"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.

---

§ I. The Dinner Party

Problem. --The Governor of Kgovjni wants to give a very small dinner party, and invites 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. Find the number of guests.

Answer.--One.

In this genealogy, males are denoted by capitals, and females by small let- ters.

The Governor is E and his guest is C.

Ten answers have been received. Of these, one is wrong, Galanthus Nivalis Major, who insists on inviting two guests, one being the Governor's wife's brother's father. If she had taken his sister's husband's father instead, she would have found it possible to reduce the guests to one.

Of the nine who send right answers, Sea-Breeze is the very faintest breath that ever bore the name! She simply states that the Governor's uncle might fulfil all the conditions "by intermarriages"! "Wind of the western sea", you have had a very narrow escape! Be thankful to appear in the Class List at all! Bog-Oak and Bradshaw of the Future use genealogies which require 16 people instead of 14, by inviting the Governor's father's sister's husband instead of his father's wife's brother. I cannot think this so good a solution as one that requires only 14. Caius and Valentine deserve special mention as the only two who have supplied genealogies.

Class List.

I.

Bee.			Matthew Matticks.
Caius.			Old Cat.
M. M.			Valentine.

II.

Bog-Oak.

Bradshaw of the Future.

III.

Sea-Breeze.

§ 2. The Lodgings

Problem.--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?

Answer.--From No. 9.

[I appologize for the interruption, but as of yet I have been unable to find a satisfactory substitute for the square root sign, and so have substituted -/. If you know how to make a better representation of a square root sign, PLEASE do not hesitate to E-mail me.]

Let A be No. 9, B No. 25, C No. 52,and D NO. 73.

Then AB = -/(I2² + 5²) = -/I69 = 13;
AC =2I;
AD = -/(9² + 8²) = -/I45 = 12 +
(N.B. i.e, "between 12 and 13")
BC = -/(I6² + 12²) = -/400 = 20;
BD = -/(3² + 21²) = -/450 = 21 + ;
CD = -/(9² + 13²) = -/250 = 15 +;

Hence the sum of distances from A is between 46 and 47; from B, between 54 and 55; from C, between 56 and 57; from D, between 48 and 51. (Why not "between 48 and 49"? Make this out for yourselves.) Hence the sum is least for A.

Twenth-five solutions have been received. Of these, 15 must be marked "zero", 5 are partly right, and 5 right'. Of the 15, I may dismiss Alphabetical Phantom, Bog-Oak, Dinah Mite, Fifee, Galanthus Nivalis Major (I fear the cold spring has blighted our Snowdrop), Guy, H.M.S. Pinafore, Janet, and Valentine with the simple remark that they insist on the unfortunate lodgers keeping to the pavement. (I used the words "crossed to Number Seventy-three" for the special purpose of showng that short cuts were possible.) Sea-Breeze does the same, and adds that "the result would be the same" even if they crossed the Square, but gives no proof of this. M. M. draws a diagram, and says that No. 9 is the house, "as the diagram shows". I cannot see how it does so. Old Cat assumes that the house must be No. 9 or No. 73. She does not explain how she estimates the distances. Bee's arithmetic is faulty: she makes -/I69+-/442+-/I30=74I. (I suppose you mean -/742, which would be a little nearer the truth. But roots cannot be added in this manner. Do you think -/9 + -/I6 is 25, or even -/25!) But Ayr's state is more perilous still: she draws illogical conclusions with a frightful calmness. After pointing out (rightly) that AC is less than BD, she says, "therefore the nearest house to the other three must be A or C." And again, after pointing out (rightly) that B and D are both within the half-square containing A, she says, "therefore" AB +AD must be less than BC + CD. (There is no logical force in either "therefore". For the first, try Nos. 1, 21, 60, 70: this will make your premiss true, and your con- clusion false. Similarly, for the second, try Nos. 1, 30, 91, 71.)

Of the five partly-right solutions, Rags and Tatters and Mad Hatter (who send one answer between them) make No. 25 6 units from the corner instead of 5. Cheam, E. R. D. L., and Meggy Potts leave openings at the corner's of the square, which are not in the data: moreover Cheam gives values for the distances without any hint that they are only approximations, Crophi and Mophi make the bold and unfounded assumption that there were really 21 houses on each side, instead of 20 as stated by Balbus. "We may assume", they add, "that the doors of Nos. 21, 42, 63, 84, are invisible from the centre of the Square"! What is there, I wonder, that Crophi and Mophi would not assume?

Of the five who are wholly right, I think Bradshaw of the Future, Caius, Clifton C., and Martreb deserve special praise for their full analytical solutions. Matthew Matticks picks out No. 9, and proves it to be the right house in two ways, very neatly and ingeniously, but why he picks it out does not appear. It is an excellent synthetical proof, but lacks the analysis which the other four supply.

Class List.

I.

Bradshaw of the Future.			Clifton C.
Caius.			Martreb.

II.

Matthew Matticks.

III.

Cheam.			Meggy Potts.
Crophi and Mophi.			Rags and Tatters.
E. R. D. L.			Mad Hatter.

A remonstrance has reached me from Scrutator on the subject of Knot 1, which he declares was "no problem at all". "Two questions", he says, "are put. To solve one there is no data: the other answers itself." As to the first point, Scrutator is mistaken; there are (not "is") data sufficient to answer the question. As to the other, it is interesting to know that the question "answers itself", and I am sure it does the question great credit: still I fear I cannot enter it on the list of winners, as this competition is only open to human beings.