Chapter V: Hand Shadows and how to work them. Illustrated (1)

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THE TWENTIETH CENTURY
=STANDARD PUZZLE
BOOK=
THREE PARTS IN ONE VOLUME

EDITED BY
A. CYRIL PEARSON, M.A.
AUTHOR OF
‘_100 Chess Problems_,’ ‘_Anagrams, Ancient and Modern_,’ _etc._

_PROFUSELY ILLUSTRATED_

SECOND IMPRESSION

LONDON
GEORGE ROUTLEDGE & SONS, LTD.
NEW YORK: E. P. DUTTON & CO.

=Also in Three Parts=

I.--MAGIC SQUARES, PICTURE PUZZLES, ENIGMAS, CHARADES, RIDDLES,
CONUNDRUMS, NUTS TO CRACK, SOLUTIONS.

II.--OPTICAL ILLUSIONS, FREAKS OF FIGURES, CHESS CAMEOS, SCIENCE AT
PLAY, CURIOUS CALCULATIONS, WORD AND LETTER PUZZLES, SOLUTIONS.

III.--WORD PUZZLES, MISSING WORDS, LETTER PUZZLES, ANAGRAMS, PICTURE
PUZZLES, PALINDROMES, SOLUTIONS.

=Also by the same Author=

PICTURED PUZZLES AND WORD PLAY. Profusely Illustrated. Crown 8vo.
Cloth.

=PART I.=

CONTENTS

PAGE

MAGIC SQUARES, PUZZLES, TRICKS, ENIGMAS I-1

CHARADES, ETC. I-80

RIDDLES AND CONUNDRUMS I-104

NUTS TO CRACK I-115

SOLUTIONS I-148

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:--

+--+--+--+--+
| 1|15|14| 4|
+--+--+--+--+
|12| 6| 7| 9|
+--+--+--+--+
| 8|10|11| 5|
+--+--+--+--+
|13| 3|2 |16|
+--+--+--+--+

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

+--+--+--+--+--+
| 1|20|16|23| 5|
+--+--+--+--+--+
|15| 7|12| 9|22|
+--+--+--+--+--+
|24|18|13| 8| 2|
+--+--+--+--+--+
| 4|17|14|19|11|
+--+--+--+--+--+
|21| 3|10| 6|25|
+--+--+--+--+--+

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.

No. IV.--A NEST OF CENTURIES

+--+--+--+--+--+--+--+
|22|47|16|41|10|35| 4|
+--+--+--+--+--+--+--+
| 5|23|48|17|42|11|29|
+--+--+--+--+--+--+--+
|30| 6|24|49|18|36|12|
+--+--+--+--+--+--+--+
|13|31| 7|25|43|19|37|
+--+--+--+--+--+--+--+
|38|14|32| 1|26|44|20|
+--+--+--+--+--+--+--+
|21|39| 8|33| 2|27|45|
+--+--+--+--+--+--+--+
|46|15|40| 9|34| 3|28|
+--+--+--+--+--+--+--+

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.

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.

*
+--+--+--+--+--+
| 1| 3| 5| 2| 4|
+--+--+--+--+--+
| 5| 2| 4| 1| 3|
+--+--+--+--+--+
| 4| 1| 3| 5| 2|
+--+--+--+--+--+
| 3| 5| 2| 4| 1|
+--+--+--+--+--+
| 2| 4| 1| 3| 5|
+--+--+--+--+--+

PREPARATORY SQUARE NO. 2.

*
+--+--+--+--+--+
| 5|15| 0|10|20|
+--+--+--+--+--+
|10|20| 5|15| 0|
+--+--+--+--+--+
|15| 0|10|20| 5|
+--+--+--+--+--+
|20| 5|15| 0|10|
+--+--+--+--+--+
| 0|10|20| 5|15|
+--+--+--+--+--+

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.

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.

+--+
| 1|
+--+--+--+
| 2| | 6|
+==+==+==+==+==+
∥ 3|20| 7|24|11∥
+--+--+--+--+--+--+--+
| 4∥16| 8|25|12| 4∥16|
+--+--+--+--+--+--+--+--+--+
| 5| ∥ 9|21|13| 5|17∥ |21|
+--+--+--+--+--+--+--+--+--+
|10∥22|14| 1|18|10∥22|
+--+--+--+--+--+--+--+
∥15| 2|19| 6|23∥
+==+==+==+==+==+
|20| |24|
+--+--+--+
|25|
+--+

