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 a.s.sertion 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 p.r.o.noun), "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," oh 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 3 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 3rd 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 compet.i.tor 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 cla.s.s-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 shuttlec.o.c.k, 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 Cla.s.s-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."

CLa.s.s LIST.

I.

GUY.

OLD CAT.

SEA-BREEZE.

II.

AYR.

BRADSHAW OF THE FUTURE.

F. LEE.

H. VERNON.

III.

CAT.

ANSWERS TO KNOT VI.

_Problem 1._--_A_ and _B_ began the year with only 1,000_l._ a-piece.

They borrowed nought; they stole nought. On the next New-Year's Day they had 60,000_l._ between them. How did they do it?

_Solution._--They went that day to the Bank of England. _A_ stood in front of it, while _B_ went round and stood behind it.

Two answers have been received, both worthy of much honour. ADDLEPATE makes them borrow "0" and steal "0," and uses both cyphers by putting them at the right-hand end of the 1,000_l._, thus producing 100,000_l._, which is well over the mark. But (or to express it in Latin) AT SPES INFRACTA has solved it even more ingeniously: with the first cypher she turns the "1" of the 1,000_l._ into a "9," and adds the result to the original sum, thus getting 10,000_l._: and in this, by means of the other "0," she turns the "1" into a "6," thus. .h.i.tting the exact 60,000_l._

CLa.s.s LIST

I.

AT SPES INFRACTA.

II.

ADDLEPATE.

_Problem 2._--_L_ makes 5 scarves, while _M_ makes 2: _Z_ makes 4 while _L_ makes 3. Five scarves of _Z_'s weigh one of _L_'s; 5 of _M_'s weigh 3 of _Z_'s. One of _M_'s is as warm as 4 of _Z_'s: and one of _L_'s as warm as 3 of _M_'s. Which is best, giving equal weight in the result to rapidity of work, lightness, and warmth?

_Answer._--The order is _M_, _L_, _Z_.

_Solution._--As to rapidity (other things being constant) _L_'s merit is to _M_'s in the ratio of 5 to 2: _Z_'s to _L_'s in the ratio of 4 to 3.

In order to get one set of 3 numbers fulfilling these conditions, it is perhaps simplest to take the one that occurs _twice_ as unity, and reduce the others to fractions: this gives, for _L_, _M_, and _Z_, the marks 1, 2/5, 4/3. In estimating for _lightness_, we observe that the greater the weight, the less the merit, so that _Z_'s merit is to _L_'s as 5 to 1. Thus the marks for _lightness_ are 1/5, 5/3, 1. And similarly, the marks for warmth are 3, 1, 1/4. To get the total result, we must _multiply_ _L_'s 3 marks together, and do the same for _M_ and for _Z_. The final numbers are 1 1/5 3, 2/5 5/3 1, 4/3 1 1/4; _i.e._ 3/5, 2/3, 1/3; _i.e._ multiplying throughout by 15 (which will not alter the proportion), 9, 10, 5; showing the order of merit to be _M_, _L_, _Z_.

Twenty-nine answers have been received, of which five are right, and twenty-four wrong. These hapless ones have all (with three exceptions) fallen into the error of _adding_ the proportional numbers together, for each candidate, instead of _multiplying_. _Why_ the latter is right, rather than the former, is fully proved in text-books, so I will not occupy s.p.a.ce by stating it here: but it can be _ill.u.s.trated_ very easily by the case of length, breadth, and depth. Suppose _A_ and _B_ are rival diggers of rectangular tanks: the amount of work done is evidently measured by the number of _cubical feet_ dug out. Let _A_ dig a tank 10 feet long, 10 wide, 2 deep: let _B_ dig one 6 feet long, 5 wide, 10 deep. The cubical contents are 200, 300; _i.e._ _B_ is best digger in the ratio of 3 to 2. Now try marking for length, width, and depth, separately; giving a maximum mark of 10 to the best in each contest, and then _adding_ the results!

Of the twenty-four malefactors, one gives no working, and so has no real claim to be named; but I break the rule for once, in deference to its success in Problem 1: he, she, or it, is ADDLEPATE. The other twenty-three may be divided into five groups.

First and worst are, I take it, those who put the rightful winner _last_; arranging them as "Lolo, Zuzu, Mimi." The names of these desperate wrong-doers are AYR, BRADSHAW OF THE FUTURE, FURZE-BUSH and POLLUX (who send a joint answer), GREYSTEAD, GUY, OLD HEN, and SIMPLE SUSAN. The latter was _once_ best of all; the Old Hen has taken advantage of her simplicity, and beguiled her with the chaff which was the bane of her own chickenhood.

Secondly, I point the finger of scorn at those who have put the worst candidate at the top; arranging them as "Zuzu, Mimi, Lolo." They are GRAECIA, M. M., OLD CAT, and R. E. X. "'Tis Greece, but----."

The third set have avoided both these enormities, and have even succeeded in putting the worst last, their answer being "Lolo, Mimi, Zuzu." Their names are AYR (who also appears among the "quite too too"), CLIFTON C., F. B., FIFEE, GRIG, JANET, and MRS. SAIREY GAMP. F. B. has not fallen into the common error; she _multiplies_ together the proportionate numbers she gets, but in getting them she goes wrong, by reckoning warmth as a _de_-merit. Possibly she is "Freshly Burnt," or comes "From Bombay." JANET and MRS. SAIREY GAMP have also avoided this error: the method they have adopted is shrouded in mystery--I scarcely feel competent to criticize it. MRS. GAMP says "if Zuzu makes 4 while Lolo makes 3, Zuzu makes 6 while Lolo makes 5 (bad reasoning), while Mimi makes 2." From this she concludes "therefore Zuzu excels in speed by 1" (_i.e._ when compared with Lolo; but what about Mimi?). She then compares the 3 kinds of excellence, measured on this mystic scale. JANET takes the statement, that "Lolo makes 5 while Mimi makes 2," to prove that "Lolo makes 3 while Mimi makes 1 and Zuzu 4" (worse reasoning than MRS. GAMP'S), and thence concludes that "Zuzu excels in speed by 1/8"!

JANET should have been ADELINE, "mystery of mysteries!"

The fourth set actually put Mimi at the top, arranging them as "Mimi, Zuzu, Lolo." They are MARQUIS AND CO., MARTREB, S. B. B. (first initial scarcely legible: _may_ be meant for "J"), and STANZA.

The fifth set consist of AN ANCIENT FISH and CAMEL. These ill-a.s.sorted comrades, by dint of foot and fin, have scrambled into the right answer, but, as their method is wrong, of course it counts for nothing. Also AN ANCIENT FISH has very ancient and fishlike ideas as to _how_ numbers represent merit: she says "Lolo gains 2-1/2 on Mimi." Two and a half _what_? Fish, fish, art thou in thy duty?

Of the five winners I put BALBUS and THE ELDER TRAVELLER slightly below the other three--BALBUS for defective reasoning, the other for scanty working. BALBUS gives two reasons for saying that _addition_ of marks is _not_ the right method, and then adds "it follows that the decision must be made by _multiplying_ the marks together." This is hardly more logical than to say "This is not Spring: _therefore_ it must be Autumn."

CLa.s.s LIST.

I.

DINAH MITE.

E. B. D. L.

JORAM.

II.

BALBUS.

THE ELDER TRAVELLER.