"And what made you choose the first train, Goosey?" said Mad Mathesis, as they got into the cab. "Couldn't you count better than that?"

"I took an extreme case," was the tearful reply. "Our excellent preceptress always says, 'When in doubt, my dears, take an extreme case.' And I was in doubt."

"Does it always succeed!" her aunt inquired.

Clara sighed. "Not always," she reluctantly admitted. "And I ca'n't make out why. One day she was telling the little girls--they make such a noise at tea, you know-- The more noise you make, the less jam you will have, and vice versa.' And I thought they wouldn't know what 'vice versa' meant: so I explained it to them. I said, 'if you make an infinite noise, you'll get no jam: and if you make no noise, you'll get an infinite lot of jam.' But our excellent preceptress said that wasn't a good instance. Why wasn't it?" she added plaintively.

Her aunt evaded the question. "One sees certain objections to it," she said. "But how did you work it with the Metropolitan trains? None of them go infinitely fast, I believe."

"I called them hares and tortoises," Clara said--a little timidly, for she dreaded being laughed at. "And I thought there couldn't be so many hares as tortoises on the Line: so I took an extreme case--one hare and an infinite number of tortoises."

"An extreme case, indeed," her aunt remarked with admirable gravity: "and a most dangerous state of things!"

"And I thought, if I went with a tortoise, there would be only one hare to meet: but if I went with the hare-- you know there were crowds of tortoises!"

"It wasn't a bad idea," said the elder lady, as they left the cab, at the entrance of Burlington House. "You shall have another chance to-day. We'll have a match in marking pictures."

Clara brightened up. "I should like to try again, very much," she said. "I'll take more care this time. How are we to play?"

To this question Mad Mathesis made no reply: she was busy drawing lines down the margins of the catalogue. "See," she said after a minute, "I've drawn three columns against the names of the pictures in the long room, and I want you to fill them with oughts and crosses--crosses for good marks and oughts for bad. The first column is for choice of subject, the second for arrangement, the third for colouring. And these are the conditions of the match: You must give three crosses to two or three pictures. You must give two crosses to four or five "

"Do you mean only two crossed" said Clara. "Or may I count the three-cross pictures among the two-cross pictures,"

"Of course you may," said her aunt. "Anyone that has three eyes, may be said to have two eyes, I suppose?" Clara followed her aunt's dreamy gaze across the crowded gallery, half-dreading to find that there was a three-eyed person in sight.

"And you must give one cross to nine or ten."

"And which wins the match?" Clara asked, as she carefully entered these conditions on a blank leaf in her catalogue.

"Whichever marks fewest pictures." But suppose we marked the same number?"

"Then whichever uses most marks."

Clara considered "I don't think it's much of a match," she said. "I shall mark nine pictures, and give three crosses to three of them, two crosses to two more, and one cross to all the rest."

"Will you, indeed?" said her aunt. "Wait till you've heard all the conditions, my impetuous child. You must give three oughts to one or two pictures, two oughts to three or four, and one ought to eight or nine. I don't want you to be too hard on the R.A.'s."

Clara quite gasped as she wrote down all these fresh conditions. "It's a great deal worse than Circulating Decimals!" she said. "But I'm determined to win, all the same!"

Her aunt smiled grimly. "We can begin here," she said, as they paused before a gigantic picture, which the catalogue informed them was the "Portrait of Lieutenant Brown, mounted on his favourite elephant"

"He looks awfully conceited!" said Clara. "I don't think he was the elephant's favourite Lieutenant What a hideous picture it is! And it takes up room enough for twenty!"

"Mind what you say, my dear!" her aunt interposed "It's by an R.A.!"

But Clara was quite reckless. "I don't care who it's by!" she cried. "And I shall give it three bad marks!"

Aunt and niece soon drifted away from each other in the crowd, and for the next half-hour Clara was hard at work, putting in marks and rubbing them out again, and hunting up and down for a suitable picture This she found the hardest part of all. "I ca'n't find the one I want!" she exclaimed at last, almost crying with vexation. "

What is it you want to find, my dear?" The voice was strange to Clara, but so sweet and gentle that she felt attracted to the owner of it, even before she had seen her; and when she turned, and met the smiling looks of two little old ladies, whose round dimpled faces, exactly alike, seemed never to have known a care, it was as much as she same !" it was as much as she could do--as she confessed to Aunt Mattie afterwards--to keep herself from hugging them both. "I was looking for a picture", she said, "that has a good subject--and that's well arranged--but badly coloured."

