In this note I classify certain polyhedra which are of interest in that they have in some sense a higher degree of regularity than any other polyhedra except those which are completely regular. The definition of a regular polyhedron is that it shall possess two particular symmetries; one which cyclically permutes the vertices of any face c, and one which cyclically permutes the faces that meet at a vertex C, C being a vertex of c.

Though much work has been done on these most interesting squares, there appear to be a few points, anyhow, that are either new or not published anywhere easily accessible to the majority.

Most mathematicians have encountered the Problem of the Three Prisoners. Kraitchik gives the problem and the solution as follows:

“Three unsuccessful revolutionists managed to escape across the border, where they were interned in the lockup of a small town. The sheriff’s sympathies lay with the prisoners, and he sought a way to let them go free. One day he came into the jail brandishing three white discs and two black ones and said to the prisoners : ‘On the back of each of you I shall pin one of these discs. None of you can see his own disc, though you can see your companions ! If any of you guesses correctly that his own disc is white he will receive his freedom. If not, he will be detained indefinitely.’ With this he pinned a white disc on the back of each and left them in the care of a guard. How can each of the prisoners, A, B, and C, guess whether his disc is white?

Our common use of the word “mind” invests the term with considerable obscurity and ambiguity. This is partly because, whilst many philosophers, psychologists and scientists deny any reality to the notion, many others have regarded it as the only reality Others again have regarded both mind and matter as having reality but reality of quite different types, e.g. subjective reality and objective reality. Without going into these controversies I must begin by pointing out that for a psychologist, whatever his philosophical beliefs, the mental processes of the people he is studying are as real and as objective as the interior of a star is to an astrophysicist—and as inaccessible to direct observation. Any other assumption would make nonsense not only of psychology but of the whole of education. And it has a special bearing on the problem of determining the nature of mathematics.

This is a continuation of a note in the Gazette of May, 1954.

Before men learnt to count, the shepherd guarded against the loss of a she from his flock by making a tally of the flock. As each sheep passed through the gate in the morning, on the way to the grazing land, the shepherd cut notch in his stick. At nightfall, when the flock returned to the fold, checked his sheep against the tally by running his fìnger along the notche moving from notch to notch as each sheep passed. The tally stick served i purpose just as well as counting the flock serves the shepherd today. T tally could be passed from hand to hand almost as easily as numbers a now passed by word of mouth (or written down); it constituted a reasonat permanent flock record, and so long as flocks remained quite small there w no necessity to mother the invention of a new system of recording.