the level or
on the hill-side, 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 3-1/2 hours must be within 1/2 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 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 MO