To crack this tough nut by finding the dimensions of the tray you need to work out the angle through which the plate turns in one circuit and from that the length of its circumference that comes into contact with the edge of the tray as the plate rolls round. You are given that a circular plate has radius one unit. It rolls around inside a rectangular tray, always touching the edge, and returns to its exact starting position after five circuits of the tray. During this time it rotates about its own centre by seven complete revolutions. The motion is said to be periodic because it repeats itself over and over again after each five circuits and the length of the period is 5 circuits. Is there more than one solution?