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:--

+---+---+---+---+---+---+---+---+---+---+---+
| 23|464|459|457|109|111|108|110|132|133|130|
+---+---+---+---+---+---+---+---+---+---+---+
| 25| 41|436|435|433|432|196|195|241|242|200|
+---+---+---+---+---+---+---+---+---+---+---+
| 27| 45| 13|474|469|467| 82| 81| 72| 90| 91|
+---+---+---+===+===+===+===+===+===+===+===+
|461| 55| 15∥ 34|450|449|447|446|156|157|180|
+---+---+---+---+---+---+---+---+---+---+---+
|456| 56| 17∥ 42| 3|484|479|477| 66| 65| 68|
+---+---+---+---+---+===+===+===+===+===+===+
|137|428|471∥ 41| 5∥127|126|125|361|362|363|
+---+---+---+---+---+---+===+===+===+===+===+
|153|431|466∥ 31| 7∥347∥148|338|339|145|143|
+---+---+---+---+---+---+---+---+---+---+---+
|154|439| 98∥453|481∥325∥161|169|168|318|319|
+---+---+---+---+---+---+---+---+---+---+---+
|384|266|407∥445|476∥292∥293|191|190|299|298|
+---+---+---+---+---+---+---+---+---+---+---+
|383|268|406∥442|424∥270∥280|272|273|211|210|
+---+---+---+---+---+---+---+---+---+---+---+
|379|265|392∥172| 60∥248∥227|250|251|230|232|
+---+---+---+---+---+---+---+---+---+---+---+
|378|267|391∥173| 59∥226∥249|228|229|252|254|
+---+---+---+---+---+---+---+---+---+---+---+
|351|282|405∥176| 74∥204∥214|206|207|277|276|
+---+---+---+---+---+---+---+---+---+---+---+
|350|263|390∥177| 73∥182∥192|301|300|189|187|
+---+---+---+---+---+---+---+---+---+---+---+
|334|199| 77∥330|423∥171∥315|323|322|164|165|
+---+---+---+---+---+---+---+---+---+---+---+
|333|216| 96∥311|413∥149∥346|147|146|340|341|
+---+---+---+---+---+---+===+===+===+===+===+
|100|221| 76∥310|414∥369|359|360|124|123|122|
+---+---+---+---+---+===+===+===+===+===+===+
| 99|223| 75∥291|483| 1| 6| 8|419|420|417|
+---+---+---+---+---+---+---+---+---+---+---+
|104|202| 97∥452| 35| 36| 38| 39|329|328|305|
+---+---+---+===+===+===+===+===+===+===+===+
|105|238|473| 11| 16| 18|403|404|393|395|394|
+---+---+---+---+---+---+---+---+---+---+---+
|136|438| 49| 50| 52| 53|289|290|244|243|285|
+---+---+---+---+---+---+---+---+---+---+---+
|463| 21| 26| 28|376|374|377|375|353|352|355|
+---+---+---+---+---+---+---+---+---+---+---+

+---+---+---+---+---+---+---+---+---+---+---+
|131|373|371|357|356|372|382|370|335| 30| 22|
+---+---+---+---+---+---+---+---+---+---+---+
|225|284|287|246|245|288|261| 51| 58| 47|460|
+---+---+---+---+---+---+---+---+---+---+---+
| 83|401|400|396|398|399|397| 20| 12|440|458|
+===+===+===+===+===+===+===+===+---+---+---+
|181|326|327|306|307| 44| 37| 33∥470|430| 24|
+---+---+---+---+---+---+---+---+---+---+---+
| 67|422|421|416|415| 10| 2|443∥468|429| 29|
+===+===+===+===+===+===+---+---+---+---+---+
|364|365|366|118|117|116∥480|444∥ 14| 57|348|
+===+===+===+===+===+---+---+---+---+---+---+
|342|142|344|345|139∥138∥478|454∥ 19| 54|332|
+---+---+---+---+---+---+---+---+---+---+---+
|320|321|163|162|324∥160∥ 4|32 ∥387| 46|331|
+---+---+---+---+---+---+---+---+---+---+---+
|297|186|185|184|302∥193∥ 9| 40∥ 78|219|101|
+---+---+---+---+---+---+---+---+---+---+---+
|209|208|278|279|205∥215∥ 61| 43∥ 79|217|102|
+---+---+---+---+---+---+---+---+---+---+---+
|231|233|256|257|258∥237∥425|313∥ 93|220|106|
+---+---+---+---+---+---+---+---+---+---+---+
|253|255|234|235|236∥259∥426|312∥ 94|218|107|
+---+---+---+---+---+---+---+---+---+---+---+
|275|274|212|213|271∥281∥411|309∥ 80|203|134|
+---+---+---+---+---+---+---+---+---+---+---+
|188|296|295|294|183∥303∥412|308∥ 95|222|135|
+---+---+---+---+---+---+---+---+---+---+---+
|166|167|317|316|170∥314∥ 62|155∥408|286|151|
+---+---+---+---+---+---+---+---+---+---+---+
|144|343|141|140|337∥336∥ 72|174∥389|269|152|
+===+===+===+===+===+---+---+---+---+---+---+
|121|120|119|367|368|358∥ 71|175∥409|264|385|
+===+===+===+===+===+===+---+---+---+---+---+
|418| 63| 64| 69| 70|475|482|194∥410|262|386|
+---+---+---+---+---+---+---+---+---+---+---+
|304|159|158|179|178|441|448|451∥388|283|381|
+===+===+===+===+===+===+===+===+---+---+---+
|402| 84| 85| 89| 87| 86| 88|465|472|247|380|
+---+---+---+---+---+---+---+---+---+---+---+
|260|201|198|239|240|197|224|434|427|437|349|
+---+---+---+---+---+---+---+---+---+---+---+
|354|112|114|128|129|113|103|115|150|455|462|
+---+---+---+---+---+---+---+---+---+---+---+

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:--

+--+--+--+
|28|29|24|
+--+--+--+
|23|27|31|
+--+--+--+
|30|25|26|
+--+--+--+

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.

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.

+--+--+--+--++--+--+--+--+
| 1| 8|29|28||11|14|23|18|
+--+--+--+--++--+--+--+--+
|30|27| 2| 7||21|20| 9|16|
+--+--+--+--++--+--+--+--+
| 4| 5|32|25||10|15|22|19|
+--+--+--+--++--+--+--+--+
|31|26| 3| 6||24|17|12|13|
+--+--+--+--++--+--+--+--+

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.

