The question whether any projective plane of order ten exists or not, is an unsolved problem that has attracted some interest (see, for instance, [2]). A method, by which a plane might have been discovered, was suggested to me by a theorem in [1]: ‘If order of a plane is greater than 10, a six-arc is not complete’. Elementary arguments do not, it appears, exclude the possibility of a complete six-arc in a plane of order ten: but they do show that such a figure would be of an extreme type, and that the whole plane would fit round it in a particular way. The limitation, in fact, is so severe that it becomes feasible to consider, for a good many of the incidences in the plane, all the alternative arrangements that seem possible. With the help of the Elliott 4130 computer of the University of Leicester, I have carried out an exhaustive search, and discovered that it is impossible to build up a projective plane by this method. So I can assert: