen moves more, you will, consequently, release the ninth ring in two hundred and fifty-six moves; and, for your encouragement, your labor will diminish, by one half, with each following ring which is finally released. The eighth comes off in one hundred and twenty-eight moves, the seventh in sixty-four moves, and so on, until you arrive at the second and first rings, which come off together, making six hundred and eighty-one moves, which are necessary to take off all the rings. With the experience you will by this time have acquired, it is only necessary to say, that to replace the rings, you begin by putting up the first and second together, and follow precisely the same system as before.

46. MOVING THE KNIGHT OVER ALL THE SQUARES ALTERNATELY.