Chapter XII: Conclusion (4)

Ask in which of the rows, C or D, is the number thought of: in the case supposed it is in D.

Take up the rows C D, and put one underneath the other, as at M, taking care that the half-row in which is the number thought of, shall be above the other.

Divide it again into two rows, as at E F, on each side of B, in the same way as before. Ask again in which row it is: it is now in E.

Place one row under the other, as at N, and divide again into two rows, which will now be as G H.

You will be informed that the number is in row H, and you may then announce it to be the top number of that row.

The number thought of will always be _at the top of one of the rows after three transpositions_. If there were 32 counters it would be at the top after four transpositions.

MAGIC SQUARES.

The name "Magic Square" is given to a square divided into several smaller squares, in which numbers are placed in such a manner that every column of numbers, whether vertical, horizontal, or from corner to corner, shall amount to the same sum.

They are divided into three principal classes: 1st, Those which have an odd number of squares in each band; 2d, Those which have an even number of squares in each band, this even number being divisible exactly by 4; 3d, Where the even number of squares in each band cannot be divided by 4 without a fraction.

ODD MAGIC SQUARES.

Squares of this kind are formed thus. Imagine an exterior line of squares above the magic square you wish to form, and another exterior line on the right hand of it These two imaginary lines are shown in the figure.

Then attend to the two following rules:

1st. In placing the numbers in the squares we must go in an ascending oblique direction from left to right; any number which, by pursuing this direction, would fall into the exterior line, must be carried along that line of squares, whether vertical or horizontal, to the last square. Thus, 1 having been placed in the center of the top line, (see the first table on p. 228,) 2 would fall into the exterior square above the fourth vertical line; it must be therefore carried down to the lowest square of that line; then, ascending obliquely 3 falls into the square, but four falls out of it, to the end of a horizontal line, and it must be carried along that line to the extreme left, and there placed. Resuming our oblique ascension to the right, we place 5, where the reader sees it, and would place 6 in the middle of the top band, but finding it occupied by 1, we look for direction to the 2d Rule, which prescribes that, when in ascending obliquely, we come to a square already occupied, we must place the number, which according to the first rule should go into that occupied square, directly under the last number placed. Thus, in ascending with 4, 5, 6, the 6 must be placed directly under the 5, because the square next to 5 in an oblique direction is "engaged."

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

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

Magic squares of this class, however large in the number of compartments, can be easily filled up by attending to these two rules.

We give opposite, a seven-placed square.

There are various other kinds of magic squares; but explanations of them would be too lengthy for our work.

The invention of these contrivances has been traced back to the early ages of science, and talismanic properties were attributed to them. Modern philosophers have amused themselves in bringing them to perfection, and none has contributed so much as "the model of practical wisdom," Dr. Franklin.

THE SQUARE OF GOTHAM.

The wise men of Gotham, famous for their eccentric blunders, once undertook the management of a school; they arranged their establishment in the form of a square divided into 9 rooms. The playground occupied the center, and 24 scholars the rooms around it, 3 being in each. In spite of the strictness of discipline, it was suspected that the boys were in the habit of playing truant, and it was determined to set a strict watch. To assure themselves that all the boys were on the premises, they visited the rooms, and found three in each, or 9 in each row. Four boys then went out, and the wise men soon after visited the rooms, and finding 9 in each row, thought all was right. The four boys then came back, accompanied by four strangers; and the Gothamites, on their third round, finding still 9 in each row, entertained no suspicion of what had taken place. Then 4 more "chums" were admitted; but the clever men, on examining the establishment a fourth time, still found 9 in each row, and so came to an opinion that their previous suspicions had been unfounded. How was all this possible?

The following figures represent the contents of each room at the four different visits; the first, at the commencement of the watch; the second, when four had gone out; the third, when these 4, accompanied by another 4, had returned; and the fourth, when 4 more had joined them.