The little old ladies glanced at each other in some alarm. "Calm yourself, my dear," said the one who had spoken first, "and try to remember which it was. What was the subject?"

"Was it an elephant, for instance?" the other sister suggested. They were still in sight of Lieutenant Brown.

"I don't know, indeed" Clara impetuously replied. "You know it doesn't matter a bit what the subject is, so long as it's a good one!"

Once more the sisters exchanged looks of alarm, and one of them whispered something to the other, of which Clara caught only the one word "mad".

"They mean Aunt Mattie, of course," she said to herself--fancying, in her innocence, that London was like her native town, where everybody knew everybody else. "If you mean my aunt," she added aloud, "she's there-- just three pictures beyond Lieutenant Brown."

"Ah, well! Then you'd better go to her, my dear" her new friend said soothingly. "She'll find you the picture you want. Good-bye, dear!"

"Good-bye, dear!" echoed the other sister. "Mind you don't lose sight of your aunt!" And the pair trotted off into another room, leaving Clara rather perplexed at their manner.

"They're real darlings!" she soliloquized. "I wonder why they pity me so" And she wandered on, murmuring to herself, "It must have two good marks, and--"

---

Answers to Knot 5

Problem.--To mark pictures, giving 3 x's to 2 or 3, 2 to 4 or 5, and 1 to 9 or 10; also giving 3 o's to 1 or 2, 2 to 3 or 4, and 1 to 8 or 9; so as to mark the smallest possible number of pictures, and to give them the largest possible number of marks.

Answer.--10 pictures; 29 marks; arranged thus:

Solution.--By giving all the x's possible, putting into brackets the optional ones, we get 10 pictures marked thus:

By then assigning o's in the same way, beginning at the other end, we get 9 pictures marked thus:

All we have now to do is to run these two wedges as close together as they will go, so as to get the minimum number of pictures--erasing optional marks where by so doing we can run them closer, but otherwise letting them stand. There are 10 necessary marks in the 1st row, and in the 3rd; but only 7 in the end. Hence we erase all optional marks in the 1st and 3rd rows, but let them stand in the end.

Twenty-two answers have been received. Of these, 11 give no working; so, in accordance with what I announced in my last review of answers, I leave them unnamed, merely mentioning that 5 are right and 6 wrong.

Of the eleven answers with which some working is supplied, 3 are wrong. C. H. begins with the rash assertion that under the given conditions "the sum is impossible. For", he or she adds (these initialed correspondents are dismally vague beings to deal with: perhaps "it" would be a better pronoun), "10 is the least possible number of pictures" (granted): "therefore we must either give 2 x 's to 6, or 2 o's to 5". Why "must," O alphabetical phantom? It is nowhere ordained that every picture "must" have 3 marks! Fifee sends a folio page of solution,which deserved a better fate: she offers three answers, in each of which 10 pictures are marked, with 30 marks; in one she gives 2 x's to 6 pictures; in another to 7; in the third she gives 2 o's to 5; thus in every case ignoring the conditions. (I pause to remark that the condition "2 x 's to 4 or 5 pictures" can only mean "either to 4 or else to 5": if, as one competitor holds, it might mean any number not less than 4, the words "or 5" would be superfluous.) I. E. A. (I am happy to say that none of these bloodless phantoms appear this time in the class-list. Is it Idea with the "D" left out?) gives 2 x's to 6 pictures. She then takes me to task for using the word "ought" instead of "nought". No doubt, to one who thus rebels against the rules laid down for her guidance, the word must be distasteful. But does not I. E. A. remember the parallel case of "adder"? That creature was originally "a nadder": then the two words took to bandying the poor "n", backwards and forwards like a shuttlecock, the final state of the game being "an adder". May not "a nought" have similarly become "an ought" Anyhow, "oughts and crosses" is a very old game. I don't think I ever heard it called "noughts and crosses".

In the following Class List, I hope the solitary occupant of III will sheathe her claws when she hears how narrow an escape she has had of not being named at all. Her account of the process by which she got the answer is so meagre that, like the nursery tale of "Jack-a-Minory" (I trust I. E. A. will be merciful to the spelling), it is scarcely to be distinguished from "zero".

Class List

I.

Guy.			Old Cat.
			Sea-Breeze.

II.

Ayr.			F. Lee.
Bradshaw of the Future.			H. Vernon.

III.

Cat.