SOLUTIONS
No. V.—THE MAKING OF A MAGIC SQUARE
The perfect Magic Square, for which we have given the construction of two preparatory squares, is formed by placing one of these over the other, so that the numbers in their corresponding cells combine, as is shown below.
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 |
The Perfect Magic Square.
| 6 | 18 | 5 | 12 | 24 |
| 15 | 22 | 9 | 16 | 3 |
| 19 | 1 | 13 | 25 | 7 |
| 23 | 10 | 17 | 4 | 11 |
| 2 | 14 | 21 | 8 | 20 |
No less than 57,600 Magic Squares can be formed with twenty-five cells by varying the arrangement of these same figures, but not many are so perfect as our specimen, in which sixty-five can be counted in forty-two ways. These comprise each horizontal row; each perpendicular row; main diagonals; blended diagonals from every corner (such as 6, with 14, 17, 25, 3; or 15, 18, with 21, 4, 7); centre with any four equidistant in outer cells; any perfect St George’s cross (such as 18, 22, 1, 15, 9); and any perfect St Andrew’s cross (such as 6, 22, 13, 5, 19).
No. XII.—A CENTURY OF CELLS
Here is the solution of the ingenious Magic Square of 100 cells with 36 cells unfilled. The rows, columns, and diagonals all add up to 505.
| 91 | 2 | 3 | 97 | 6 | 95 | 94 | 8 | 9 | 100 |
| 20 | 82 | 83 | 17 | 16 | 15 | 14 | 88 | 89 | 81 |
| 21 | 72 | 73 | 74 | 25 | 26 | 27 | 78 | 79 | 30 |
| 60 | 39 | 38 | 64 | 66 | 65 | 67 | 33 | 32 | 41 |
| 50 | 49 | 48 | 57 | 55 | 56 | 54 | 43 | 42 | 51 |
| 61 | 59 | 58 | 47 | 45 | 46 | 44 | 53 | 52 | 40 |
| 31 | 69 | 68 | 34 | 35 | 36 | 37 | 63 | 62 | 70 |
| 80 | 22 | 23 | 24 | 75 | 76 | 77 | 28 | 29 | 71 |
| 90 | 12 | 13 | 87 | 86 | 85 | 84 | 18 | 19 | 11 |
| 1 | 99 | 98 | 4 | 96 | 5 | 7 | 93 | 92 | 10 |
Notice that the top and bottom rows contain all the numbers from 1 to 10 and from 91 to 100; the two rows next to these range from 11 to 20 and from 81 to 90; the two next from 21 to 30 and from 71 to 80; the two next from 31 to 39 and 60 to 70, excluding 61, but including 41; and the two central rows the numbers run from 42 to 59, with 40 and 61.
No. XXIII.—TWIN PUZZLE SQUARES
The following diagram shows how the twin Magic Squares are evolved from our diagram:—
| 1 | 5 | 6 | 2 | 3 | 7 | |
| 2 | 6 | 7 | 3 | 4 | 8 | |
| 3 | 7 | 8 | 4 | 5 | 9 | |
The sums of the corresponding rows in each square are now equal, and the sums of the squares of the corresponding cells of these rows are equal. The sums of the four diagonals are also equal, and the sum of the squares of the cells in corresponding diagonals are equal. The sum of any two numbers symmetrically placed with respect to the connecting link between the 7 and the 3 is always 10.
No. XXX.—THE UNIQUE TRIANGLE
The figures to be transposed in triangle A are 9 and 3 and 7 and 1.
A
| 5 | ||||||
| 4 | 6 | |||||
| 3 | 7 | |||||
| 2 | 1 | 9 | 8 | |||
B
| 5 | ||||||
| 4 | 6 | |||||
| 9 | 1 | |||||
| 2 | 7 | 3 | 8 | |||
Then in triangle B, the sum of the side is in each case 20, and the sums of the squares of the numbers along the sides is in each case 126.
No. XXXI.—MAGIC TRIANGLES
The subjoined diagram shows the order in which the first 18 numbers can be arranged so that they count 19, 38, or 57 in many ways, down, across, or along some angles, 19 in 6 ways, 38 in 12, and 57 in 14 ways.
