Please see the [Transcriber’s Notes] at the end of this text.

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Also in Three Parts

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Also by the same Author

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PART I.

CONTENTS

MAGIC SQUARES

No. I.—FOUR HUNDRED YEARS OLD!

In Albert Dürer’s day, as in Milton’s, “melancholy” meant thoughtfulness, and on this ground we find on his woodcut, “Melancholia, or the Genius of the Industrial Science of Mechanics,” a very early instance of a Magic Square, showing that Puzzles had a recognised place in mental gymnastics four hundred years ago.

No. II.—A SIMPLE MAGIC SQUARE

Much time was devoted in olden days to the construction and elaboration of Magic Squares. Before we go more deeply into this fascinating subject, let us study the following pretty and ingenious method of making a Magic Square of sixteen numbers, which is comparatively simple, and easily committed to memory:—

Start with the small square at the top left-hand corner, placing there the 1; then count continuously from left to right, square by square, but only insert those numbers which fall upon the diagonals—namely, 4, 6, 7, 10, 11, 13, and 16.

Then start afresh at the bottom right-hand corner, calling it 1, and fill up the remaining squares in order, from right to left, counting continuously, and so placing in their turn 2, 3, 5, 8, 9, 12, 14, and 15. Each row, column, diagonal, and almost every cluster of four has 34 as the sum of its numbers.

No. III.—ANOTHER MAGIC SQUARE

In this Magic Square the rows, columns, and diagonals add up to 65, and the sum of any two opposite and corresponding squares is 26.

ENIGMAS

1
A MYSTIC ENIGMA

He stood himself beside himself
And looked into the sea;
Within himself he saw himself,
And at himself gazed he.
Now when himself he saw himself
Within himself go round,
Into himself he threw himself,
And in himself was drowned.
Now if he had not been himself,
But other beast beside,
He would himself have cut himself
Nor in himself have died.

[Solution]

No. IV.—A NEST OF CENTURIES

The numbers in this Magic Square of 49 cells add up in all rows, columns, and diagonals to 175. The four corner cells of every square or rectangle that has cell 25 in its centre, and cells 1, 7, 49, 43, add up to 100.

2

One morning Chloe, to avoid the heat,
Sat in a corner of a shady seat.
Young Strephon, on the self-same errand bound,
This fairest flower of all the garden found.
Her peerless beauty set his heart aflame,
Three monosyllables expressed his aim.

At a respectful distance he conversed
About the weather; then became immersed
In other topics, lessening the while
The space between them, heartened by her smile.
The same three simple words, now joined in one,
Expressed their happy state at set of sun.

[Solution]

No. V.—THE MAKING OF A MAGIC SQUARE

An ideal Magic Square can be constructed thus:

Place 1, 2, 3, 4, 5 in any order in the five top cells, set an asterisk over the third column, as shown in the diagram; begin the next row with this figure, and let the rest follow in the original sequence; continue this method with the other three rows.

Preparatory Square No. 1.

Preparatory Square No. 2.

Make a similar square of 25 cells with 0, 5, 10, 15, 20, as is shown in No. 2, placing the asterisk in this case over the fourth column of cells, and proceeding as before, in an unchanging sequence. Using these two preparatory squares, try to form a Magic Square in which the same number can be counted up in forty-two different ways.

[Solution]

No. VI.—ANOTHER WAY TO MAKE A MAGIC SQUARE

Here is one of many methods by which a Magic Square of the first twenty-five numbers can readily be made.

This is done by first placing the figures from 1 to 25 in diagonal rows, as is shown above, and then introducing the numbers that are outside the square into it, by moving each of them five places right, left, up, or down. A Magic Square is thus formed, the numbers of which add up to 65 in lines, columns and diagonals, and with the centre and any four corresponding numbers on the borders.

No. VII.—A MONSTER MAGIC SQUARE

Here is what may indeed be called a Champion Magic Square:—

Its 484 cells form, as they are numbered, a Magic Square, in which all rows, columns, and diagonals add up to 5335, and it is no easy matter to determine in how many other symmetrical ways its key-number can be found.

When the cells outside each of the dark border lines are removed, three other perfect Magic Squares remain.

