illing 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 x 1/5 x 3, 2/5 x 5/3 x 1, 4/3 x 1 x
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 space by stating it here: but it can be _illustrated_ 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 sc