We distinguish between a level on a sequence of charts (e.g. top level or 5th level) and a place on a single chart (e.g. 5th place on the 22nd week). A song occupies a place by halting there prior to a continued rise or fall of its path, or by stopping there if that place is at the highest level attained. A song which passes to a level higher (or lower) than level p without halting there bye-passes level p. The upward path of a song terminates at the first stopping place and the downward path commences at the last stopping place. Note : Level No. 1 is the lowest level and level P is the uppermost one. This is contrary to popular usage but makes the mathematics more conventional.

The problem of arranging dominoes with like numbers in groups of four was proposed by Édouard Lucas. Although Lucas published only symmetrical solutions for the double-six set, generalizations of the problem have led to some interesting results. For the double-six set, it has been shown there are hundreds of shapes meeting the requirement that like numbers appear in groups of four.

If we wish to plot the results of the election on a graph, constituency by constituency, we face the problem that a sheet of paper has two degrees of freedom, but there are three political parties. We may overcome this by using only the ratio of the votes obtained by each party. In any constituency,

This September the transition from Grammar to Senior High School will be virtually complete. The transfer age is 14, and is, at the moment, by parental choice. We still have all boys, but receive our first intake of girls this year. Our first non-selective entry was in 1964, and all members of the mathematics staff, only one of whom had met them previously, have now taught the lower ability groups. We have had our problems, none insurmountable, and more of administration than of adaptation. Anyone embarking on a transition to Comprehensive education and needing advice is recommended to read Canon Eperson’s admirable article in the recent Gazette. Much of what he says we have learned by experience, and if I requote from Newsom, it is because the quotations are particularly apt.

Do you know precisely how each lesson is to develop?

If the answer to the above question is yes, it is recommended that you look carefully at your technique to see if it is in line with modern ideas. Let me be so immodest as to describe how a lesson developed the other day.

Populations of large mammals such as elephants tend to group themselves into herds. As these herds wander about, it may happen that two of them meet and amalgamate, or that a large one may split into two smaller ones, (see John Hillaby, “Elephants as a pest control problem”, New Scientist, 12 (1961), 736-8). Suppose that a given area contains n animals, and let pk (t) denotes the probability that at time t they are grouped into k herds. Conditional on this, let x1(t), …, xk(t) denote the sizes of the herds arranged say, in non-increasing order. The chance that two herds will meet and amalgamate will increase with the number of herds, and it may, for instance be taken to be proportional to the excess of this number over unity. Thus the probability of an amalgamation in the interval (t, t + δt), given that there are k herds at time t, is

In a recent article [1] in this Gazette, Collings discusses the number of arrangements of n railway trucks on k sidings under a variety of conditions. Let qk (n) be the number of ways of arranging n indistinguishable trucks on k indistinguishable sidings or, what is the same thing, the number of partitions of n into not more than k parts. Let pk (n) be the number of these arrangements which use all k sidings, that is, the number of partitions of n into exactly k parts. Collings describes the determination of pk(n) and qk(n) as an unsolved problem in partition theory; this is not quite correct.

Everybody is familiar with the game of “Snakes and Ladders”, but not so many realise that the game constitutes an interesting example of a Markov chain. We feel that a good way of introducing Markov chain theory is by solving the problem, How long does a game of Snakes and Ladders last? Doubtless the reader has found that the last game before sending a child to bed seems to drag on interminably, and we used a computer to determine the actual average number of moves when 2 people play on a typical board.