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 7-1/2 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. 7-1/2 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; [** therefore] 3_y_ = number of miles to hill-top, and _x_-3_y_ = number of miles on the other side." You bewilder me. The other side of _what_? "Of the hill," you say. But then, how did they 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 "_y_ = 6 - (_x_ - 3_y_)/6, ..... (i); _x_/4-1/2 = 6 ..... (ii)." 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 4-1/2 miles an hour. But this would only be true if the "_part_" were an 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 INERTIÆ, and YAK. Of these, F. K., FIFEE, T. B. C., and VIS INERTIÆ 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 4_x_/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 4_x_/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 assumes 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.