MYSTIC FIGURES
Very interesting and curious are the properties of the figures 142857, used in varied order but always in similar sequence, in connection with 7 and 9:—
| 1 | 4 | 2 | 8 | 5 | 7 | × | 7 | = | 9 | 9 | 9 | 9 | 9 | 9 | ÷ | 9 | = | 1 | 1 | 1 | 1 | 1 | 1 | |
| 2 | 8 | 5 | 7 | 1 | 4 | × | 7 | = | 1 | 9 | 9 | 9 | 9 | 9 | 8 | ÷ | 9 | = | 2 | 2 | 2 | 2 | 2 | 2 |
| 4 | 2 | 8 | 5 | 7 | 1 | × | 7 | = | 2 | 9 | 9 | 9 | 9 | 9 | 7 | ÷ | 9 | = | 3 | 3 | 3 | 3 | 3 | 3 |
| 5 | 7 | 1 | 4 | 2 | 8 | × | 7 | = | 3 | 9 | 9 | 9 | 9 | 9 | 6 | ÷ | 9 | = | 4 | 4 | 4 | 4 | 4 | 4 |
| 7 | 1 | 4 | 2 | 8 | 5 | × | 7 | = | 4 | 9 | 9 | 9 | 9 | 9 | 5 | ÷ | 9 | = | 5 | 5 | 5 | 5 | 5 | 5 |
| 8 | 5 | 7 | 1 | 4 | 2 | × | 7 | = | 5 | 9 | 9 | 9 | 9 | 9 | 4 | ÷ | 9 | = | 6 | 6 | 6 | 6 | 6 | 6 |
No. XC.—THE FIVE TRIANGLES
The subjoined diagram shows how a square with sides that measure each 12 yards can be divided into five triangles, no two of which are of equal area, and of which the sides and areas can be expressed in yards by whole numbers:—
The areas of these triangles are 6, 12, 24, 48, and 54 square yards respectively, and the sum of these, 144 square yards, is the area of the square.
CURIOUS COINCIDENCES
Our readers may remember the remarkable fact that the figures of the sum, £12, 12s. 8d., when written thus, 12,128, exactly represent the number of farthings it contains. Now this, so far as we know, is the only instance of the peculiarity, but there are at least five other cases which come curiously near to it. They are these:—