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### Knot III – Mad Mathesis

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Perhaps the best example of both sides of Lewis Carroll is the somewhat obscure series “A Tangled Tale”, originally a serial in the Monthly Packet beginning in April 1880. Each story (or knot), while amusing in its own right, contained one or more math problems for the readers to solve. You may read the separate knots, or jump straight to the questions. Attempt them at your own risk. Lewis Carroll’s replies to the original correspondents are also preserved for those who wish to compare their performance.

* I waited for the train *

“Well, they call me so because I am a little mad, I suppose,” she said, good-humouredly, in answer to Clara’s cautiously worded question as to how she came by so strange a nickname “You see, I never do what sane people are expected to do nowadays. I never wear long trains (talking of trains, that’s the Charing Cross Metropolitan Station I’ve something to tell you about that), and I never play lawn-tennis. I can’t cook an omelette. I can’t even set a broken limb! There’s an ignoramus for you!”

Clara was her niece, and full twenty years her junior; in fact, she was still attending a High School — an institution of which Mad Mathesis spoke with undisguised aversion. “Let a woman be meek and lowly!” she would say. “None of your High Schools for me!” But it was vacation-time just now, and Clara was her guest, and Mad Mathesis was showing her the sights of that Eighth Wonder of the world — London.

“The Charing Cross Metropolitan Station!” she resumed, waving her hand towards the entrance as if she were introducing her niece to a friend. “The Bayswater and Birmingham Extension is just completed, and the trains now run round and round continuously — skirting the border of Wales, just touching at York, and so round by the east coast back to London. The way the trains run is most peculiar. The westerly ones go round in two hours; the easterly ones take three; but they always manage to start two trains from here, opposite ways, punctually every quarter of an hour.”

“They part to meet again,” said Clara, her eyes filling with tears at the romantic thought.

“No need to cry about it!” her aunt grimly remarked. “They don’t meet on the same line of rails, you know. Talking of meeting, an idea strikes me!” she added, changing the subject with her usual abruptness. “Let’s go opposite ways round, and see which can meet most trains. No need for a chaperon — ladies’ saloon, you know. You shall go whichever way you like, and we’ll have bet about it!”

“I never make bets,” Clara said very gravely. “Our excellent preceptress has often warned us — ”

“You’d be none the worse if you did!” Mad Mathesis interrupted. “In fact, you’d be the better, I’m certain!”

“Neither does our excellent preceptress approve of puns,” said Clara. “But we’ll have a match, if you like. Let me choose my train,” she added after a brief mental calculation, “and I’ll engage to meet exactly half as many again as you do.”

“Not if you count fair,” Mad Mathesis bluntly interrupted. “Remember, we only count the trains we meet on the way. You mustn’t count the one that starts as you start, nor the one that arrives as you arrive.”

“That will only make the difference of one train,” said Clara, as they turned and entered the station. “But I never travelled alone before. There’ll be no one to help me to alight. However, I don’t mind. Let’s have a match.”

A ragged little boy overheard her remark, and came running after her. “Buy a box of cigar-lights, Miss!” he pleaded, pulling her shawl to attract her attention Clara stopped to explain.

“I never smoke cigars,” she said in a meekly apologetic tone. “Our excellent preceptress —” But Mad Mathesis impatiently hurried her on, and the little boy was left gazing after her with round eyes of amazement. The two ladies bought their tickets and moved slowly down the central platform. Mad Mathesis prattling on as usual — Clara silent, anxiously reconsidering the calculation on which she rested her hopes of winning the match.

“Mind where you go, dear!” cried her aunt, checking her just in time. “One step more, and you’d have been in that pail of cold water!”

“I know, I know,” Clara said dreamily. “The pale, the cold, and the moony — ”

“Take your places on the spring-boards!” shouted a porter.

“What are they for?” Clara asked in a terrified whisper.

“Merely to help us into the trains.” The elder lady spoke with the nonchalance of one quite used to the process. “Very few people can get into a carriage without help in less than three seconds, and the trains only stop for one second.” At this moment the whistle was heard, and two trains rushed into the station. A moment’s pause, and they were gone again; but in that brief interval several hundred passengers had been shot into them, each flying straight to his place with the accuracy of a Minie bullet — while an equal number were showered out upon the side-platforms.

Three hours had passed away, and the two friends met again on the Charing Cross platform, and eagerly compared notes. Then Clara turned away with a sigh. To young impulsive hearts, like hers, disappointment is always a bitter pill. Mad Mathesis followed her, full of kindly sympathy.

“Try again, my love” she said cheerily. “Let us vary the experiment. We will start as we did before, but not begin counting till our trains meet. When we see each other, we will say ‘one!’, and so count on till we come here again.”

Clara brightened up. “I shall win that”, she exclaimed eagerly, “if I may choose my train!”

Another shriek of engine whistles, another upheaving of spring-boards, another living avalanche plunging into two trains as they flashed by and the travelers were off again.

Each gazed eagerly from her carriage window, holding up her handkerchief as a signal to her friend. A rush and a roar. Two trains shot past each other in a tunnel, and two travelers leaned back in their corners with a sigh — or rather with *two* sighs — of relief. “One!” Clara murmured to herself. “Won! It’s a word of good omen. This time, at any rate, the victory will be mine!”

But *was* it?

Question:

(1) Two travelers, starting at the same time, went opposite ways round a circular railway. Trains start each way every 15 minutes, the easterly ones going round in 3 hours, the westerly in 2. How many trains did each meet on the way, not counting trains met at the terminus itself? [Trains met at the terminus count when met elsewhere.]

(2)They went round, as before, each traveler counting as “one” the train containing the other traveler [*i.e.* only starting the count when they meet the other traveler]. How many did each meet?

Answer, Solution, and Correspondence: Do not click until you have solved the questions or given up!

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