The n th roots of unity play a key role in abstract algebra, providing a rich link between groups, vectors, regular n-gons, and algebraic factorizations. This richness permits extensive study. A historical example of this interest is the 1938 challenge levelled by the Soviet mathematician N. G. Chebotarëv (see [1] or [2]). His question was ‘Are the coefficients of the irreducible factors in Z[n] of xn − 1 always from the set {−1, 0, 1}?’ Massive tables of data were compiled, but attempts to prove the results for all n failed. Three years later, V. Ivanov [3] proved that all polynomials xn 1, where n < 105, had the property that when fully factored over the integers all coefficients were in the set {−1, 0, 1}. However, one of the factors of x105 − 1 contains two coefficients that are −2. Ivanov further proved for which n such factorisations would occur and which term in the factor would have the anomalous coefficients. A twist that makes this historical episode more intriguing is that Bloom credited Bang with making this discovery in 1895, predating the Chebotarëv challenge by more than four decades.

Consider the functional equation

Now let and These are elements of a group, G1 say, which is isomorphic to C2, the cyclic group of order 2, as can be seen from the group table:

The values of many functions, for example sinx or cosx, can be computed efficiently for arbitrary values of x from their known power series expansions. However, for the logarithmic function this is not so straightforward. There does not exist an expansion of ln x as a power series in x, the closest to it mathematically being an infinite power series in of the form

where (see [1]).

In [1] the author investigated a method of measuring how good the spread of lessons is for a class in a school timetable. The method considered only the maximum number of days between two consecutive lessons. However, it was observed that not all configurations with the same maximum gap are equally good. For example, the configuration for a tenday cycle in Figure 1(a) would be preferable to that in Figure 1(b); the filled circles represent days with a lesson and the unfilled circles days without. Here we propose a model that takes into account the overall configuration when measuring how good the spread of lessons is.