Between 1880 and 1885 Lewis Carroll published ten very short stories in The Monthly Packet, each of which posed mathematical problems. Carroll referred to these stories as “knots” and invited readers to send in answers to the problems in each. Carroll printed the answers he received and at times chided the readers who answered incorrectly. The ten stories along with their solutions were later published as a book, which was illustrated by Arthur B. Frost. All ten knots with solutions but unfortunately without Frost’s illustrations are available here.
In Knot 2, “Eligible Apartments,” a tutor named Balbus and his two students Hugh and Lambert. As they look for lodgings Hugh repeats a question from a letter they had received from their father—this is the first of the knot’s two mathematical puzzles:
“Well, yes. The Governor of — what-you-may-call-it-wants to give a very small dinner-party, and he means to ask his father’s brother-in-law, his brother’s father-in-law, his father-in-law’s brother, and his brother-in-law’s father: and we’re to guess how many guests there will be.”
The second puzzle is woven into the trio’s search for an available apartment. As Carroll summarized it in the solutions:
A Square has 20 doors on each side, which contains 21 equal parts. They are numbered all round, beginning at one corner. From which of the four, Nos. 9, 25, 52, 73, is the sum of the distances, to the other three, least?
Here is the entire story (if you simply want to learn the solutions, you can find them here, along with Carroll’s replies):
Knot 2 — Eligible Apartments
“Let’s ask Balbus about it,” said Hugh.
“All right,” said Lambert.
“He can guess it,” said Hugh.
“Rather,” said Lambert.
No more words were needed: the two brothers understood each other perfectly.
Balbus was waiting for them at the hotel: the journey down had tired him, he said: so his two pupils had been the round of the place, in search of lodgings, without the old tutor who had been their inseparable companion from their childhood. They had named him after the hero of their Latin exercise-book, which overflowed with anecdotes of that versatile genius — anecdotes whose vagueness in detail was more than compensated by their sensational brilliance. “Balbus has overcome all his enemies” had been marked by their tutor, in the margin of the book, “Successful Bravery.” In this way he had tried to extract a moral from every anecdote about Balbus — sometimes one of warning, as in, “Balbus had borrowed a healthy dragon,” against which he had written, “Rashness in Speculation” — sometimes of encouragement, as in the words, “Influence of Sympathy in United Action,” which stood opposite to the anecdote, “Balbus was assisting his mother-in-law to convince the dragon” — and sometimes it dwindled down to a single word, such as “Prudence”, which was all he could extract from the touching record that “Balbus, having scorched the tail of the dragon, went away”. His pupils liked the short morals best, as it left them more room for marginal illustrations, and in this instance they required all the space they could get to exhibit the rapidity of the hero’s departure.
Their report of the state of things was discouraging. That most fashionable of watering-places, Little Mendip, was “chock-full” (as the boys expressed it) from end to end. But in one Square they had seen no less than four cards, in different houses, all announcing in flaming capitals, “ELIGIBLE APARTMENTS.” “So there’s plenty of choice, after all, you see,” said spokesman Hugh in conclusion.
“That doesn’t follow from the data,” said Balbus, as he rose from the easy-chair, where he had been dozing over The Little Mendip Gazette. “They may be all single rooms. However, we may as well see them. I shall be glad to stretch my legs a bit.”
An unprejudiced bystander might have objected that the operation was needless, and that this long lank creature would have been all the better with even shorter legs: but no such thought occurred to his loving pupils. One on each side, they did their best to keep up with his gigantic strides, while Hugh repeated the sentence in their father’s letter, just received from abroad, over which he and Lambert had been puzzling. “He says a friend of his, the Governor of — what was that name again, Lambert?” (“Kgovjni,” said Lambert.) “Well, yes. The Governor of — what-you-may-call-it-wants to give a very small dinner-party, and he means to ask his father’s brother-in-law, his brother’s father-in-law, his father-in-law’s brother, and his brother-in-law’s father: and we’re to guess how many guests there will be.”
There was an anxious pause. “How large did he say the pudding was to be!” Balbus said at last. “Take its cubical contents, divide by the cubical contents of what each man can eat, and the quotient — ”
“He didn’t say anything about pudding,” said Hugh, “ — and here’s the Square,” as they turned a corner and came into sight of the “eligible apartments”.