No. X.--A BORDERED MAGIC SQUARE

Here is a notable specimen of a Magic Square:--

+--+--+--+--+--+--+--+
| 4| 5| 6|43|39|38|40|
+--+==+==+==+==+==+--+
|49∥15|16|33|30|31∥ 1|
+--+--+==+==+==+--+--+
|48∥37∥22|27|26∥13∥ 2|
+--+--+--+==+--+--+--+
|47∥36∥29∥25∥21∥14∥ 3|
+--+--+--+==+--+--+--+
| 8∥18∥24|23|28∥32∥42|
+--+--+==+==+==+--+--+
| 9∥19|34|17|20|35∥41|
+--+==+==+==+==+==+--+
|10|45|44| 7|11|12|46|
+--+--+--+--+--+--+--+

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.

No. XI.--A LARGER BORDERED MAGIC SQUARE

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

+--+--+--+--+--+--+--+--+--+
| 5|80|59|73|61| 3|63|12|13|
+--+==+==+==+==+==+==+==+--+
| 1∥20|55|30|57|28|71|26∥81|
+--+--+==+==+==+==+==+--+--+
| 4∥14∥31|50|29|60|35∥68∥78|
+--+--+--+==+==+==+--+--+--+
|76∥58∥46∥38|45|40∥36∥24∥ 6|
+--+--+--+--+--+--+--+--+--+
| 7∥65∥33∥43|41|39∥49∥17∥75|
+--+--+--+--+--+--+--+--+--+
|74∥64∥48∥42|37|44∥31∥18∥ 8|
+--+--+--+==+==+==+--+--+--+
|67∥10∥47|32|53|22|51∥72∥15|
+--+--+==+==+==+==+==+--+--+
|66∥56|27|52|25|54|11|62∥16|
+--+==+==+==+==+==+==+==+--+
|69| 2|23| 9|21|79|19|70|77|
+--+--+--+--+--+--+--+--+--+

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?

+----+----+----+----+----+----+----+----+----+-----+
|_91_| _2_| _3_|_97_| _6_|_95_|_94_| _8_| _9_|_100_|
+----+----+----+----+----+----+----+----+----+-----+
|_20_| | | |_16_|_15_| | | | _81_|
+----+----+----+----+----+----+----+----+----+-----+
|_21_| | | |_25_|_26_| | | | _30_|
+----+----+----+----+----+----+----+----+----+-----+
|_60_| | | |_66_|_65_| | | | _41_|
+----+----+----+----+----+----+----+----+----+-----+
|_50_|_49_|_48_|_57_|_55_|_56_|_54_|_43_|_42_| _51_|
+----+----+----+----+----+----+----+----+----+-----+
|_61_|_59_|_58_|_47_|_45_|_46_|_44_|_53_|_52_| _40_|
+----+----+----+----+----+----+----+----+----+-----+
|_31_| | | |_35_|_36_| | | | _70_|
+----+----+----+----+----+----+----+----+----+-----+
|_80_| | | |_75_|_76_| | | | _71_|
+----+----+----+----+----+----+----+----+----+-----+
|_90_| | | |_86_|_85_| | | | _11_|
+----+----+----+----+----+----+----+----+----+-----+
| _1_|_99_|_98_| _4_|_96_| _5_| _7_|_93_|_92_| _10_|
+----+----+----+----+----+----+----+----+----+-----+

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.

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.

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:--

+--+--+--+--+--+--+--+--+
|18|63| 4|61| 6|59| 8|41|
+--+--+--+--+--+--+--+--+
|49|32|51|14|53|12|39|10|
+--+--+--+--+--+--+--+--+
| 2|47|36|45|22|27|24|57|
+--+--+--+--+--+--+--+--+
|33|16|35|46|21|28|55|26|
+--+--+--+--+--+--+--+--+
|31|50|29|20|43|38| 9|40|
+--+--+--+--+--+--+--+--+
|64|17|30|19|44|37|42| 7|
+--+--+--+--+--+--+--+--+
|15|34|13|52|11|54|25|56|
+--+--+--+--+--+--+--+--+
|48| 1|62| 3|60| 5|58|23|
+--+--+--+--+--+--+--+--+

(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.

+---+---+---+
|637|630|635|
+---+---+---+
|632|634|636|
+---+---+---+
|633|638|631|
+---+---+---+

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?

No. XV.--SQUARING ANOTHER YEAR

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

+------+------+------+------+
|_A_ |_B_ |_C_ |_D_ |
| _476_| _469_| _477_| _484_|
+------+------+------+------+
|_E_ |_F_ |_G_ |_H_ |
| _483_| _478_| _470_| _475_|
+------+------+------+------+
|_I_ |_J_ |_K_ |_L_ |
| _471_| _474_| _482_| _479_|
+------+------+------+------+
|_M_ |_N_ |_O_ |_P_ |
| _480_| _481_| _473_| _472_|
+------+------+------+------+

Add the rows --ABCD, EFGH, IJKL, MNOP.
or the squares --ABEF, CDGH, IJMN, KLOP.
or semi-diagonals--AFIN, BEJM, CHKP, DGLO,
AFCH, BEGD, INKP, MJOL.

