mpelled either to let him make the "thirty-one" or to go
yourself beyond, and so lose the game.

You thus see that your method of certainly winning breaks down utterly,
by what may be called the "method of exhaustion." I will give the key to
the game, showing how you may always win; but I will not here say whether
you must play first or second: you may like to find it out for yourself.

80.--_The Chinese Railways._

[Illustration]

Our illustration shows the plan of a Chinese city protected by pentagonal
fortifications. Five European Powers were scheming and clamouring for a
concession to run a railway to the place; and at last one of the
Emperor's more brilliant advisers said, "Let every one of them have a
concession!" So the Celestial Government officials were kept busy
arranging the details. The letters in the diagram show the different
nationalities, and indicate not only just where each line must enter the
city, but also where the station belonging to that line must be located.
As it was agreed that the line of one company must never cross the line
of another, the representatives of the various countries concerned were
engaged so many weeks in trying to find a solution to the problem, that
in the meantime a change in the Chinese Government was brought about, and
the whole scheme fell through. Take your pencil and trace out the route
for the line A to A, B to B, C to C, and so on, without ever allowing one
line to cross another or pass through another company's station.

81.--_The Eight Clowns._

[Illustration]

This illustration represents a troupe of clowns I once saw on the
Continent. Each clown bore one of the numbers 1 to 9 on his body. After
going through the usual tumbling, juggling, and other antics, they
generally concluded with a few curious little numerical tricks, one of
which was the rapid formation of a number of magic squares. It occurred
to me that if clown No. 1 failed to appear (as happens in the
illustration), this last item of their performance might not be so easy.
The reader is asked to discover how these eight clowns may arrange
themselves in the form of a square (one place being vacant), so that
every one of the three columns, three rows, and each of the two diagonals
shall add up the same. The vacant place may be at any part of the square,
but it is No. 1 that must be absent.

82.--_The Wizard's Arithmetic._

Once upon a time a knight went to consult a certain fam