PART I

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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.

[Illustration]

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