Chapter 3

Two brothers wisely kept apart,Together ne’er employed;Though to one purpose we are bentEach takes a different side.We travel much, yet prisoners are,And close confined to boot,Can with the fleetest horse keep pace,Yet always go on foot.

Two brothers wisely kept apart,Together ne’er employed;Though to one purpose we are bentEach takes a different side.

We travel much, yet prisoners are,And close confined to boot,Can with the fleetest horse keep pace,Yet always go on foot.

Solution

On similar lines to Magic Squares, but as a distinct variety, we give below a specimen of a Magic Oblong.

The four rows of this Oblong add up in each case to 132, and its eight columns to 66. Two of its diagonals, from 10 to 5 and from 28 to 23, also total 66, as do the four squares at the right-hand ends of the top and bottom double rows.

My name declares my date to beThe morning of a Christian year;And motherless, as all agree,And yet a mother, too, ’tis clear.A father, too, which none dispute,And when my son comes I’m a fruit.And, not to puzzle overmuch,’Twas I took Holland for the Dutch.

Solution

My head is ten times ten,My body is but one.Add just five hundred more, and thenMy history is done.Although I own no royal throne,Throughout the sunny South in fame I stand alone.

Solution

Much more complicated than the Magic Square is the Magic Cube.

First Layer from Top.1212783147010611174879441001571135310940912287187410531Second Layer from Top.25811445963692235411075101328819841566122281184980662

First Layer from Top.1212783147010611174879441001571135310940912287187410531

First Layer from Top.

Second Layer from Top.25811445963692235411075101328819841566122281184980662

Second Layer from Top.

Third Layer from Top.33892071102671232985117676311950115419735924551063793Fourth Layer from Top.64120467789846011142107389425511672103349030811268124

Third Layer from Top.33892071102671232985117676311950115419735924551063793

Third Layer from Top.

Fourth Layer from Top.64120467789846011142107389425511672103349030811268124

Fourth Layer from Top.

Lowest Layer.

Those who enjoy such feats with figures will find it interesting to work out the many ways in which, when the layers are placed one upon another, and form a cube, the number 315 is obtained by adding together the cell-numbers that lie in lines in the length, breadth, and thickness of the cube.

Sad offspring of a blighted race,Pale Sorrow was my mother;I’ve never seen the smiling faceOf sister or of brother.Of all the saddest things on earth,There’s none more sad than I,No heart rejoices at my birth.And with a breath I die!

Sad offspring of a blighted race,Pale Sorrow was my mother;I’ve never seen the smiling faceOf sister or of brother.

Of all the saddest things on earth,There’s none more sad than I,No heart rejoices at my birth.And with a breath I die!

Solution

The Magic Circle below has this particularproperty:—

Magic circle

The 14 numbers ranged in smaller circles within its circumference are such that the sum of the squares of any adjacent two of them is equal to the sum of the squares of the pair diametrically opposite.

Add a hundred and nothing to ten,And the same to a hundred times more,Catch a bee, send it after them, thenMake an end of a fop and a bore.

Solution

We have had some good specimens of Magic Squares. Here is a very curious and most interesting Magic Circle, in which particular numbers, from 12 to 75 inclusive, are arranged in 8 concentric circular spaces and in 8 radiating lines, with the central 12 common to them all.

Magic circle

The sum of all the numbers in any of the concentric circular spaces, with the 12, is 360, which is the number of degrees in a circle.

The sum of the numbers in each radiating line with the 12, is also 360.

The sum of the numbers in the upper or lower half of any of the circular spaces, with half of 12, is 180, the degrees of a semi-circle.

The sum of any outer or inner four of the numbers on the radiating lines, with the half of 12, is also 180.

In the following triangle, if two couples of the figures on opposite sides are transposed, the sums of the sides become equal, and also the sums of the squares of the numbers that lie along the sides. Which are the figures that must be transposed?

Magic triangle

Solution

They did not climb in hope of gain,But at stern duty’s call;They were united in their aim,Divided in their fall.