Thus, for examples—
| 7 | + | 12 | = | 14 | + | 5 | = | 4 | + | 15 | = | 19 |
| 7 | + | 11 | + | 14 | + | 6 | ——— | = | 38 | |||
| 7 | + | 14 | + | 4 | + | 5 | + | 12 | + | 15 | = | 57 |
No. XXXIV.—MAGIC HEXAGON IN A CIRCLE
The figures in the Magic Hexagon must be arranged as is shown in this diagram:—
| 126 | ||||||||||||||||||
| 5 | 2 | 7 | 3 | 8 | 5 | |||||||||||||
| 8 | 2 | 4 | 6 | 3 | 7 | |||||||||||||
| 114 | 114 | |||||||||||||||||
| 4 | 6 | 9 | 1 | 8 | 2 | |||||||||||||
| 3 | 1 | 9 | 7 | 5 | 4 | 1 | 9 | 6 | ||||||||||
| 1 | 3 | 7 | 9 | 8 | 6 | 7 | 3 | 4 | ||||||||||
| 8 | 2 | 3 | 9 | 1 | 9 | |||||||||||||
| 126 | 126 | |||||||||||||||||
| 6 | 4 | 4 | 1 | 8 | 2 | |||||||||||||
| 5 | 5 | 6 | 7 | 2 | 5 | |||||||||||||
| 114 | ||||||||||||||||||
It will be seen that the sum of the four digits on each side of each triangle is twenty, and that, while their arrangements vary, the total of the added squares of the numbers on the alternate sides of the hexagon are equal.
No. XXXVI.—A CHARMING PUZZLE
| ✦ | ✦ | ✦ |
| ✦ | ✦ | ✦ |
| ✦ | ✦ | ✦ |
To pass through these nine dots with four continuous straight lines, start at the top right-hand corner, and draw a line along the top of the square and beyond its limits, until its end is in line with the central dots of the side and base. Draw the second line through these, continuing it until its end is below and in line with the right-hand side of the square; draw the third line up to the starting-point, and the fourth as a diagonal, which completes the course.
No. XXXVII.—LEAP-FROG
On a chess or draught-board three white men are placed on squares marked a and three black men on squares marked b in the diagram—
| a | a | a | b | b | b | |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
Every a can move from left to right one square at a time, and every b from right to left, and any piece can leap over one of another colour on to an unoccupied square. They can reverse their positions thus:—
If we number the cells or squares consecutively, and notice that at starting the vacant cell is No. 4, then in the successive moves the vacant cells will be 3, 5, 6, 4, 2, 1, 3, 5, 7, 6, 4, 2, 3, 5, 4. Of the moves thus indicated six are simple, and nine are leaps.
No. XXXVIII.—SORTING THE COUNTERS
The counters are changed in four moves only, moving two at a time as follows:—
| Move | 2 and 3 | to | 9 and 10. |
| „ | 5 and 6 | to | 2 and 3. |
| „ | 8 and 9 | to | 5 and 6. |
| „ | 1 and 2 | to | 8 and 9. |
No. XXXIX.—A TRANSFORMATION
To change the ten-pointed star of wooden matches into one of five points without touching it, let a little water fall into the very centre, as it lies on quite a smooth surface, and in a few moments, under the action of the water, it will gradually assume the shape shown in the second diagram, of a five-pointed star.
This is a very simple and effective after-dinner trick. Small matches move best.
No. XLI.—FAST AND LOOSE
The twelve counters or draughtsmen lying loosely at the bottom of a shallow box can be arranged so that they wedge themselves together and against the side thus:—
Temporary centre
Place one for the moment in the centre, and six round it. Hold these firmly in their places with the left hand, and fix the other five round them, as is shown in the diagram. Then remove the temporary centre, and fill in with it the vacant place. All will then be in firm contact, and the box may be turned upside down without displacing them.
No. XLIII.—FOR CLEVER PENCILS
This diagram, shows how a continuous course is possible without taking pencil from paper, or going twice over any line.
We have purposely left spaces wide enough to make the solution perfectly clear.
No. XLVIII.—A BOTTLED BUTTON
The diagram below shows how the thread within the bottle is severed so that the button falls, without uncorking the bottle or breaking it.
Nothing is needed but a lens to focus the rays of the sun, which pass through the glass without heating it, and burn the thread.
No. XLIX.—CLEARING THE WAY
In order to cause the coin to fall into the bottle without touching coin, match, or bottle, let a drop or two of water fall upon the bent middle of the match.
Very soon, under the action of the water, the two ends of the match will open out so that the coin which was resting on them falls between them into the bottle.
No. LII.—BILLIARD MAGIC
The diagram we give below shows the ingenious trick by which the plain white, if struck gently with a cue, will, aided by the tumbler, pot the spot white ball without in any way disturbing the red.
The balls to start with are an eighth of an inch apart, and there is not room for a ball to pass between the cushions and the red. Place the tumbler close to spot white.
