Magic square

“ VII.—ANOTHER BORDERED MAGIC SQUARE
Here is quite a good example of a bordered magic square of sixty-four cells:—
Image
It is a perfect specimen itself, and as each border is removed a fresh perfect magic square is revealed.
4. A CHARADE
Take for my first a quadruped,
Transpose one for my second;
My whole, a biped, quick or dead,
Is dainty reckoned.
Solution
5. BYRON’S ENIGMA
I am not in youth, nor in manhood, nor age,
But in infancy ever am known;
I’m a stranger alike to the fool and the sage,
And though I’m distinguish’d in history’s page
I always am greatest alone. ”

“ A boy eight or nine years of age had been chosen at random from a number who happened to be passing in the street, and the magician, taking hold of his right hand, drew in the palm of it a magic square, that is to say, one square inscribed within another, and in the space between certain Arabic numerals ”

“ Magic squares have been so called on account of the wonderful powers which the ancients thought they possessed. According to this idea, a square containing one cell represented the Deity, the product of unity by itself always being unity. ”

“ If a square be formed of the former class, one place may be left vacant, but only under particular conditions. In the case of our puzzle there would be no difficulty in making the magic square with 9 missing; but with 1 missing (that is, using 2, 3, 4, 5, 6, 7, 8, and 9) it is not possible. But a glance at the original illustration will show that the numbers we have to deal with are not actually those just mentioned. The clown that has a 9 on his body is portrayed just at the moment when two balls which he is juggling are in mid-air. ”

“ Our puzzle, therefore, resolves itself into forming a magic square with nine different prime numbers.
In Diagram A we have a magic square in prime numbers, and it is the one giving the smallest constant sum that is possible. As to the little trap I mentioned, it is clear that Diagram A is barred out by the words "every basket contained plums," for one plum is not plums. And as we were referred to the baskets, "as shown in the illustration," it is perfectly evident, without actually attempting to count the plums, that there are at any rate more than 7 plums in every basket. ”

“ Solution
[10]
No. X.—A DOMINO MAGIC SQUARE
In this magic square the rows, columns, and diagonals add up always to 33.
Image
Can you rearrange it so that the first stone (three-ace) shall occupy the centre, now filled by the double six, and it shall still add up in all ways to 33?
Solution
6. SHIFTING LETTERS
I am bright as a whole
Till you cut off my head;
Then as black as a coal,
Or a mortal instead.
Shaken up and recast
We with science are found,
Read us back from the last
And we live underground. ”

“ In Magic the play is just the same as in One-ers, except that instead of only throwing up a single stone and catching it as the others are in turn picked up, the whole number, except those remaining to be picked up, are thrown and caught. In Four Squares, four of the stones are arranged in a square, each of them is then picked up, whilst the remaining stone is flung upwards and caught ”

“ Can you construct a square of sixteen different numbers so that it shall be magic (that is, adding up alike in the four rows, four columns, and two diagonals) , whether you turn the diagram upside down or not? You must not use a 3, 4, or 5, as these figures will not reverse; but a 6 may become a 9 when reversed, a 9 a 6, a 7 a 2, and a 2 a 7. The 1, 8, and 0 will read the same both ways. Remember that the constant must not be changed by the reversal. ”

“ The numbers here given to the squares are the same as in draughts. The point to be borne in mind in playing the game is to try to unite a perpendicular with a horizontal line of squares, or with a diagonal line. By careful play three men can be got on one of these lines, and three on a communicating line, each having a vacant square. The player of the opposite colour can fill only one of these squares, and consequently cannot prevent the formation of the quaternion on the other line. ”

“ If it were possible so to multiply squares thus inscribed that a figure coincident with the circumference of a circle might be constructed, such a figure would not be a square but a circle. The manifested universe is like that—the process of inscribing squares within a circle. It is ever becoming, evolving, developing, but never quite attains. ”

“ The puzzle is to move the single rook over the whole board, so that it shall visit every square of the board once, and only once, and end its tour on the square from which it starts. You have to do this in as few moves as possible, and unless you are very careful you will take just one move too many. Of course, a square is regarded equally as "visited" whether you merely pass over it or make it a stopping-place, and we will not quibble over the point whether the original square is actually visited twice. We will assume that it is not. ”

“ If the reader will imagine this series of squares to represent a portable piece of open trellis-work, which might be set up at a window or in the open field, between the spectator and any object of interest at a distance—each square representing a number corresponding with a code in universal use—it will be obvious, that by noticing the squares which the outline of a hill would cover, and telegraphing the numbers of the squares, something in the way of form and outline may be quickly communicated from the other side of the world. ”

“ Before placing the men upon the board, it must be decided whether the white or the black squares are to be played on, as the whole must be put on one color only. If the white squares are selected, there must be a black square in the right-hand corner ”

“ Ibsen's Peer Gynt, Shaw's Back to Methuselah and Hilliard's adaptation of Heinlein's Children of Methuselah, the Capek brothers' Insect People, O'Neill's The Fountain, Flecker's Hassan, Camino Real, Children of the Moon, The Beggar's Opera, Mary of Scotland, Berkeley Square, The Road to Rome. ”