“It is a Square!” was Balbus’s first cry of delight, as he gazed around him. “Beautiful! Beau-ti-ful Equilateral! And rectangular!”
The boys looked round with less enthusiasm. “Number Nine is the first with a card,” said prosaic Lambert; but Balbus would not so soon awake from his dream of beauty.
“See, boys!” he cried. “Twenty doors on a side! What symmetry! Each side divided into twenty-one equal parts! It’s delicious!”
“Shall I knock, or ring?” said Hugh, looking in some perplexity at a square brass plate which bore the simple inscription, “RING ALSO.”
“Both,” said Balbus. “That’s an Ellipsis, my boy. Did you never see an Ellipsis before?”
“I couldn’t hardly read it,” said Hugh evasively. “Its no good having an Ellipsis, if they don’t keep it clean.”
“Which there is one room, gentlemen,” said the smiling landlady. “And a sweet room too! As snug a little back room — ”
“We will see it,” said Balbus gloomily, as they followed her in. “I knew how it would be! One room in each house! No view, I suppose?”
“Which indeed there is, gentlemen!” the landlady indignantly protested, as she drew up the blind, and indicated the back-garden.
“Cabbages, I perceive,” said Balbus. “Well, they’re green, at any rate.”
“Which the greens at the shops”, their hostess explained, “are by no means dependable upon. Here you has them on the premises, and of the best.”
“Does the window open!” was always Balbus’s first question in testing a lodging: and, “Does the chimney smoke?” his second. Satisfied on all points, he secured the refusal of the room, and they moved on to Number Twenty-five.
This landlady was grave and stern. “I’ve nobbut one room left,” she told them: “and it gives on the back-gyarden.”
“But there are cabbages?” Balbus suggested.
The landlady visibly relented. “There is, sir,” she said: “and good ones, though I say it as shouldn’t. We can’t rely on the shops for greens. So we grows them ourselves.”
“A singular advantage,” said Balbus; and, after the usual questions, they went on to Fifty-two.
“And I’d gladly accommodate you all, if I could,” was the greeting that met them. “We are but mortal” (“Irrelevant!”) muttered Balbus), “and I’ve let all my rooms but one.”
“Which one is a back-room and looking out on — on cabbages, I presume?”
“Yes, indeed, sir,” said their hostess. “Whatever other folks may do, we grows our own. For the shops — ”
“An excellent arrangement,” Balbus interrupted. “Then one can really depend on their being good. Does the window open?”
The usual questions were answered satisfactorily: but this time Hugh added one of his own invention — “Does the cat scratch?”
The landlady looked round suspiciously, as if to make sure the cat was not listening. “I will not deceive you, gentlemen,” she said. “It do scratch, but not without you pulls its whiskers! It’ll never do it”, she repeated slowly, with a visible effort to recall the exact words of some written agreement between herself and the cat, “without you pulls its whiskers!”
“Much may be excused in a cat so treated,” said Balbus, as they left the house and crossed to Number Seventy-three, leaving the landlady curtseying on the doorstep, and still murmuring to herself her parting words, as if they were a form of blessing, “ — not without you pulls its whiskers”
At Number Seventy-three they found only a small shy girl to show the house, who said “yes’m” in answer to all questions .
“The usual room,” said Balbus, as they marched in “the usual back-garden, the usual cabbages. I suppose you can’t get them good at the shops?”
“Yes’m,” said the girl.
“Well, you may tell your mistress we will take the room, and that her plan of growing her own cabbages is simply admirable!”
“Yes’m,” said the girl, as she showed them out.
“One day-room and three bedrooms,” said Balbus, as they returned to the hotel. “We will take as our day-room the one that gives us the least walking to do to get to it.”
“Must we walk from door to door, and count the steps?” said Lambert.
“No, no Figure it out, my boys, figure it out,” Balbus gayly exclaimed, as he put pens, ink, and paper before his hapless pupils, and left the room.
“I say! It’ll be a job!’ said Hugh.
“Rather!” said Lambert.