With regard to Knot V., I beg to express to VIS INERTIae and to any others who, like her, understood the condition to be that _every_ marked picture must have _three_ marks, my sincere regret that the unfortunate phrase "_fill_ the columns with oughts and crosses" should have caused them to waste so much time and trouble. I can only repeat that a _literal_ interpretation of "fill" would seem to _me_ to require that _every_ picture in the gallery should be marked. VIS INERTIae would have been in the First Cla.s.s if she had sent in the solution she now offers.

ANSWERS TO KNOT VII.

_Problem._--Given that one gla.s.s of lemonade, 3 sandwiches, and 7 biscuits, cost 1_s._ 2_d._; and that one gla.s.s of lemonade, 4 sandwiches, and 10 biscuits, cost 1_s._ 5_d._: find the cost of (1) a gla.s.s of lemonade, a sandwich, and a biscuit; and (2) 2 gla.s.ses of lemonade, 3 sandwiches, and 5 biscuits.

_Answer._--(1) 8_d._; (2) 1_s._ 7_d._

_Solution._--This is best treated algebraically. Let _x_ = the cost (in pence) of a gla.s.s of lemonade, _y_ of a sandwich, and _z_ of a biscuit.

Then we have _x_ + 3_y_ + 7_z_ = 14, and _x_ + 4_y_ + 10_z_ = 17. And we require the values of _x_ + _y_ + _z_, and of 2_x_ + 3_y_ + 5_z_. Now, from _two_ equations only, we cannot find, _separately_, the values of _three_ unknowns: certain _combinations_ of them may, however, be found.

Also we know that we can, by the help of the given equations, eliminate 2 of the 3 unknowns from the quant.i.ty whose value is required, which will then contain one only. If, then, the required value is ascertainable at all, it can only be by the 3rd unknown vanis.h.i.+ng of itself: otherwise the problem is impossible.

Let us then eliminate lemonade and sandwiches, and reduce everything to biscuits--a state of things even more depressing than "if all the world were apple-pie"--by subtracting the 1st equation from the 2nd, which eliminates lemonade, and gives _y_ + 3_z_ = 3, or _y_ = 3-3_z_; and then subst.i.tuting this value of _y_ in the 1st, which gives _x_-2_z_ = 5, _i.e._ _x_ = 5 + 2_z_. Now if we subst.i.tute these values of _x_, _y_, in the quant.i.ties whose values are required, the first becomes (5 + 2_z_) + (3-3_z_) + _z_, _i.e._ 8: and the second becomes 2(5 + 2_z_) + 3(3-3_z_) + 5_z_, _i.e._ 19. Hence the answers are (1) 8_d._, (2) 1_s._ 7_d._

The above is a _universal_ method: that is, it is absolutely certain either to produce the answer, or to prove that no answer is possible.

The question may also be solved by combining the quant.i.ties whose values are given, so as to form those whose values are required. This is merely a matter of ingenuity and good luck: and as it _may_ fail, even when the thing is possible, and is of no use in proving it _im_possible, I cannot rank this method as equal in value with the other. Even when it succeeds, it may prove a very tedious process. Suppose the 26 compet.i.tors, who have sent in what I may call _accidental_ solutions, had had a question to deal with where every number contained 8 or 10 digits! I suspect it would have been a case of "silvered is the raven hair" (see "Patience") before any solution would have been hit on by the most ingenious of them.

Forty-five answers have come in, of which 44 give, I am happy to say, some sort of _working_, and therefore deserve to be mentioned by name, and to have their virtues, or vices as the case may be, discussed.

Thirteen have made a.s.sumptions to which they have no right, and so cannot figure in the Cla.s.s-list, even though, in 10 of the 13 cases, the answer is right. Of the remaining 28, no less than 26 have sent in _accidental_ solutions, and therefore fall short of the highest honours.

I will now discuss individual cases, taking the worst first, as my custom is.

FROGGY gives no working--at least this is all he gives: after stating the given equations, he says "therefore the difference, 1 sandwich + 3 biscuits, = 3_d._": then follow the amounts of the unknown bills, with no further hint as to how he got them. FROGGY has had a _very_ narrow escape of not being named at all!