Solution

Forsaken in some desert vast,Where never human being dwelt,Or on some lonely island cast,Unseen, unheard, I still am felt.Brimful of talent, sense, and wit,I cannot speak or understand;I’m out of sight in Church, and yetGrace many temples in the land.

Forsaken in some desert vast,Where never human being dwelt,Or on some lonely island cast,Unseen, unheard, I still am felt.

Brimful of talent, sense, and wit,I cannot speak or understand;I’m out of sight in Church, and yetGrace many temples in the land.

Solution

Here is a nest of concentric triangles. Can you arrange the first 18 numbers at their angles, and at the centres of their sides, so that they count 19, 38, or 57 in many ways, down, across, or along some angles?

Concentric triangles

This curiosity is found in an old document of the Mathematical Society of Spitalfields, dated 1717.

Solution

Allow me, pray, to go as first,And then as number two;Then after these, why, there you are,To follow as is due.But lest you never guess this queerAnd hyperbolic fable,Pray let there follow after thatWhatever may be able.

Allow me, pray, to go as first,And then as number two;Then after these, why, there you are,To follow as is due.

But lest you never guess this queerAnd hyperbolic fable,Pray let there follow after thatWhatever may be able.

Solution

The numbers outside these twin triangles give the sum of the squares of the four figures of the adjacentsides:—

Triangles

The twins are also closely allied on thesepoints:—

18 is the common difference of 99, 117, 135, and of 119, 137, 155.

19 is the sum of each side of the upper triangle.

20 is the common difference of any two sums of squares symmetrically placed, both being on a line through the central spot.

21 is the sum of each side of the lower triangle.

10 is the sum of any two figures in the two triangles that correspond.

254 is the sum of 135, 119, of 117, 137, and of 90, 155.

By transposing in each triangle the figures joined by dotted lines, the nine digits run in natural sequence.

We have dealt with Magic Squares, Circles, and Triangles. Here is a Magic Hexagon, or a nest of Hexagons, in which the numbers from 1 to 73 are arranged about the common centre 37.

Each of these Hexagons always gives the same sum, when counted along the six sides, or along the six diameters which join its corners, or along the six which are at right angles to its sides. These sums are 259, 185, and 111.

When I am in, its four legs have no motion;When I am out, as fish it swims the ocean.Then, if transposed, it strides across a stream,Or adds its quality to eyes that gleam.

Solution

Inscribe six equilateral triangles in a circle, as shown in this diagram, so as to form a regular hexagon.

Circle and hexagons

Now place the nine digits round the sides of each of the triangles, so that their sum on each side may be 20, and so that, while there are no two triangles exactly alike in arrangement, the squares of the sums on the other sides may be alternately equal.

Solution

We can but see his sad reverse,And while we say alas!We hail his work so keen and terse,With just a touch of gas.

Solution

There are 33 different combinations of four of the numbers in the cells of this magic cross which total up in each case to 26.

Those who care to work them out on separate crosses will find that there is a very regular correspondence in the positions which the numbers occupy.

What boy can live on with a prospect of age,If you cut off his head at an early stage?

Solution

Here’s plenty of water, you’ll all of you say;And minus theha thing used every day;And here is nice beverage; put them together—What is it with claws, but with never a feather?

Solution

Here is quite a charming little puzzle, which is by no means easy ofaccomplishment:—

Start from one of these nine dots, and without taking the pen from the paper draw four straight lines which pass through them all. Each line, after the first, must start where the preceding one ends.

Solution

The deil jumpedthe clouds so highThat he bounded almostrightthe sky.the treesgates and fields andHe dodged with his taildraggingall these,But, alas! made a terriblebl,For a twist in his taila rail,hookedAnd broke that appendageas.

Solution

Place on a chess or draught-board three white men on the squares markeda, and three black men on the squares markedb.

The pieces markedacan only move one square at a time, from left to right, and those markedbone square at a time, from right to left, on to unoccupied squares; and any piece can leap over one of the other colour, on to an unoccupied square. What is the least number of moves in which the positions of the white and the black men can be reversed, so that each square now occupied by a white is occupied by a black, and each now occupied by a black holds a white piece?

