The Busy Beaver, the Placid Platypus and other Crazy Creatures

The busy beaver is an example of a function which is not computable. It is based on a particular class of Turing machines, and is defined as the largest number of 1’s that can be printed by a terminating machine with n states. Whilst there have been various quests to determine the precise value of this function (which is known precisely only for n ≤ 4), our aim is not to determine this value per se, but to investigate the properties of this class of machines. On the one hand, these are remarkably simple (and, intuitively, form perhaps the simplest class of computationally complete machines); on the other hand, as some of the machines for n = 6 show, they are capable of representing phenomenally large numbers. We describe our quest to better understand these machines, including the placid platypus problem, ie. to determine the minimum number of states needed by a machine of this type to print a given number of 1’s.