Collectors should take particular note of this masterpiece.

No. VIII.—A NOVEL MAGIC SQUARE

A Magic Square of nine cells can be built up by taking any number divisible by 3, and placing, as a start, its third in the central cell. Thus:—

Say that 81 is chosen for the key number. Place 27 in the centre; 28, 29, in cells 1, 2; 30 in cell 7; 31 in 6; and then fill up cells 3, 4, 8, and 9 with the numbers necessary to make up 81 in each row, column, and diagonal.

Any number above 14 that is divisible by 3 can be dealt with in this way.

3

Enriched I am with much that’s fat,
Yet money I possess not;
Enlightening all who come to me,
True wisdom I express not.
I may be wicked, but protest
That sinful none have found me;
Though I destroy myself to be
Of use to those around me.

[Solution]

No. IX.—TWIN MAGIC SQUARES

Among the infinite number of Magic Squares which can be constructed, it would be difficult to find a more remarkable setting of the numbers 1 to 32 inclusive than this, in which two squares, each of 16 cells, are perfect twins in characteristics and curious combinations.

There are at least forty-eight different ways in which 66 is the sum of four of these numbers. Besides the usual rows, columns, and diagonals, any square group of four, both corner sets, all opposite pairs on the outer cells, and each set of corresponding cells next to the corners, add up exactly to 66.

4

Of Spanish extraction, my hue
Is as dark as a negro can be;
I am solid, and yet it is true
That in part I am wet as the sea,
My second and first are the same
In all but condition and name;
My second can burst
The abode of my first,
And my whole from the underground came.

[Solution]

No. X.—A BORDERED MAGIC SQUARE

Here is a notable specimen of a Magic Square:—

The rows, columns, and diagonals all add up to exactly 175 in the full square. Strip off the outside cells all around, and a second Magic Square remains, which adds up in all such ways to 125.

Strip off another border, as is again indicated by the darker lines, and a third Magic Square is left, which adds up to 75.

5
AN OLD ENIGMA
By Hannah More

I’m a strange contradiction: I’m new and I’m old,
I’m sometimes in tatters and sometimes in gold,
Though I never could read, yet letter’d I’m found,
Though blind I enlighten, though free I am bound.

I’m English, I’m German, I’m French, and I’m Dutch;
Some love me too dearly, some slight me too much.
I often die young, though I sometimes live ages,
And no Queen is attended by so many pages.

[Solution]

No. XI.—A LARGER BORDERED MAGIC SQUARE

Here is another example of what is called a “bordered” Magic Square:—

These 81 cells form a complete magic square, in which rows, columns, and diagonals add up to 369. As each border is removed fresh Magic Squares are formed, of which the distinctive numbers are 287, 205, and 123. The central 41 is in every case the greatest common divisor.

No. XII.—A CENTURY OF CELLS

Can you complete this Magic Square, so that the rows, columns, and diagonals add up in every case to 505?

We have given you a substantial start, and, as a further hint, as all the numbers in the first and last columns end in 0 or 1, so in the two next columns all end in 2 or 9, in the two next in 3 or 8, in the two next in 4 or 7, and in the two central columns in 5 or 6.

[Solution]

6
HALLAM’S UNSOLVED ENIGMA

I sit on a rock while I’m raising the wind,
But the storm once abated I’m gentle and kind.
I’ve Kings at my feet, who await but my nod
To kneel in the dust on the ground I have trod.
Though seen to the world, I am known to but few,
The Gentile detests me, I’m pork to the Jew.
I never have passed but one night in the dark,
And that was with Noah alone in the ark.
My weight is three pounds, my length is a mile.
And when I’m discovered you’ll say, with a smile,
That my first and my last are the pride of this isle.

[Solution]

No. XIII.—A SINGULAR MAGIC SQUARE

In this Magic Square, not only do the rows, columns, and diagonals add up to 260, but this same number is produced in three other and quite unusual ways:—

(1) Each group of 8 numbers, ranged in a circle round the centre; there are six of these, of which the smallest is 22, 28, 38, 44, 19, 29, 35, 45, and the largest is 8, 10, 56, 58, 1, 15, 49, 63. (2) The sum of the 4 central numbers and 4 corners. (3) The diagonal cross of 4 numbers in the middle of the board.

