“A Knot,” said Alice. “Oh, do let me help to undo it!”

Problem. — Two travelers spend from 3 o’clock till 9 in walking along a level road, up a hill, and home again: their pace on the level being 4 miles an hour, up hill 3, and down hill 6. Find the distance walked: also (within half an hour) time of reaching top of hill.

Answer. — 24 miles: half-past 6.

Solution. — A level mile takes 1/4 of an hour, up hill 1/3, down hill 1/6. Hence to go and return over the same mile, whether on the level or on the hillside, takes 1/2 an hour. Hence in 6 hours they went 12 miles out and 12 back. If the 12 miles out had been nearly all level, they would have taken a little over 3 hours; if nearly all up hill, a little under 4. Hence 3½ hours must be within a an hour of the time taken in reaching the peak; thus, as they started at 3, they got there within 1/2 an hour of 1/2 past 6.

Twenty-seven answers have come in. Of these, 9 are right, 16 partially right, and 2 wrong. The 16 give the distance correctly, but they have failed to grasp the fact that the top of the hill might have been reached at any moment between 6 o’clock and 7.

The two wrong answers are from Gerty Vernon and A Nihilist. The former makes the distance “23 miles”, while her revolutionary companion puts it at “27” Gerty Vernon says, “they had to go 4 miles along the plain, and got to the foot of the hill at 4 o’clock.” They might have done so, I grant; but you have no ground for saying they did so. “It was 7½ miles to the top of the hill, and they reached that at 1/4 before 7 o’clock.” Here you go wrong in your arithmetic, and I must, however reluctantly, bid you farewell. 7½ miles, at 3 miles an hour, would not require 2 3/4 hours. A Nihilist says, “let x denote the whole number of miles; y the number of hours to hill-top; ∴ 3y = number of miles to hill-top, and x-3y = number of miles on the other side.” You bewilder me. The other side of what? “Of the hill,” you say. But then, how did the y get home again? However, to accommodate your views we will build a new hostelry at the foot of the hill on the opposite side, and also assume (what I grant you is possible, though it is not necessarily true) that there was no level road at all. Even then you go wrong. You say:

          x - 3y
    y = 6 —————. . . .  (i);
            6

     x
    ——— = 6. . . .  (ii);
    4½

I grant you (i), but I deny (ii): it rests on the assumption that to go part of the time at 3 miles an hour, and the rest at 6 miles an hour, comes to the same result as going the whole time at 4½ miles an hour. But this would only be true if the “part” were in exact half, i.e. if they went up hill for 3 hours, and down hill for the other 3: which they certainly did not do.

The sixteen who are partially right, are Agnes Bailey, F. K. Fifee, G. E. B., H. P., Kit, M. E. T., Mysie, A Mother’s Son, Nairam, A Redruthian, A Socialist, Spear Maiden, T. B. C., Vis Inertiae, and Yak. Of these, F. K., Fifee, T. B. C., and Vis Inertiae do not attempt the second part at all. F. K. and H. P. give no working. The rest make particular assumptions, such as that there was no level road — that there were 6 miles of level road — and so on, all leading to particular times being fixed for reaching the hill-top. The most curious assumption is that of Agnes Bailey, who says, “Let x = number of hours occupied in ascent; then x/2 = hours occupied in descent; and 4x/3 = hours occupied on the level.” I suppose you were thinking of the relative rates, up hill and on the level; which we might express by saying that, if they went x miles up hill in a certain time, they would go 4x/3 miles on the level in the same time. You have, in fact, assumed that they took the same time on the level that they took in ascending the hill. Fifee assumed that, when the aged knight said they had gone “four miles in the hour” on the level, he meant that four miles was the distance gone, not merely the rate. This would have been — if Fifee will excuse the slang expression — a “sell”, ill-suited to the dignity of the hero.

And now, “descend, ye classic Nine!” who have solved the whole problem, and let me sing your praises. Yo............