No. LIII.—THE NIMBLE COIN
The most effective way to transfer the coin from the top of the circular band of paper into the bottle is to strike a smart blow with a cane, or any small stick, on the inside of the paper band. There is not time for the coin to be influenced in the same direction, and it falls plumb into the neck of the bottle.
No. LVIII.—WHAT WILL HAPPEN?
When the boy shown in this picture blows hard at the bottle which is between his mouth and the candle flame, the divided air current flows round the bottle, reunites, and extinguishes the flame.
No. LX.—VIS INERTIÆ
If, by a strong pull of my finger, I launch the draughtsman that is on the edge of the table against the column of ten in front of it, the black man, which is just at the height to receive the full force of the blow, will be knocked clean out of its place, while the others will not fall. This is another illustration of the vis inertiæ.
No. LXI.—CUT AND COME AGAIN
A block of ice would never be divided completely by a loop of wire on which hangs a 5 ℔ weight. For as the wire works its way through, the slit closes up by refreezing, and the weight falls to the ground with the wire, leaving the ice still in a single block.
No. LXIII.—CATCHING THE DICE
It is quite easy to throw the upper of this pair of dice into the air and catch it in the cup, but the other is more elusive. As you throw it upward with sufficient force you will also throw the die that has been already caught out of the cup.
The secret of success lies in dropping the hand and cup rapidly downwards, quitting hold at the same moment of the die, which then falls quietly into the cup held to receive it.
No. LXIV.—WILL THEY FALL?
When the single domino shown in the diagram in front of the double archway, is quite smartly tipped up by the forefinger carefully inserted through the lower arch, the stone which lies flat below another is knocked clean out, while none of the other stones fall, another practical illustration of vis inertiæ.
For this very curious trick, club dominoes, thick and large, should be used. Some patience and experience is needed, but success at last is certain.
No. LXV.—A TRANSPOSITION
You will be able to place the shaded coin between the other two in a straight line without touching one of these, and without moving the other, if you place a finger firmly on the king’s head and then move the shaded coin an inch or two to the right, and flick it back against the coin you hold. The other “tail” coin will then spring away far enough to allow the space that is required.
No. LXVI.—COIN COUNTING
After reaching and turning the coin which you first call “four,” miss three coins, and begin then a fresh set of four; repeat this process to the end.
No. LXVIII.—NUTS TO CRACK
Hold a cup of water so that it will wet the handle of the knife, then remove it, and place the nut exactly on the spot where the drop of water falls from the handle.
No. LXXI.—WHAT IS THIS?
The photographic enlargement is simply a much magnified reproduction of Mr Chamberlain’s eye and eyeglass, exactly as they appear in the picture which we give below, taken from its negative. A strong condensing lens will reproduce the original effect, which can also be obtained by holding the enlargement at a distance.
No. LXXV.—THE SEAL OF MAHOMET
This double crescent may be drawn by one continuous line, without passing twice over any part, by starting at A, passing along the curve AGD, from D along DEB, from B along BFC, and from C along CEA.
No. LXXVI.—MOVE THE MATCHES
If fifteen matches are arranged thus—
and six are removed, ten is the number that remains, thus—
or one hundred may remain, thus:—
No. LXXVII.—LINES ON AN OLD SAMPLER
This diagram shows the arrangement in which seventeen trees can be planted in twenty-eight rows, three trees in each row:—
No. LXXXI.—COUNTING THEM OUT
Here is an arrangement of dominoes which enables us to count out the first twelve numbers, one after the other, by their spelling:—
Start with the double five, and, touching each stone in turn, say o, n, e, one; remove the stone with one pip, and go on, t, w, o, two; remove the two, and say t, h, r, e, e, three, and so on till you reach at last the twelve.
Playing cards can be used, counting knave, queen, as eleven, twelve. It makes quite a good trick if you place the cards face downwards in the proper order, and then, saying that you will call up each number in turn, move the cards one at a time to the other end, spelling out each number as before, either aloud or not, and turning up and throwing out each as you hit upon it. If you do not call the letters aloud it adds to the mystery if you are blindfolded.
No. LXXXII.—TRICKS WITH DOMINOES
This is the other combination of stones and their pips which fulfils the conditions, and forms the word AGES.
In both cases a complete set of stones is used, which are arranged in proper domino sequence, and everyone of the eight letters carries exactly forty-two pips.
No. XCI—THE STOLEN PEARLS
The dishonest jeweller reset the pearls in a cross so that its arms were a stage higher up. It will be seen that by this arrangement nine pearls can still be counted in each direction.
