Amusements in Mathematics HTML version

There are some curious facts concerning the movements of wheels that are apt to perplex
the novice. For example: when a railway train is travelling from London to Crewe certain
parts of the train at any given moment are actually moving from Crewe towards London.
Can you indicate those parts? It seems absurd that parts of the same train can at any time
travel in opposite directions, but such is the case.
In the accompanying illustration we have two wheels. The lower one is supposed to be
fixed and the upper one running round it in the direction of the arrows. Now, how many
times does the upper wheel turn on its own axis in making a complete revolution of the
other wheel? Do not be in a hurry with your answer, or you are almost certain to be
wrong. Experiment with two pennies on the table and the correct answer will surprise
you, when you succeed in seeing it.
In the illustration eighteen matches are shown arranged so that they enclose two spaces,
one just twice as large as the other. Can you rearrange them (1) so as to enclose two four-
sided spaces, one exactly three times as large as the other, and (2) so as to enclose two
five-sided spaces, one exactly three times as large as the other? All the eighteen matches
must be fairly used in each case; the two spaces must be quite detached, and there must
be no loose ends or duplicated matches.
Here is a new little puzzle with matches. It will be seen in the illustration that thirteen
matches, representing a farmer's hurdles, have been so placed that they enclose six sheep-
pens all of the same size. Now, one of these hurdles was stolen, and the farmer wanted
still to enclose six pens of equal size with the remaining twelve. How was he to do it? All
the twelve matches must be fairly used, and there must be no duplicated matches or loose