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:--

+--+--+--+--+--+--+--+--+--+--+--+
| 2|13|24|10|16| 2|13|24|10|16| 2|
+--+--+--+--+--+--+--+--+--+--+--+
| 9|20| 1|12|23| 9|20| 1|12|23| 9|
+--+--+--+--+--+==+==+==+==+==+--+
|11|22| 8|19| 5∥11|22| 8|19| 5∥11|
+--+--+--+--+--+--+--+--+--+--+--+
|18| 4|15|21| 7∥18| 4|15|21| 7∥18|
+--+--+--+--+--+--+--+--+--+--+--+
|25| 6|17| 3|14∥25| 6|17| 3|14∥25|
+--+--+--+--+--+--+--+--+--+--+--+
| 2|13|24|10|16∥ 2|13|24|10|16∥ 2|
+--+--+==+==+==+==+==+--+--+--+--+
| 9|20∥ 1|12|23∥ 9|20∥ 1|12|23∥ 9|
+--+--+--+--+--+==+==+==+==+==+--+
|11|22∥ 8|19| 5|11|22∥ 8|19| 5|11|
+--+--+--+--+--+--+--+--+--+--+--+
|18| 4∥15|21| 7|18| 4∥15|21| 7|18|
+--+--+--+--+--+--+--+--+--+--+--+
|25| 6∥17| 3|14|25| 6∥17| 3|14|25|
+--+--+--+--+--+--+--+--+--+--+--+
| 2|13∥24|10|16| 2|13∥24|10|16| 2|
+--+--+==+==+==+==+==+--+--+--+--+
| 9|20| 1|12|23| 9|20| 1|12|23| 9|
+--+--+--+--+--+--+--+--+--+--+--+
|11|22| 8|19| 5|11|22| 8|19| 5|11|
+--+--+--+--+--+--+--+--+--+--+--+

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.

No. XVII.--LARGER AUXILIARY MAGIC SQUARES

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

+---+---+---+---+---+---+---+---+---+---+---+
| 1| 2| 3| 4| 5| 6| 7| 8| 9| 10| 11|
+---+---+---+---+---+---+---+---+---+---+---+
| 3| 4| 5| 6| 7| 8| 9| 10| 11| 1| 2|
+---+---+---+---+---+---+---+---+---+---+---+
| 5| 6| 7| 8| 9| 10| 11| 1| 2| 3| 4|
+---+---+---+---+---+---+---+---+---+---+---+
| 7| 8| 9| 10| 11| 1| 2| 3| 4| 5| 6|
+---+---+---+---+---+---+---+---+---+---+---+
| 9| 10| 11| 1| 2| 3| 4| 5| 6| 7| 8|
+---+---+---+---+---+---+---+---+---+---+---+
| 11| 1| 2| 3| 4| 5| 6| 7| 8| 9| 10|
+---+---+---+---+---+---+---+---+---+---+---+
| 2| 3| 4| 5| 6| 7| 8| 9| 10| 11| 1|
+---+---+---+---+---+---+---+---+---+---+---+
| 4| 5| 6| 7| 8| 9| 10| 11| 1| 2| 3|
+---+---+---+---+---+---+---+---+---+---+---+
| 6| 7| 8| 9| 10| 11| 1| 2| 3| 4| 5|
+---+---+---+---+---+---+---+---+---+---+---+
| 8| 9| 10| 11| 1| 2| 3| 4| 5| 6| 7|
+---+---+---+---+---+---+---+---+---+---+---+
| 10| 11| 1| 2| 3| 4| 5| 6| 7| 8| 9|
+---+---+---+---+---+---+---+---+---+---+---+

+---+---+---+---+---+---+---+---+---+---+---+
| 0| 11| 22| 33| 44| 55| 66| 77| 88| 99|110|
+---+---+---+---+---+---+---+---+---+---+---+
| 33| 44| 55| 66| 77| 88| 99|110| 0| 11| 22|
+---+---+---+---+---+---+---+---+---+---+---+
| 66| 77| 88| 99|110| 0| 11| 22| 33| 44| 55|
+---+---+---+---+---+---+---+---+---+---+---+
| 99|110| 0| 11| 22| 33| 44| 55| 66| 77| 88|
+---+---+---+---+---+---+---+---+---+---+---+
| 11| 22| 33| 44| 55| 66| 77| 88| 99|110| 0|
+---+---+---+---+---+---+---+---+---+---+---+
| 44| 55| 66| 77| 88| 99|110| 0| 11| 22| 33|
+---+---+---+---+---+---+---+---+---+---+---+
| 77| 88| 99|110| 0| 11| 22| 33| 44| 55| 66|
+---+---+---+---+---+---+---+---+---+---+---+
|110| 0| 11| 22| 33| 44| 55| 66| 77| 88| 99|
+---+---+---+---+---+---+---+---+---+---+---+
| 22| 33| 44| 55| 66| 77| 88| 99|110| 0| 11|
+---+---+---+---+---+---+---+---+---+---+---+
| 55| 66| 77| 88| 99|110| 0| 11| 22| 33| 44|
+---+---+---+---+---+---+---+---+---+---+---+
| 88| 99|110| 0| 11| 22| 33| 44| 55| 66| 77|
+---+---+---+---+---+---+---+---+---+---+---+

+---+---+---+---+---+---+---+---+---+---+---+
| 1| 13| 25| 37| 49| 61| 73| 85| 97|109|121|
+---+---+---+---+---+---+---+---+---+---+---+
| 36| 48| 60| 72| 84| 96|108|120| 11| 12| 24|
+---+---+---+---+---+---+---+---+---+---+---+
| 71| 83| 95|107|119| 10| 22| 23| 35| 47| 59|
+---+---+---+---+---+---+---+---+---+---+---+
|106|118| 9| 21| 33| 34| 46| 58| 70| 82| 94|
+---+---+---+---+---+---+---+---+---+---+---+
| 20| 32| 44| 45| 57| 69| 81| 93|105|117| 8|
+---+---+---+---+---+---+---+---+---+---+---+
| 55| 56| 68| 80| 92|104|116| 7| 19| 31| 43|
+---+---+---+---+---+---+---+---+---+---+---+
| 79| 91|103|115| 6| 18| 30| 42| 54| 66| 67|
+---+---+---+---+---+---+---+---+---+---+---+
|114| 5| 17| 29| 41| 53| 65| 77| 78| 90|102|
+---+---+---+---+---+---+---+---+---+---+---+
| 28| 40| 52| 64| 76| 88| 89|101|113| 4| 16|
+---+---+---+---+---+---+---+---+---+---+---+
| 63| 75| 87| 99|100|112| 3| 15| 27| 39| 51|
+---+---+---+---+---+---+---+---+---+---+---+
| 98|110|111| 2| 14| 26| 38| 50| 62| 74| 86|
+---+---+---+---+---+---+---+---+---+---+---+

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.