Of those who are wrong, VIS INERTIae has sent in a piece of incorrect working. Peruse the horrid details, and shudder! She takes _x_ (call it "_y_") as the cost of a sandwich, and concludes (rightly enough) that a biscuit will cost (3-_y_)/3. She then subtracts the second equation from the first, and deduces 3_y_ + 7 (3-_y_)/3-4_y_ + 10 (3-_y_)/3 = 3.

By making two mistakes in this line, she brings out _y_ = 3/2. Try it again, oh VIS INERTIae! Away with INERTIae: infuse a little more VIS: and you will bring out the correct (though uninteresting) result, 0 = 0!

This will show you that it is hopeless to try to coax any one of these 3 unknowns to reveal its _separate_ value. The other compet.i.tor, who is wrong throughout, is either J. M. C. or T. M. C.: but, whether he be a Juvenile Mis-Calculator or a True Mathematician Confused, he makes the answers 7_d._ and 1_s._ 5_d._ He a.s.sumes, with Too Much Confidence, that biscuits were 1/2_d._ each, and that Clara paid for 8, though she only ate 7!

We will now consider the 13 whose working is wrong, though the answer is right: and, not to measure their demerits too exactly, I will take them in alphabetical order. ANITA finds (rightly) that "1 sandwich and 3 biscuits cost 3_d._," and proceeds "therefore 1 sandwich = 1-1/2_d._, 3 biscuits = 1-1/2_d._, 1 lemonade = 6_d._" DINAH MITE begins like ANITA: and thence proves (rightly) that a biscuit costs less than a 1_d._: whence she concludes (wrongly) that it _must_ cost 1/2_d._ F. C. W. is so beautifully resigned to the certainty of a verdict of "guilty," that I have hardly the heart to utter the word, without adding a "recommended to mercy owing to extenuating circ.u.mstances." But really, you know, where _are_ the extenuating circ.u.mstances? She begins by a.s.suming that lemonade is 4_d._ a gla.s.s, and sandwiches 3_d._ each, (making with the 2 given equations, _four_ conditions to be fulfilled by _three_ miserable unknowns!). And, having (naturally) developed this into a contradiction, she then tries 5_d._ and 2_d._ with a similar result. (N.B. _This_ process might have been carried on through the whole of the Tertiary Period, without gratifying one single Megatherium.) She then, by a "happy thought," tries half-penny biscuits, and so obtains a consistent result. This may be a good solution, viewing the problem as a conundrum: but it is _not_ scientific. JANET identifies sandwiches with biscuits!

"One sandwich + 3 biscuits" she makes equal to "4." Four _what_? MAYFAIR makes the astounding a.s.sertion that the equation, _s_ + 3_b_ = 3, "is evidently only satisfied by _s_ = 3/2, _b_ = 1/2"! OLD CAT believes that the a.s.sumption that a sandwich costs 1-1/2_d._ is "the only way to avoid unmanageable fractions." But _why_ avoid them? Is there not a certain glow of triumph in taming such a fraction? "Ladies and gentlemen, the fraction now before you is one that for years defied all efforts of a refining nature: it was, in a word, hopelessly vulgar. Treating it as a circulating decimal (the treadmill of fractions) only made matters worse. As a last resource, I reduced it to its lowest terms, and extracted its square root!" Joking apart, let me thank OLD CAT for some very kind words of sympathy, in reference to a correspondent (whose name I am happy to say I have now forgotten) who had found fault with me as a discourteous critic. O. V. L. is beyond my comprehension. He takes the given equations as (1) and (2): thence, by the process [(2)-(1)] deduces (rightly) equation (3) viz. _s_ + 3_b_ = 3: and thence again, by the process [x3] (a hopeless mystery), deduces 3_s_ + 4_b_ = 4. I have nothing to say about it: I give it up. SEA-BREEZE says "it is immaterial to the answer" (why?) "in what proportion 3_d._ is divided between the sandwich and the 3 biscuits": so she a.s.sumes _s_ = l-1/2_d._, _b_ = 1/2_d._ STANZA is one of a very irregular metre. At first she (like JANET) identifies sandwiches with biscuits. She then tries two a.s.sumptions (_s_ = 1, _b_ = 2/3, and _s_ = 1/2 _b_ = 5/6), and (naturally) ends in contradictions. Then she returns to the first a.s.sumption, and finds the 3 unknowns separately: _quod est absurdum_.

