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 3? 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 partially2 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? 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? 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);
4?
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? 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., Kit4, M. E. T., Mysie, A Mother’s Son, Nairam, A Redruthian, A Socialist5, Spear Maiden6, T. B. C., Vis Inertiae, and Yak7. 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 fixed8 for reaching the hill-top. The most curious assumption is that of Agnes Bailey, who says, “Let x = number of hours occupied in ascent9; 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 ascending10 the hill. Fifee assumed that, when the aged11 knight12 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 Blithe13, 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 joint14 answer.) Roe15 and Simple Susan and Co. do not actually state that the hill-top was reached sometime between 6 and 7, but, as they have clearly grasped the fact that a mile, ascended16 and descended17, took the same time as two level miles, I mark them as “right”. A Marlborough Boy and Putney Walker deserve honourable18 mention for their algebraic 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 chivalry19! 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 precisely20 to an inch how much level and how much hill there was on the road.” “Fair damsel,” the aged knight replies, “ — if, as I surmise21, 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 parley22. If now thou wilt23 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.
Rose.
E.W.
Sea-Breeze.
L.B.
Simple Susan.
O. V. L.
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 say,” said the youth, as a gleam of inspiration lighted up the relaxing muscles of his quiescent24 features. “Stay. Methinks it matters little when we reached that summit, the crown of our toil25. For in the space of time wherein we clambered up one mile and bounded down the same on our return, we could have trudged26 the twain on the level. We have plodded27, then, four-and-twenty miles in these six mortal hours; for never a moment did we stop for catching28 of fleeting29 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 sometime between six and seven of the clock. Now mark me! For every five minutes that had fled since six of the clock when we stoorl on yonder peak, so many miles had we toiled30 upwards31 on the dreary32 mountain-side!”
The youth moaned and rushed into the hostel3.
BLITHE.
The elder and the younger knight
?They sallied forth33 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 query34 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, Ywixt 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 tangled35 now no more.
Simple Susan.
Money-Spinner.
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1
undo
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| vt.解开，松开；取消，撤销 | |
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partially
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| adv.部分地，从某些方面讲 | |
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3
hostel
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| n.（学生）宿舍，招待所 | |
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kit
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| n.用具包，成套工具；随身携带物 | |
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socialist
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| n.社会主义者；adj.社会主义的 | |
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maiden
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| n.少女，处女；adj.未婚的，纯洁的，无经验的 | |
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7
yak
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| n.牦牛 | |
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8
fixed
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| adj.固定的，不变的，准备好的；(计算机)固定的 | |
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9
ascent
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| n.(声望或地位)提高；上升，升高；登高 | |
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ascending
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| adj.上升的，向上的 | |
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aged
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| adj.年老的，陈年的 | |
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12
knight
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| n.骑士，武士；爵士 | |
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13
blithe
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| adj.快乐的，无忧无虑的 | |
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14
joint
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| adj.联合的，共同的；n.关节，接合处；v.连接，贴合 | |
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15
roe
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| n.鱼卵；獐鹿 | |
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16
ascended
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| v.上升，攀登( ascend的过去式和过去分词 ) | |
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17
descended
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| a.为...后裔的，出身于...的 | |
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18
honourable
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| adj.可敬的；荣誉的，光荣的 | |
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19
chivalry
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| n.骑士气概，侠义；（男人）对女人彬彬有礼，献殷勤 | |
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20
precisely
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| adv.恰好，正好，精确地，细致地 | |
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21
surmise
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| v./n.猜想，推测 | |
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22
parley
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| n.谈判 | |
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23
wilt
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| v.(使)植物凋谢或枯萎；(指人)疲倦，衰弱 | |
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24
quiescent
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| adj.静止的，不活动的，寂静的 | |
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25
toil
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| vi.辛劳工作，艰难地行动；n.苦工，难事 | |
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26
trudged
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| vt.& vi.跋涉，吃力地走（trudge的过去式与过去分词形式） | |
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27
plodded
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| v.沉重缓慢地走（路）( plod的过去式和过去分词 )；努力从事；沉闷地苦干；缓慢进行（尤指艰难枯燥的工作） | |
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28
catching
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| adj.易传染的，有魅力的，迷人的，接住 | |
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29
fleeting
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| adj.短暂的，飞逝的 | |
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30
toiled
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| 长时间或辛苦地工作( toil的过去式和过去分词 )； 艰难缓慢地移动，跋涉 | |
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31
upwards
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| adv.向上，在更高处...以上 | |
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32
dreary
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| adj.令人沮丧的，沉闷的，单调乏味的 | |
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33
forth
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| adv.向前；向外，往外 | |
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34
query
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| n.疑问，问号，质问；vt.询问，表示怀疑 | |
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35
tangled
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| adj. 纠缠的，紊乱的 动词tangle的过去式和过去分词 | |
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