+---+---+---+---+---+---+---+---+---+---+---+
|112|124|136|148|160|172|184|196|208|220|232|
+---+---+---+---+---+---+---+---+---+---+---+
|147|159|171|183|195|207|219|231|122|123|135|
+---+---+---+---+---+---+---+---+---+---+---+
|182|194|206|218|230|121|133|134|146|158|170|
+---+---+---+---+---+---+---+---+---+---+---+
|217|229|120|132|144|145|157|169|181|193|205|
+---+---+---+---+---+---+---+---+---+---+---+
|131|143|155|156|168|180|192|204|216|228|119|
+---+---+---+---+---+---+---+---+---+---+---+
|166|167|179|191|203|215|227|118|130|142|154|
+---+---+---+---+---+---+---+---+---+---+---+
|190|202|214|226|117|129|141|153|165|177|178|
+---+---+---+---+---+---+---+---+---+---+---+
|225|116|128|140|152|164|176|188|189|201|213|
+---+---+---+---+---+---+---+---+---+---+---+
|139|151|163|175|187|199|200|212|224|115|127|
+---+---+---+---+---+---+---+---+---+---+---+
|174|186|198|210|211|223|114|126|138|150|162|
+---+---+---+---+---+---+---+---+---+---+---+
|209|221|222|113|125|137|149|161|173|185|197|
+---+---+---+---+---+---+---+---+---+---+---+

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.

No. XIX.--A MAGIC SQUARE OF SEVEN

+----+
| _1_|
+----+----+----+
| _8_| | _2_|
+----+----+----+----+----+
|_15_| | _9_| | _3_|
+====+====+====+====+====+====+====+
∥_22_|_47_|_16_|_41_|_10_|_35_| _4_∥
+----+----+----+----+----+----+----+----+----+
|_29_∥ _5_|_23_|_48_|_17_|_42_|_11_|_29_∥ _5_|
+----+----+----+----+----+----+----+----+----+----+----+
|_36_| ∥_30_| _6_|_24_|_49_|_18_|_36_|_12_∥ | _6_|
+----+----+----+----+----+----+----+----+----+----+----+----+----+
|_43_| |_37_∥_13_|_31_| _7_|_25_|_43_|_19_|_37_∥_13_| | _7_|
+----+----+----+----+----+----+----+----+----+----+----|----+----+
|_44_| ∥_38_|_14_|_32_| _1_|_26_|_44_|_20_∥ |_14_|
+----+----+----+----+----+----+----+----+----+----+----+
|_45_∥_21_|_39_| _8_|_33_| _2_|_27_|_45_∥_21_|
+----+----+----+----+----+----+----+----+----+
∥_46_|_15_|_40_| _9_|_34_| _3_|_28_∥
+====+====+====+====+====+====+====+
|_47_| |_41_| |_35_|
+----+----+----+----+----+
|_48_| |_42_|
+----+----+----+
|_49_|
+----+

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.

No. XX.--CURIOUS SQUARES

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

+--+--+--+ +--+--+--+
| 6|13| 8| | 7|14| 9|
+--+--+--+ +--+--+--+
|11| 9| 7| |12|20| 8|
+--+--+--+ +--+--+--+
|10| 5|12| |11| 6|13|
+--+--+--+ +--+--+--+

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!

No. XXI.--A MOORISH MAGIC SQUARE

Among Moorish Mussulmans 78 is a mystic number.

+--+--+--+--+
|40|10|20| 8|
+--+--+--+--+
| 7|21| 9|41|
+--+--+--+--+
|12|42| 6|18|
+--+--+--+--+
|19| 5|43|11|
+--+--+--+--+

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:--

+==+==+==+==+==+==+==+==+==+
|31|36|29∥76|81|74∥13|18|11|
+--+--+--+--+--+--+--+--+--+
|30|32|34∥75|77|79∥12|14|16|
+--+--+--+--+--+--+--+--+--+
|35|28|33∥80|73|78∥17|10|15|
+==+==+==+==+==+==+==+==+==+
|22|27|20∥40|45|38∥58|63|56|
+--+--+--+--+--+--+--+--+--+
|21|23|25∥39|41|43∥57|59|61|
+--+--+--+--+--+--+--+--+--+
|26|19|24∥44|37|42∥62|55|60|
+==+==+==+==+==+==+==+==+==+
|67|72|65∥ 4| 9| 2∥49|54|47|
+--+--+--+--+--+--+--+--+--+
|66|68|70∥ 3| 5| 7∥48|50|52|
+--+--+--+--+--+--+--+--+--+
|71|64|69∥ 8| 1| 6∥53|46|51|
+==+==+==+==+==+==+==+==+==+

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.

XXIII.--THE TWIN PUZZLE SQUARES

+-+-+-+ +-+-+-+
|1|2|3| | |2|3|
+-+-+-+ +-+-+-+
| |5|6+====+4|5| |
+-+-+-+ +-+-+-+
|7|8| | |7|8|9|
+-+-+-+ +-+-+-+

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.

