Problem 4

Suppose you have 10 circular discs, each of equal radius, and each with one letter of MATHSBOMBE written on it. Arrange the discs in an arbitrary chain spelling out the word MATHSBOMBE. Each disc can only touch its neighbouring discs, and must only touch tangentially. No discs can overlap each other. The discs must close up so that the final E just touches the first M. See the illustration for an example configuration.

Now suppose you have another disc (labelled with a bomb) of exactly half the radius of the MATHSBOMBE discs. Place the bomb disc at some place along the chain so that it is just touching (without overlapping) one of the discs and is on the outside of the chain (for example, as illustrated).

Now roll the bomb disc once around the chain clockwise (without slipping and without sliding, with the bomb disc maintaining contact with the chain) until it returns to exactly where it started. The shape of the chain is such that the bomb disc touches every one of the letter discs during its motion.

How many times will the image on the bomb disc have rotated when it returns to exactly where the disc started?

Enter your answer correct to two decimal places. For example, if you think that the bomb disc will have rotated 6.543... times then you should enter 6.54.