STILETTO identifies sandwiches and biscuits, as "articles." Is the word ever used by confectioners? I fancied "What is the next article, Ma'am?"

was limited to linendrapers. TWO SISTERS first a.s.sume that biscuits are 4 a penny, and then that they are 2 a penny, adding that "the answer will of course be the same in both cases." It is a dreamy remark, making one feel something like Macbeth grasping at the spectral dagger.

"Is this a statement that I see before me?" If you were to say "we both walked the same way this morning," and _I_ were to say "_one_ of you walked the same way, but the other didn't," which of the three would be the most hopelessly confused? TURTLE PYATE (what _is_ a Turtle Pyate, please?) and OLD CROW, who send a joint answer, and Y. Y., adopt the same method. Y. Y. gets the equation _s_ + 3_b_ = 3: and then says "this sum must be apportioned in one of the three following ways." It _may_ be, I grant you: but Y. Y. do you say "must"? I fear it is _possible_ for Y. Y. to be _two_ Y's. The other two conspirators are less positive: they say it "can" be so divided: but they add "either of the three prices being right"! This is bad grammar and bad arithmetic at once, oh mysterious birds!

Of those who win honours, THE SHETLAND SNARK must have the 3rd cla.s.s all to himself. He has only answered half the question, viz. the amount of Clara's luncheon: the two little old ladies he pitilessly leaves in the midst of their "difficulty." I beg to a.s.sure him (with thanks for his friendly remarks) that entrance-fees and subscriptions are things unknown in that most economical of clubs, "The Knot-Untiers."

The authors of the 26 "accidental" solutions differ only in the number of steps they have taken between the _data_ and the answers. In order to do them full justice I have arranged the 2nd cla.s.s in sections, according to the number of steps. The two Kings are fearfully deliberate! I suppose walking quick, or taking short cuts, is inconsistent with kingly dignity: but really, in reading THESEUS'

solution, one almost fancied he was "marking time," and making no advance at all! The other King will, I hope, pardon me for having altered "Coal" into "Cole." King Coilus, or Coil, seems to have reigned soon after Arthur's time. Henry of Huntingdon identifies him with the King Coel who first built walls round Colchester, which was named after him. In the Chronicle of Robert of Gloucester we read:--

"Aftur Kyng Aruirag, of wam we habbeth y told, Marius ys sone was kyng, quoynte mon & bold.

And ys sone was aftur hym, _Coil_ was ys name, Bothe it were quoynte men, & of n.o.ble fame."

BALBUS lays it down as a general principle that "in order to ascertain the cost of any one luncheon, it must come to the same amount upon two different a.s.sumptions." (_Query._ Should not "it" be "we"? Otherwise the _luncheon_ is represented as wis.h.i.+ng to ascertain its own cost!) He then makes two a.s.sumptions--one, that sandwiches cost nothing; the other, that biscuits cost nothing, (either arrangement would lead to the shop being inconveniently crowded!)--and brings out the unknown luncheons as 8_d._ and 19_d._, on each a.s.sumption. He then concludes that this agreement of results "shows that the answers are correct." Now I propose to disprove his general law by simply giving _one_ instance of its failing. One instance is quite enough. In logical language, in order to disprove a "universal affirmative," it is enough to prove its contradictory, which is a "particular negative." (I must pause for a digression on Logic, and especially on Ladies' Logic. The universal affirmative "everybody says he's a duck" is crushed instantly by proving the particular negative "Peter says he's a goose," which is equivalent to "Peter does _not_ say he's a duck." And the universal negative "n.o.body calls on her" is well met by the particular affirmative "_I_ called yesterday." In short, either of two contradictories disproves the other: and the moral is that, since a particular proposition is much more easily proved than a universal one, it is the wisest course, in arguing with a Lady, to limit one's _own_ a.s.sertions to "particulars,"