Solution

To a word of assent join the first half of fright,Then add what will never be seen in the night.By such a conjunction we quickly attainWhat most men have seen, but can’t see again.

Solution

My first is stately, proud, and grave,My next will guard your treasure;My whole, a slow but sturdy slave,Will wait upon your pleasure.

Solution

In the upper row of this diagram four white and four black counters are placed alternately.

Counters

It is possible, by moving these counters two at a time, to arrange them in four moves as they stand on the lower row. Can you do this? Draughtsmen are handy for solving this puzzle, on a paper ruled as above.

Solution

I am a word of letters six,First link me with your mind;Then shuffle me, and lo! I mixWith grief of noisy kind.Shake me again, and you may fixA cloak that hangs behind.

Solution

We are of use to every manIn walking, riding, rambling;We join the gambols of the knave,And play the knave in gambling!

Solution

Take five wooden matches, and bend each of them into a V. Place them together, as is shown in the diagram, so that they take the form of an asterisk, or a ten-pointed star.

Matches

Lay them on some smooth surface, and without touching them transform them into a star with five points.

Solution

Strange that a straggling tiresome weedWill change its meaning quite,And turn into a sign of griefIf we transpose it right;And, stranger still, transposed againWill tell of ease from grief or pain.

Solution

Find me two English verbs that everIn a united state will blend,Let one say “join,” the other “sever,”While I divide them to the end.

Solution

It is possible, with plenty of patience, to build up a whole set of dominoes, so that they are safely supported on only two stones set up on end.

Dominoes

This, which might well seem impossible, is done by placing, as a foundation, dominoes in the positions indicated by dotted lines. The arch is then carefully constructed, as shown in the diagram, and for the finish the four stones between the two foundation arches are drawn out, and placed in pairs on end above, and finally, with the utmost care, the other four are drawn away, and built in on the top. Thus the stones indicated by the dotted lines at the base take their place within the dotted lines above.

This diagram represents a shallow box, on the bottom of which twelve counters or draughtsmen are lying loose.

Shallow box

How can they be readjusted so that they will wedge themselves together, and against the side of the box, and it can be turned upside down without displacing them?

Solution

Taken entireI’m full of fire.With head awayA tax I pay.If tail you barI turn from tar.Headless again,With tail restored.Goddess of pain,I sow discord.

Solution

The diagram below is an exact reproduction of an old-fashioned maze, cut in the ground near Nottingham. It is eighteen yards square, and the black line represents the pathway, which is 535 feet in length.

Maze

The point of this convoluted path is not so much to puzzle people, as to show how much ground may be covered without diverging far from a centre, or going over the same ground twice. As we advance along the line there are no obstructions, and we find ourselves, after passing over the whole of it, on the spot whence we set out.

Thrice three pins in shining lineMary meant to fix;Why did Mary turn the nineInto thirty-six?

Solution

Start atA, and trace these figures with one continuous line, finishing atB.

Maze

You must not take your pencil from the paper, or go over any line twice.

Solution

A ring and a wing and three-fourths of a fog,Will bring to your view a most obstinate dog.

Solution

Add fifty-seven to two-thirds of one,Then take a fiddle,And it will help to show you what is done,To make this riddle.

Solution

I am a fish so neat and clever,In pools and crystal streams I play,To find me out my name you severAs near the middle as you may.

Solution

Those who have not seen it will find some real fun in the following little experiment. Fix three matches as shown in the diagram, light the cross match in the middle, and watch to see which of the ends will first catch fire, or what will happen.

Matchbox

I stand stock still, let who will hurry,You cannot put me in a flurry,Nor stir my stumps, for all your worry.I am in haste, let none delay meAs fleetest couriers convey me.You must transpose me ere you stay me.

I stand stock still, let who will hurry,You cannot put me in a flurry,Nor stir my stumps, for all your worry.

I am in haste, let none delay meAs fleetest couriers convey me.You must transpose me ere you stay me.

Solution

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