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.

No: XXIV.--MAGIC FRACTIONS

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

+-------+-------+-------+
| _³⁄₈_ | _⁵⁄₁₂_| _⁵⁄₂₄_|
+-------+-------+-------+
| _¹⁄₆_ | _¹⁄₃_ | _¹⁄₂_ |
+-------+-------+-------+
|_¹¹⁄₂₄_| _¹⁄₄_ | _⁷⁄₂₄_|
+-------+-------+-------+

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.

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.

+-------+-------+-------+-------+-------+
|_¹⁹⁄₈₀_| _⁷⁄₂₀_| _¹⁄₄₀_|_¹¹⁄₈₀_| _¹⁄₄_ |
+-------+-------+-------+-------+-------+
|_¹³⁄₄₀_| _¹⁄₈_ | _⁹⁄₈₀_| _⁹⁄₄₀_|_¹⁷⁄₈₀_|
+-------+-------+-------+-------+-------+
| _¹⁄₁₀_| _⁷⁄₈₀_| _¹⁄₅_ | _⁵⁄₁₆_| _³⁄₁₀_|
+-------+-------+-------+-------+-------+
| _³⁄₁₆_| _⁷⁄₄₀_|_²³⁄₈₀_|_¹¹⁄₄₀_| _³⁄₄₀_|
+-------+-------+-------+-------+-------+
| _³⁄₂₀_|_²¹⁄₈₀_| _³⁄₈_ | _¹⁄₂₀_|_¹³⁄₈₀_|
+-------+-------+-------+-------+-------+

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.

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.

+--+--+--+--+--+--+--+--+
| 1|10|11|29|28|19|18|16|
+--+--+--+--+--+--+--+--+
| 9| 2|30|12|20|27| 7|25|
|--+--+--+--+--+--+--+--|
|24|31| 3|21|13| 6|26| 8|
|--+--+--+--+--+--+--+--|
|32|23|22| 4| 5|14|15|17|
+--+--+--+--+--+--+--+--+

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.

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.

No. XXVII.--A MAGIC CUBE

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

First Layer from Top.

+---+---+---+---+---+
|121| 27| 83| 14| 70|
+---+---+---+---+---+
| 10| 61|117| 48| 79|
+---+---+---+---+---+
| 44|100| 1| 57|113|
+---+---+---+---+---+
| 53|109| 40| 91| 22|
+---+---+---+---+---+
| 87| 18| 74|105| 31|
+---+---+---+---+---+

Second Layer from Top.

+---+---+---+---+---+
| 2 | 58|114| 45| 96|
+---+---+---+---+---+
| 36| 92| 23| 54|110|
+---+---+---+---+---+
| 75|101| 32| 88| 19|
+---+---+---+---+---+
| 84| 15| 66|122| 28|
+---+---+---+---+---+
|118| 49| 80| 6| 62|
+---+---+---+---+---+

Third Layer from Top.

+---+---+---+---+---+
| 33| 89| 20| 71|102|
+---+---+---+---+---+
| 67|123| 29| 85| 11|
+---+---+---+---+---+
| 76| 7| 63|119| 50|
+---+---+---+---+---+
|115| 41| 97| 3| 59|
+---+---+---+---+---+
| 24| 55|106| 37| 93|
+---+---+---+---+---+

Fourth Layer from Top.

+---+---+---+---+---+
| 64|120| 46| 77| 8|
+---+---+---+---+---+
| 98| 4| 60|111| 42|
+---+---+---+---+---+
|107| 38| 94| 25| 51|
+---+---+---+---+---+
| 16| 72|103| 34| 90|
+---+---+---+---+---+
| 30| 81| 12| 68|124|
+---+---+---+---+---+

Lowest Layer.

+---+---+---+---+---+
| 95| 21| 52|108| 39|
+---+---+---+---+---+
|104| 35| 86| 17| 73|
+---+---+---+---+---+
| 13| 69|125| 26| 82|
+---+---+---+---+---+
| 47| 78| 9| 65|116|
+---+---+---+---+---+
| 56|112| 43| 99| 5|
+---+---+---+---+---+

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!

No. XXVIII.--A MAGIC CIRCLE

The Magic Circle below has this particular property:--

32
61 94
52 38
191 4
+
28 193
26 44
98 67
16

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.

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.

57 24 15 14
31 71 64 72
48 17 22 23
38 69 66 65
50 19 20 21
36 60 75 67
59 26 13 12
29 12 12 74
12 12
+
12 12
42 12 12 61
44 45 58 27
35 43 28 68
53 52 51 18
33 34 37 70
55 54 49 16
40 32 39 63
46 47 56 25
41 30

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?

/\
/ \
/ 5 \
/ \
/ 4 6 \
/ \
/ 3 7 \
/ \
/ 2 1 9 0 \
------------------

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.

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.

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.

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.

No. XXXII.--TWIN TRIANGLES

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

/\
/ \
/ 7 \
/ \
135 / 2 3 \ 99
/ . . \
/ 9 . . 5 \
/ . . \
/ 1 8 6 4 \
------------------
. 117 .
. .
. .
. .
..
. .
. .
. .
. 137 .
------------------
\ 6 4 2 9 /
\ . . /
\ 5 . . 1 /
\ . . /
119 \ 7 8 / 155
\ /
\ 3 /
\ /
\/

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.

