The other day I received a long tube from a Canadian stranger containing a large poster featuring over a hundred polyhedra, including ‘all 92 Johnson polyhedra’. This term, though probably unfamiliar this side of the pond, was not completely unknown to me; it means convex polyhedra, excluding the regular and Archimedean ones, all of whose faces are regular polygons. Of course, as usual, we have to exclude those naughty polyhedra whose faces go around in pairs collecting squares (prisms) or equilateral triangles (antiprisms) and don’t know when to stop. The word convex is vital, otherwise there would be another infinite set. A lot of them are rather trivial, like sticking pyramids on the faces of a dodecahedron, but they include the deltahedra and many other interesting members. But they have at least one imitator who didn't quite make the grade. Trying to discover why, and how to coach him so that he would, I found that my spherical trigonometry was getting rather rusty so I set out to make one and see what was happening. I thought perhaps other readers would like to share this piece of antiresearch.

Almost everybody has a light in their house that can be switched on and off from two or more different places. If your electrician set up the wiring properly, you are able to toggle the light with every single switch, independent of the others' current configuration.

We may think of the switches as implementing a parity function. If we label the two positions of a switch by 0 and 1, the light is on if the sum of switch positions is even and off if this sum is odd (or vice versa). So in the common case of only two switches, ‘light on’ is just the binary relation xor. (Not all real-life installations are of this kind. We also find the binary and implemented in a few houses—which can be very annoying, and even dangerous, at nights.)

In this paper we analyse a game that was published in the 1996 Burger King Kid’s Club calendar. This calendar included an activity for each month, and for the month of September the activity was called the Mountain Bike Rally. To begin the game, each player places a marker on day 1. A turn consists of each player flipping a fair coin. Each player whose coin comes up heads moves his marker forward one day. If a player’s coin shows tails, then he moves his marker back one day, unless his marker is on day 1, in which case it stays there. The object is to be the first player to reach square 31. We never played the game because it was clear that it would take a long time to complete.

When identical copies of a solid can be packed together indefinitely to cover the whole of space and leave no gaps it is said to be space-filling. Usually only convex solids are considered in this context since, as Grünbaum and Shephard point out, non-convex space-filling solids can be generated from convex examples by making compensating additions and reductions, analogous to Escher’s modifications of plane tessellations. Nevertheless some non-convex space-filling solids are not obvious derivatives, and are interesting in their own right. Although the examples considered here have all been generated in the same way (to be described), their relation to other space-filling objects will also be considered.

Probability theory abounds in counterintuitive results, perhaps the most celebrated being the answer to the birthday problem: what is the least value of n such that p (n, 2) > ½ where p (n, 2) denotes the probability that at least two out of n randomly chosen people have the same birthday? The question assumes birthdays are uniformly and independently distributed with leap years being ignored. The solution, n = 23, never fails to startle beginning students, and very often triggers an interest in the subject of probability. It is derived from the well-known observation that by the principle of complementation p (n, 2) is one minus the probability that no two have the same birthday, i.e.

This number arises naturally, of course, as the hypotenuse of a right triangle with legs of length 1. Notoriously, √2 was found to be irrational by the Pythagoreans in around 500 BC. Their mathematics and philosophy was based upon natural numbers and small number ratios, and thus this discovery would have been very troubling: the Pythagoreans wouldn’t be the only ones to react badly to the imposition of the irrational.

Dedicated to Professor S. Kurepa on the occasion of his 73rd birthday. We are familiar with many trigonometric formulas (identities)

leading to five more identities

and so on, where #x211D; is the set of reals.