* * * * *

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 quantities 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
competitors, 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 assumptions to which they have no right, and so
cannot figure in the Class-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 x (3-_y_)/3-4_y_ + 10 x (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