and leave _her_ to prove the "universal" contradictory, if she can. You will thus generally secure a _logical_ victory: a _practical_ victory is not to be hoped for, since she can always fall back upon the crus.h.i.+ng remark "_that_ has nothing to do with it!"--a move for which Man has not yet discovered any satisfactory answer. Now let us return to BALBUS.) Here is my "particular negative," on which to test his rule. Suppose the two recorded luncheons to have been "2 buns, one queen-cake, 2 sausage-rolls, and a bottle of Zoedone: total, one-and-ninepence," and "one bun, 2 queen-cakes, a sausage-roll, and a bottle of Zoedone: total, one-and-fourpence." And suppose Clara's unknown luncheon to have been "3 buns, one queen-cake, one sausage-roll, and 2 bottles of Zoedone:" while the two little sisters had been indulging in "8 buns, 4 queen-cakes, 2 sausage-rolls, and 6 bottles of Zoedone." (Poor souls, how thirsty they must have been!) If BALBUS will kindly try this by his principle of "two a.s.sumptions," first a.s.suming that a bun is 1_d._ and a queen-cake 2_d._, and then that a bun is 3_d._ and a queen-cake 3_d._, he will bring out the other two luncheons, on each a.s.sumption, as "one-and-nine-pence" and "four-and-ten-pence" respectively, which harmony of results, he will say, "shows that the answers are correct." And yet, as a matter of fact, the buns were 2_d._ each, the queen-cakes 3_d._, the sausage-rolls 6_d._, and the Zoedone 2_d._ a bottle: so that Clara's third luncheon had cost one-and-sevenpence, and her thirsty friends had spent four-and-fourpence!

Another remark of BALBUS I will quote and discuss: for I think that it also may yield a moral for some of my readers. He says "it is the same thing in substance whether in solving this problem we use words and call it Arithmetic, or use letters and signs and call it Algebra." Now this does not appear to me a correct description of the two methods: the Arithmetical method is that of "synthesis" only; it goes from one known fact to another, till it reaches its goal: whereas the Algebraical method is that of "a.n.a.lysis": it begins with the goal, symbolically represented, and so goes backwards, dragging its veiled victim with it, till it has reached the full daylight of known facts, in which it can tear off the veil and say "I know you!"

Take an ill.u.s.tration. Your house has been broken into and robbed, and you appeal to the policeman who was on duty that night. "Well, Mum, I did see a chap getting out over your garden-wall: but I was a good bit off, so I didn't chase him, like. I just cut down the short way to the Chequers, and who should I meet but Bill Sykes, coming full split round the corner. So I just ups and says 'My lad, you're wanted.' That's all I says. And he says 'I'll go along quiet, Bobby,' he says, 'without the darbies,' he says." There's your _Arithmetical_ policeman. Now try the other method. "I seed somebody a running, but he was well gone or ever _I_ got nigh the place. So I just took a look round in the garden. And I noticed the foot-marks, where the chap had come right across your flower-beds. They was good big foot-marks sure-ly. And I noticed as the left foot went down at the heel, ever so much deeper than the other. And I says to myself 'The chap's been a big hulking chap: and he goes lame on his left foot.' And I rubs my hand on the wall where he got over, and there was soot on it, and no mistake. So I says to myself 'Now where can I light on a big man, in the chimbley-sweep line, what's lame of one foot?' And I flashes up permiscuous: and I says 'It's Bill Sykes!' says I." There is your _Algebraical_ policeman--a higher intellectual type, to my thinking, than the other.

LITTLE JACK'S solution calls for a word of praise, as he has written out what really is an algebraical proof _in words_, without representing any of his facts as equations. If it is all his own, he will make a good algebraist in the time to come. I beg to thank SIMPLE SUSAN for some kind words of sympathy, to the same effect as those received from OLD CAT.

HECLA and MARTREB are the only two who have used a method _certain_ either to produce the answer, or else to prove it impossible: so they must share between them the highest honours.

CLa.s.s LIST.

I.

HECLA.

MARTREB.

II.

-- 1 (2 _steps_).

ADELAIDE.

CLIFTON C....

E. K. C.

GUY.

L'INCONNU.

LITTLE JACK.

NIL DESPERANDUM.

SIMPLE SUSAN.

YELLOW-HAMMER.

WOOLLY ONE.

-- 2 (3 _steps_).

A. A.

A CHRISTMAS CAROL.

AFTERNOON TEA.

AN APPRECIATIVE SCHOOLMA'AM.

BABY.

BALBUS.

BOG-OAK.

THE RED QUEEN.

WALL-FLOWER.

-- 3 (4 _steps_).

HAWTHORN.

JORAM.

S. S. G.

-- 4 (5 _steps_).