1 5 6 70 60 59 58

63 8

62 19 53 46 22 45 9

61 20 24 64

2 48 31 42 38 49 57

3 47 39 40 44 56

67 51 41 37 33 23 7

66 50 34 35 54 11

65 25 36 32 43 26 12

10 30 27 13

17 29 21 28 52 55 72

18 71

16 69 68 4 14 15 73

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.

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.

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.

24

A PERSONAL ENIGMA

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.

No. XXXV.--A MAGIC CROSS

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.

+----+----+
| _1_|_12_|
+----+----+----+----+
| _9_| _8_| _5_| _4_|
+----+----+----+----+
| _2_| _7_| _6_|_11_|
+----+----+----+----+
|_10_| _3_|
+----+----+

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.

25

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

26

BY LORD MACAULAY

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

No. XXXVI.--A CHARMING PUZZLE

Here is quite a charming little puzzle, which is by no means easy of accomplishment:--

✦ ✦ ✦

✦ ✦ ✦

✦ ✦ ✦

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.

27

A BROKEN TALE

The deil jumped
the clouds so high
That he bounded almost
right
the sky.
the trees
gates and fields and
He dodged with his tail
dragging
all these,
But, alas! made a terrible
bl,
For a twist in his tail
a rail,
hooked
And broke that appendage
as.

No. XXXVII.--LEAP-FROG

Place on a chess or draught-board three white men on the squares marked _a_, and three black men on the squares marked _b_.

+---+---+---+---+---+---+---+
|_a_|_a_|_a_| |_b_|_b_|_b_|
+---+---+---+---+---+---+---+

The pieces marked _a_ can only move one square at a time, from left to right, and those marked _b_ one 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?

28

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 attain
What most men have seen, but can’t see again.

29

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.

No. XXXVIII.--SORTING THE COUNTERS

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

1 2 3 4 5 6 7 8 9 10
+---+---+---+---+---+---+---+---+---+---+
| ○ | ● | ○ | ● | ○ | ● | ○ | ● | | |
+---+---+---+---+---+---+---+---+---+---+

1 2 3 4 5 6 7 8 9 10
+---+---+---+---+---+---+---+---+---+---+
| | | ● | ● | ● | ● | ○ | ○ | ○ | ○ |
+---+---+---+---+---+---+---+---+---+---+

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.

30

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

31

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

No. XXXIX.--A TRANSFORMATION

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.

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

32

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

33

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

No. XL.--DOMINO BUILDING

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.

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.

No. XLI.--FAST AND LOOSE

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

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?

34

Taken entire
I’m full of fire.
With head away
A tax I pay.
If tail you bar
I turn from tar.
Headless again,
With tail restored.
Goddess of pain,
I sow discord.

No. XLII.--MAZY PROGRESS.

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.

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.

35

Thrice three pins in shining line
Mary meant to fix;
Why did Mary turn the nine
Into thirty-six?

No. XLIII.--FOR CLEVER PENCILS

Start at _A_, and trace these figures with one continuous line, finishing at _B_.

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

36

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

37

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.

38

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

No. XLIV.--TEST AND TRY

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.

39

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 me
As fleetest couriers convey me.
You must transpose me ere you stay me.

40

Two to one, in case we hide,
You will find us in our site;
We are harmless side by side,
Parted we prepare to bite.
When united we divide,
When divided we unite.

No. XLV.--A CURIOUS PHENOMENON

Equal volumes of alcohol and water, when mixed, occupy less space than when separate, to the extent indicated in this picture:

If the sum of the volume of the two separate liquids is 100, the volume of the mixture will be only 94. It is thought that the molecules of the two liquids accommodate themselves to each other, so as to reduce the pores and diminish the volume of the mixture.

41

Cut off my head, I’m every inch a king,
A warrior formed to deal a heavy blow;
Halve what remains, my second is a thing
Which nothing but my third can e’er make go.
My whole will vary as you take your line,
This less than human, that way all divine.

42

One half of me in solid earth you find,
The other half in ocean’s ample bed:
When in my whole we see these parts combined,
The earth remains, but all the sea is fled.

No. XLVI.--A HOME-MADE MICROSCOPE

The simplest and cheapest of all microscopes can easily be made at home. The only materials needed are a thin slip of glass, on to which one or two short paper tubes, coated with black sealing wax, are cemented with the wax, a small stick, and a tumbler half full of water.

Water is dropped gradually by aid of the stick into the cells, until lenses are formed of the desired convexity, and objects held below the glass will be more or less magnified.

43

Not ever changed unless unchanged,
Nor hanged unless beheaded;
Quick eyes may find in me arranged
Almost an angel bedded.

No. XLVII.--A PRETTY EXPERIMENT

For this curious experiment a glass bottle or decanter about half full of water and a sound stalk of straw are needed.

Bend the straw without breaking it, and put it, as is shown, into the bottle, which can then be lifted steadily and safely by the straw, if it is a sound one.

44

“WHAT THE DICKENS IS HIS NAME?”
_Merry Wives of Windsor._

A Russian nobleman had three sons. Rab, the eldest, became a lawyer, his brother Mary was a soldier, and the youngest was sent to sea. What was his name?

No. XLVIII.--A BOTTLED BUTTON

The button in a clear glass bottle, as is shown below, hangs attached by a thread to the cork, which is securely sealed at the top.

How can you sever the thread so that the button falls to the bottom without uncorking or breaking the bottle?

45

A NEW “LIGHT BRIGADE”

Six before six before
Five times a hundred;
This must be brilliant, or
Solvers have blundered.

No. XLIX.--CLEARING THE WAY

Here is a pretty trick which requires an empty bottle, a lucifer match, and a small coin.

Break the wooden match almost in half, and place it and the coin in the position shown above. Now consider how you can cause the coin to drop into the bottle, if no one touches it, or the match, or the bottle.

