Tangled Tale — Answers to Knot I

FARE-WELL!

APPENDIX.

"A knot!"

said Alice.

"Oh, do let me help to undo it!"

ANSWERS TO KNOT I.

_Problem.

_--"Two travellers 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 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 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 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.

And now "descend, ye classic Nine!"

who have solved the whole problem, and let me sing your praises.

Your names are BLITHE, E. W.

, L.

B.

, A

MARLBOROUGH BOY, O.

V. L. , PUTNEY WALKER, ROSE, SEA BREEZE, SIMPLE

SUSAN, and MONEY SPINNER.

(These last two I count as one, as they send a joint answer.

) ROSE and SIMPLE SUSAN and CO. do not actually state that the hill-top was reached some time between 6 and 7, but, as they have clearly grasped the fact that a mile, ascended and descended, took the same time as two level miles, I mark them as "right."

A MARLBOROUGH BOY

and PUTNEY WALKER deserve honourable mention for their algebraical solutions being the only two who have perceived that the question leads to _an indeterminate equation_.

E.

W.

brings a charge of untruthfulness against the aged knight--a serious charge, for he was the very pink of chivalry!

She says "According to the data given, the time at the summit affords no clue to the total distance.

It does not enable us to state precisely to an inch how much level and how much hill there was on the road."

"Fair damsel," the aged knight replies, "--if, as I surmise, thy initials denote Early Womanhood--bethink thee that the word 'enable' is thine, not mine.

I did but ask the time of reaching the hill-top as my _condition_ for further parley.

If _now_ thou wilt not grant that I am a truth-loving man, then will I affirm that those same initials denote Envenomed Wickedness!"

CLASS LIST.

I. A MARLBOROUGH BOY.

PUTNEY WALKER.

II. BLITHE.

E.

W.

L. B.

O.

V. L. ROSE.

SEA BREEZE.

{SIMPLE SUSAN.

{MONEY-SPINNER.

BLITHE has made so ingenious an addition to the problem, and SIMPLE SUSAN and CO. have solved it in such tuneful verse, that I record both their answers in full.

I have altered a word or two in BLITHE'S--which I trust she will excuse; it did not seem quite clear as it stood.

* * * * * "Yet stay," said the youth, as a gleam of inspiration lighted up the relaxing muscles of his quiescent features.

"Stay.

Methinks it matters little _when_ we reached that summit, the crown of our toil.

For in the space of time wherein we clambered up one mile and bounded down the same on our return, we could have trudged the _twain_ on the level.

We have plodded, then, four-and-twenty miles in these six mortal hours; for never a moment did we stop for catching of fleeting breath or for gazing on the scene around!"

"Very good," said the old man.

"Twelve miles out and twelve miles in.

And we reached the top some time between six and seven of the clock.

Now mark me!

For every five minutes that had fled since six of the clock when we stood on yonder peak, so many miles had we toiled upwards on the dreary mountainside!"

The youth moaned and rushed into the hostel.

BLITHE.

The elder and the younger knight, They sallied forth at three;

How far they went on level ground It matters not to me;

What time they reached the foot of hill, When they began to mount, Are problems which I hold to be Of very small account.

The moment that each waved his hat Upon the topmost peak-- To trivial query such as this No answer will I seek.

Yet can I tell the distance well They must have travelled o'er: On hill and plain, 'twixt three and nine, The miles were twenty-four.

Four miles an hour their steady pace Along the level track, Three when they climbed--but six when they Came swiftly striding back Adown the hill; and little skill It needs, methinks, to show, Up hill and down together told, Four miles an hour they go.

For whether long or short the time Upon the hill they spent, Two thirds were passed in going up, One third in the descent.

Two thirds at three, one third at six, If rightly reckoned o'er, Will make one whole at four--the tale Is tangled now no more.

SIMPLE SUSAN.

MONEY SPINNER.

ANSWERS TO KNOT II.