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Amusements in Mathematics

201.—HOW TO MAKE CISTERNS.—solution
Here is a general formula for solving this problem. Call the two sides of the rectangle a
and b. Then
( a + b - (a2 + b2 - ab)½ )/6
equals the side of the little square pieces to cut away. The measurements given were 8 ft.
by 3 ft., and the above rule gives 8 in. as the side of the square pieces that have to be cut
away. Of course it will not always come out exact, as in this case (on account of that
square root), but you can get as near as you like with decimals.
202.—THE CONE PUZZLE.—solution
The simple rule is that the cone must be cut at one-third of its altitude.
203.—CONCERNING WHEELS.—solution
If you mark a point A on the circumference of a wheel that runs on the surface of a level
road, like an ordinary cart-wheel, the curve described by that point will be a common
cycloid, as in Fig. 1. But if you mark a point B on the circumference of the flange of a
locomotive-wheel, the curve will be a curtate cycloid, as in Fig. 2, terminating in nodes.
Now, if we consider one of these nodes or loops, we shall see that "at any given moment"
certain points at the bottom of the loop must be moving in the opposite direction to the
train. As there is an infinite number of such points on the flange's circumference, there
must be an infinite number of these loops being described while the train is in motion. In
fact, at any given moment certain points on the flanges are always moving in a direction
opposite to that in which the train is going.
In the case of the two wheels, the wheel that runs round the stationary one makes two
revolutions round its own centre. As both wheels are of the same size, it is obvious that if
at the start we mark a point on the circumference of the upper wheel, at the very top, this
point will be in contact with the lower wheel at its lowest part when half the journey has
been made. Therefore this point is again at the top of the moving wheel, and one
revolution has been made. Consequently there are two such revolutions in the complete
204.—A NEW MATCH PUZZLE.—solution