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Amusements in Mathematics

Unicursal And Route Problems
"I see them on their winding way."
It is reasonable to suppose that from the earliest ages one man has asked another such
questions as these: "Which is the nearest way home?" "Which is the easiest or pleasantest
way?" "How can we find a way that will enable us to dodge the mastodon and the
plesiosaurus?" "How can we get there without ever crossing the track of the enemy?" All
these are elementary route problems, and they can be turned into good puzzles by the
introduction of some conditions that complicate matters. A variety of such complications
will be found in the following examples. I have also included some enumerations of more
or less difficulty. These afford excellent practice for the reasoning faculties, and enable
one to generalize in the case of symmetrical forms in a manner that is most instructive.
For years I have been perpetually consulted by my juvenile friends about this little
puzzle. Most children seem to know it, and yet, curiously enough, they are invariably
unacquainted with the answer. The question they always ask is, "Do, please, tell me
whether it is really possible." I believe Houdin the conjurer used to be very fond of giving
it to his child friends, but I cannot say whether he invented the little puzzle or not. No
doubt a large number of my readers will be glad to have the mystery of the solution
cleared up, so I make no apology for introducing this old "teaser."
The puzzle is to draw with three strokes of the pencil the diagram that the little girl is
exhibiting in the illustration. Of course, you must not remove your pencil from the paper
during a stroke or go over the same line a second time. You will find that you can get in a
good deal of the figure with one continuous stroke, but it will always appear as if four
strokes are necessary.
Another form of the puzzle is to draw the diagram on a slate and then rub it out in three
The illustration is a rough sketch somewhat resembling the British flag, the Union Jack. It
is not possible to draw the whole of it without lifting the pencil from the paper or going
over the same line twice. The puzzle is to find out just how much of the drawing it is
possible to make without lifting your pencil or going twice over the same line. Take your
pencil and see what is the best you can do.