46

Scorned by the meek and humble mind,
And often by the vain possessed,
Heard by the deaf, seen by the blind,
I give the troubled spirit rest.

47

To fifty for my half append
Two-thirds of one;
The other third my whole will end
When you have done.

No. L.--IS WATER POROUS?

Our belief that two portions of matter cannot occupy the same space at the same time is almost shaken by the following experiment:

If we introduce slowly some fine powdered sugar into a tumblerful of warm water a considerable quantity may be dissolved in the water without increasing its bulk.

It is thought that the atoms of the water are so disposed as to receive the sugar between them, as a scuttle filled with coal might accommodate a quantity of sand.

48

A DOUBLE SHUFFLE

See, the letters that I bring
Change their meaning quite;
Spell a hard and heavy thing,
Spell a soft and light.

No. LI.--A TEST OF GRAVITY

Set a stool, as is shown in the diagram below, about nine or ten inches from the wall.

Clasp it firmly by its two side edges, plant your feet well away from it, and rest your head against the wall. Now lift the stool, and then try, without moving your feet, to recover an upright position.

It will be as impossible as it is to stand on one leg while the foot of that leg rests sideways against a wall or door.

No. LII.--BILLIARD MAGIC

Place a set of billiard balls as is shown in the diagram, the spot ball overhanging a corner pocket, and the red and the plain white in a straight line with it, leaving an eighth of an inch between the balls.

How can you pot the spot white with the plain white, using a cue, and without touching, or in any way disturbing, the red ball? There is not room to pass on either side between the red ball and the cushion.

No. LIII.--THE NIMBLE COIN

Prepare a circular band of stiff paper, as is shown in the diagram, and balance it, with a coin on the top, on the lip of a bottle.

How can you most effectively transfer the coin into the bottle?

49

AN ENIGMA BY MUTATION

Search high or low, you’d find me where you list;
For not a place without me can exist.
I lose my head, and, seen with shoulders fair,
Become the very fairest of the fair.
Again I lose it, and, like some staunch hound,
The first and best amongst a pack am found.
And if at first both head and tail I lose
I am a portion such as all would choose.

No. LIV.--HIT IT HARD!

Place a strip of thin board, or a long wide flat ruler, on the edge of a table, so that it just balances itself, and spread over it an ordinary newspaper, as is shown in the illustration.

You may now hit it quite hard with your doubled fist, or with a stick, and the newspaper will hold it down, and remain as firmly in its place as if it were glued to the table over it. You are more likely to break the stick with which you strike than to displace the strip of wood or the paper. Try the experiment.

50

AN ENIGMA BY SWIFT

We are little airy creatures
All of different voice and features.
One of us in glass is set,
One of us is found in jet.
Another you may see in tin,
And the fourth a box within.
If the others you pursue
They can never fly from you.

No. LV.--THE BRIDGE OF KNIVES

Here is an after-dinner balancing trick, which it is well to practise with something less brittle than the best glass:--

It will be seen that the blades of the knives are so cunningly interlaced as to form quite a firm support.

51

A MEDLEY

Twice six is six, and so
Six is but three;
Three is just five you know,
What can we be?
Would you count more of us,
Nine are but four of us,
Ten are but three.

No. LVI.--DIFFERENT DENSITIES

Here is a pretty little experiment, which shows the effect of liquids of different densities.

Drop an egg into a glass vessel half full of water, it sinks to the bottom. Drop it into strong brine, it floats. Introduce the brine through a long funnel at the bottom of the pure water, and the water and the egg will be lifted, so that the egg floats between the water and the brine in equilibrium. The egg is denser than the water, and the brine is denser than the egg.

52

THE MISSING LINK

A friend to all the human race
From emperor to peasant,
None is more missed when out of place,
More opportune when present.

Obedient to the general will
I yield to due control;
And yet the public twist me, till
They put me in a hole!

No. LVII.--COLUMBUS OUTDONE

Here is a very simple and effective little trick. Offer to balance an egg on its end on the lip of a glass bottle.

The picture shows how it is done, with the aid of a cork and a couple of silver forks.

(From “La Science Amusante”).

53

Two words of equal length we here indite,
Which hold a famous father and his mate.
Embracing five, with fifty left and right,
The mother, looking both ways, keeps things straight.

Her husband, following a thousand quite,
With them has changed his sex, a funny fate,
And if this lady lose her head, she might,
Being a man, oppose the water-rate.

No. LVIII.--WHAT WILL HAPPEN?

The boy in this picture is blowing hard against the bottle, which is between his mouth and the candle flame.

What will happen?

54

In marble walls as white as milk,
Lined with a skin as soft as silk,
Within a fountain crystal clear
A golden apple doth appear.
No doors are there to this stronghold,
Yet thieves break in, and steal the gold.

No. LIX.--THE FLOATING NEEDLE

Here is a simple way to make a needle float on water:--

Fill a wineglass or tumbler with water, and on this lay quite flat a cigarette paper; place a needle gently on this, and presently the paper will sink, and the needle will float on the water.

55

A one-syllable adjective I,
Indeterminate, misty, obscure:
Reduce me by five, and then try
How you like my attacks to endure.

I’m now a two-syllable noun,
My victims are hot and are cold;
In country more rife than in town,
I’m not such a pest as of old.

No. LX.--VIS INERTIÆ

Here is a pile of ten draughtsmen--one black among nine white.

If I take another draughtsman, and with a strong pull of my finger send it spinning against the column, what will happen?

56

DR. WHEWELL’S ENIGMA