“ You need only consider squares of one colour, for whatever can be done in the case of the white squares can always be repeated on the black, and they are here quite independent of one another. This equality, of course, is in consequence of the fact that the number of squares on an ordinary chessboard, sixty-four, is an even number. If a square chequered board has an odd number of squares, then there will always be one more square of one colour than of the other. ”

“ If you will look at the lettered square you will understand that there are only ten really differently placed squares on a chessboard—those enclosed by a dark line—all the others are mere reversals or reflections. For example, every A is a corner square, and every J a central square. Consequently, as the solution shown has a turning-point at the enclosed D square, we can obtain a solution starting from and ending at any square marked D—by just turning the board about. ”

“ Again, was I not taught by my Lord that as in a Line there are two bounding Points, and in a Square there are four bounding Lines, so in a Cube there must be six bounding Squares? Behold once more the confirming Series, 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught me to believe, “strictly according to Analogy”? ”

“ If in the external universe any one constructive principle can be detected, it is the geometrical; and in every part of the world in which magic pretends to a written character, I find that its hieroglyphics are geometrical figures. Is it not laughable that the most positive of all the sciences should thus lend its angles and circles to the use of—what shall I call it?—the ignorance?—ay, that is the word—the ignorance of dealers in magic?”
He took up the paper, on which I had hastily described the triangles and the circle, and left the room, chanting the serpent-charmer’s song. ”

“ As already noted, the great difficulty with most circle-squarers is that they are ignorant both of the nature of the problem to be solved and of the results which have been already attained. Sometimes we see it explained as the drawing of a square inside a circle and at other times as the drawing of a square around a circle, but both these problems are amongst the very simplest in practical geometry, the solutions being given in the sixth and seventh propositions of the Fourth Book of Euclid. ”

“ I do not think I have mentioned the bright form of quadrature in which a square is made equal to a circle by making each side equal to a quarter of the circumference. The last squarer of this kind whom I have seen figures in the last number of the Athenæum for 1855: he says the thing is no longer a problem, but an axiom. He does not know that the area of the circle is greater than that of any other figure of the same circuit. ”

“ To conceive a round square, or to conceive a body all black and yet all white, would only be to conceive two different sensations as produced in us simultaneously by the same object—a conception familiar to our experience—and we should probably be as well able to conceive a round square as a hard square, or a heavy square, if it were not that in our uniform experience, at the instant when a thing begins to be round, it ceases to be square, so that the beginning of the one impression is inseparably associated with the departure or cessation of the other. ”

“ As Lethaby reminds us, modern people take squareness very much for granted as being a self-evident form, but the discovery of the square was a great step in geometry. [4] It opened a new era in the story of the builders. Early inventions must have seemed like revelations, as indeed they were; and it is not strange that skilled craftsmen were looked upon as magicians. If man knows as much as he does, the discovery of the Square was a great event to the primitive mystics of the Nile. ”

“ The solution to the puzzle is shown in Figs. 25 and 26. The direction of the cut dividing A and C in the first diagram is very obvious, and the second cut is made at right angles to it. That the four pieces should fit together and form a square will surprise the novice, who will do well to study the puzzle with some care, as it is most instructive.
I will now explain the beautiful rule by which we determine the size of a square that shall have the same area as a Greek cross, for it is applicable, and necessary, to the solution of almost every dissection puzzle that we meet with. ”

“ For all straight lines which meet at any point on the horizon are perspectively parallel to each other, just as two geometrical parallels crossing two others at any angle, as at Fig. 49. Note also (Fig. 48) that all squares formed between the two vanishing lines AS, BS, and by the aid of these diagonals, are also equal, and further, that any number of squares such as are shown in this figure (Fig. 50) , formed in the same way and having equal bases, are also equal; and the nine squares contained in the square abcd being equal, they divide each side of the larger square into three equal parts. ”

“ However, given that he had the experience of a cube being pushed through his plane, he would know there was some figure, whereof his square was part, which was bounded by his square on one side, and by a White square on another side. [108] We have supposed him to make models of these boundaries, a Black square and a White square. The Black square, which is his solid matter, is only one boundary of a figure in Higher Space. ”

“ Seat yourself comfortably and don't work hurriedly.
The first thing necessary in this system of squares and circles is to know [132]
How to Cut a Circle Quickly
easily, and accurately, and always without a pattern. Here is a method which never fails:
Cut a square the size you wish to make your circle. That is, if you want a circle with a diameter of four inches, cut a four-inch square (Fig. 226) . Fold the square diagonally through the centre according to the dotted line on Fig. 226, and you have a triangle (Fig. 227) . ”

“ A later stage was to repeat the squares with varieties of colour; and also to introduce details into the squares, and so make them compound patterns, as in the XVIIth dynasty at El Kab, where the sequence of the blue, green, and red lines makes a brilliant effect from these simple elements. Not only a square, but also a hexagon, was worked into the same design. This, from the nature of it, suggests a rush-work screen, and probably it was plaited with rushes in three directions, and hence the production of this particular angle. The previous zigzag patterns all suggest weaving ”

“ From this and the last article it appears, that, whether the sides of a rectangle be a whole or a fractional number of inches, the number of square inches in its surface is the product of the numbers of inches in its sides. The square itself is a rectangle whose sides are all equal, and therefore the number of square inches which a square contains is found by multiplying the number of inches in its side by itself. For example, a square whose side is 13 inches in length contains 13 × 13 or 169 square inches. ”