No. XIV.—SQUARING THE YEAR

On [another page] we give an interesting Magic Square of 121 cells based upon the figures of the year 1892. Here, in much more condensed form, is one more up to date.

The rows, columns, and diagonals of these nine cells add up in all cases to the figures of the year 1902.

The central 634 is found by dividing 1902 by its lowest factor greater than 2, and this is taken as the middle term of nine numbers, which are thus arranged to form a Magic Square.

7
RANK TREASON
By an Irish Rebel, 1798

The pomps of Courts and pride of Kings
I prize above all earthly things;
I love my country, but the King
Above all men his praise I sing.
The royal banners are displayed,
And may success the standard aid!

I fain would banish far from hence
The “Rights of Men” and “Common Sense;”
Confusion to his odious reign,
That Foe to princes, Thomas Payne.
Defeat and ruin seize the cause
Of France, its liberties and laws!

Where does the treason come in?

[Solution]

No. XV.—SQUARING ANOTHER YEAR

The following square of numbers is interesting in connection with the year 1906.

and the sum, in every case, is 1906.

No. XVI.—MANIFOLD MAGIC SQUARES

Here is quite a curious nest of clustered Magic Squares, which is worth preserving:—

Every square of every possible combination of 25 of these numbers in their cells, such as the two with darker borders, is a perfect Magic Square, with rows, columns, and diagonals that add up in all cases to 65.

8
AN ENIGMA FOR CHRISTMAS HOLIDAYS

Formed half beneath and half above the earth,
We owe, as twins, to art our second birth.
The smith’s and carpenter’s adopted daughters,
Made upon earth, we travel on the waters.
Swifter we move as tighter we are bound,
Yet never touch the sea, or air, or ground.
We serve the poor for use, the rich for whim,
Sink if it rains, and if it freezes swim.

[Solution]

No. XVII.—LARGER AUXILIARY MAGIC SQUARES

A very interesting method of constructing a Magic Square is shown in these three diagrams:—

It will be noticed that each row after the first, in the two upper auxiliary squares, begins with a number from the same column in the row above it, and maintains the same sequence of numbers. When the corresponding cells of these two squares are added together, and placed in the third square, a Magic Square is formed, in which 671 is the sum of all rows, columns, and diagonals.

No. XVIII.—SQUARING BY ANNO DOMINI

Here is a curious form of Magic Square. The year 1892 is taken as its basis.

Within this square 1892 can be counted up in all the usual ways, and altogether in 44 variations. Thus any two rows that run parallel to a diagonal, and have between them eleven cells, add up to this number, if they are on opposite sides of the diagonal.

9

The sun, the sun is my delight!
I shun a gloomy day,
Though I am often seen at night
To dart across the way.
Sometimes you see me climb a wall
As nimble as a cat,
Then down into a pit I fall
Like any frightened rat.
Catch me who can—woman or man—
None have succeeded who after me ran.

[Solution]

No. XIX.—A MAGIC SQUARE OF SEVEN

This Magic Square of 49 cells is constructed with a diagonal arrangement of the numbers from 1 to 49 in their proper order. Those that fall outside the central square are written into it in the seventh cell inwards from where they stand. It is interesting to find out the many combinations in which the number 175 is made up.

10
WHAT MOVED HIM?

I grasped it, meaning nothing wrong,
And moved to meet my friend,
When lo! the stalwart man and strong
At once began to bend.
The biped by the quadruped
No longer upright stood,
But bowed the knee and bent his head
Before the carved wood.

[Solution]

No. XX.—CURIOUS SQUARES

These are two interesting Magic Squares found on an antique gong, at Caius College, Cambridge:—

In the one nine numbers are so arranged that they count up to 27 in every direction; and in the other the outer rows total 30, while the central rows and diagonals make 40.

11
RINGING THE CHANGES

My figure, singular and slight,
Measures but half enough at sight.
I rode the waters day and night.
I tell the new in Time’s quick flight,
Or how old ages rolled in might.
Cut off my tail, it still is on!
Put on my head, and there is none!

