* * * * *

The trains one way took 180 minutes, the other way 120. Let us take the
L. C. M., 360, and divide the railway into 360 units. Then one set of
trains went at the rate of 2 units a minute and at intervals of 30
units; the other at the rate of 3 units a minute and at intervals of 45
units. An easterly train starting has 45 units between it and the first
train it will meet: it does 2-5ths of this while the other does 3-5ths,
and thus meets it at the end of 18 units, and so all the way round. A
westerly train starting has 30 units between it and the first train it
will meet: it does 3-5ths of this while the other does 2-5ths, and thus
meets it at the end of 18 units, and so all the way round. Hence if the
railway be divided, by 19 posts, into 20 parts, each containing 18
units, trains meet at every post, and, in (1), each traveller passes 19
posts in going round, and so meets 19 trains. But, in (2), the easterly
traveller only begins to count after traversing 2-5ths of the journey,
_i.e._, on reaching the 8th post, and so counts 12 posts: similarly the
other counts 8. They meet at the end of 2-5ths of 3 hours, or 3-5ths of
2 hours, _i.e._, 72 minutes.

* * * * *

Forty-five answers have been received. Of these 12 are beyond the reach
of discussion, as they give no working. I can but enumerate their names.
ARDMORE, E. A., F. A. D., L. D., MATTHEW MATTICKS, M. E. T., POO-POO,
and THE RED QUEEN are all wrong. BETA and ROWENA have got (1) right and
(2) wrong. CHEEKY BOB and NAIRAM give the right answers, but it may
perhaps make the one less cheeky, and induce the other to take a less
inverted view of things, to be informed that, if this had been a
competition for a prize, they would have got no marks. [N.B.--I have
not ventured to put E. A.'s name in full, as she only gave it
provisionally, in case her answer should prove right.]

Of the 33 answers for which the working is given, 10 are wrong; 11
half-wrong and half-right; 3 right, except that they cherish the
delusion that it was _Clara_ who travelled in the easterly train--a
point which the data do not enable us to settle; and 9 wholly right.

The 10 wrong answers are from BO-PEEP, FINANCIER, I. W. T., KATE B., M.
A. H., Q. Y. Z., SEA-GULL, THISTLEDOWN, TOM-QUAD, and an unsigned one.
BO-PEEP rightly says that the easterly traveller met all trains which
started during the 3 hours of her trip, as well as all