THE SOCIABLE SCHOOLGIRLS
On how many days can fifteen schoolgirls go out for a walk so arranged in rows of three, that no two are together more than once?
Fifteen schoolgirls can go out for a walk on seven days so arranged in rows of three that no two are together more than once.
It is said, on high authority, that there are no less than 15,567,522,000 different solutions to this problem. Here is one of them, given in Ball’s Mathematical Recreations, in which k stands for one of the girls, and a, b, c, d, e, f, g, in their modifications, for her companions on the seven different days:—
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
|---|---|---|---|---|---|---|
| ka1a2 | kb1b2 | kc1c2 | kd1d2 | ke1e2 | kf1f2 | kg1g2 |
| b1d1f1 | a1d2e2 | a1d1e1 | a2b2c2 | a2b1c1 | a1b2c1 | a1b1c2 |
| b2e1g1 | a2f2g2 | a2f1g1 | a1f2g1 | a1f1g2 | a2d2e1 | a2d1e2 |
| c1d2g2 | c1d1g1 | b1d2f2 | b1e1g2 | b2d1f2 | b1e2g1 | b2d2f1 |
| c2e2f2 | c2e1f1 | b2e2g2 | c1e2f1 | c2d2g1 | c2d1g2 | c1e1f2 |
It is an excellent game of patience, for those who have time and inclination, to place the figures 1 to 15 inclusive in seven such columns, so as to fulfil the conditions.
No. XCVII.—THE THREE CROSSES
It is possible from a Greek cross to cut off four equal pieces which, when put together, will form another Greek cross exactly half the size of the original, and by this process to leave a third Greek cross complete.
This is how to do it:—