[Solution]

No. XXI.—A MOORISH MAGIC SQUARE

Among Moorish Mussulmans 78 is a mystic number.

Here is a cleverly-constructed Magic Square, to which this number is the key.

The number 78 can be arrived at in twenty-three different combinations—namely, ten rows, columns, or diagonals; four corner squares of four cells; one central square of four cells; the four corner cells; two sets of corresponding diagonal cells next to the corners; and two sets of central cells on the top and bottom rows, and on the outside columns.

No. XXII.—A CHOICE MAGIC SQUARE

Here is a Magic Square of singular charm:—

The 81 cells of this remarkable square are divided by parallel lines into 9 equal parts, each made up of 9 consecutive numbers, and each a Magic Square in itself within the parent square. Readers can work out for themselves the combinations in the larger square and in the little ones.

12
CANNING’S ENIGMA

There is a noun of plural number,
Foe to peace and tranquil slumber.
Now almost any noun you take
By adding “s” you plural make.
But if you add an “s” to this
Strange is the metamorphosis.
Plural is plural now no more,
And sweet what bitter was before.

[Solution]

XXIII.—THE TWIN PUZZLE SQUARES

Fill each square by repeating two of its figures in the vacant cells. Then rearrange them all, so that the sums of the corresponding rows in each square are equal, and the sums of the squares of the corresponding cells of these rows are also equal; and so that the sums of the four diagonals are equal, and the sum of the squares of the cells in corresponding diagonals are equal.

[Solution]

13

There is an old-world charm about this Enigma:—

In the ears of young and old
I repeat what I am told;
And they hear me, old and young,
Though I have no busy tongue.
When a thunder-clap awakes me
Not a touch of terror takes me;
Yet so tender is my ear
That the softest sound I fear.
Call me not with bated breath,
For a whisper is my death.

[Solution]

No: XXIV.—MAGIC FRACTIONS

Here is an arrangement of fractions which form a perfect Magic Square:—

If these fractions are added together in any one of the eight directions, the result in every case is unity. Thus ³⁄₈ + ¹⁄₃ + ⁷⁄₂₄ = 1, ¹⁄₆ + ¹⁄₃ + ¹⁄₂ = 1, and so on throughout the rows, columns, and diagonals.

14
“DOUBLE, DOUBLE, TOIL AND TROUBLE!”

“By hammer and hand
All arts do stand”—
So says an ancient saw;
But hammer and hand
Will work or stand
By my unwritten law.
Behold me, as sparks from the anvils fly,
But fires lie down at my bitter cry.

[Solution]

No. XXV.—MORE MAGIC FRACTIONS

We are indebted to a friend for the following elaborate Magic Square of fractions, on the lines of that on the preceding page.

The composer claims that there are at least 160 combinations of 5 cells in which these fractions add up to unity, including, of course, the usual rows, columns, and diagonals.

15

Two brothers wisely kept apart,
Together ne’er employed;
Though to one purpose we are bent
Each 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]

No. XXVI.—A MAGIC OBLONG

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.

16

My name declares my date to be
The 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]

17

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

[Solution]

No. XXVII.—A MAGIC CUBE

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

First Layer from Top.

Second Layer from Top.

Third Layer from Top.

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.

18

Sad offspring of a blighted race,
Pale Sorrow was my mother;
I’ve never seen the smiling face
Of 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]

No. XXVIII.—A MAGIC CIRCLE

The Magic Circle below has this particular property:—

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.

19

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

[Solution]

No. XXIX.—MAGIC CIRCLE OF CIRCLES

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.

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.

No. XXX.—THE UNIQUE TRIANGLE

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?

[Solution]

20

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]

21

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 yet
Grace many temples in the land.

[Solution]

No. XXXI.—MAGIC TRIANGLES

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?

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

[Solution]

22

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 queer
And hyperbolic fable,
Pray let there follow after that
Whatever may be able.

[Solution]

No. XXXII.—TWIN TRIANGLES

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

The twins are also closely allied on these points:—

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.

No. XXXIII.—A MAGIC HEXAGON

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.

23

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]

No. XXXIV.—MAGIC HEXAGON IN A CIRCLE

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