Necklaces are the equivalence classes of words under the action of the cyclic group. Let a transition in a word be any change between two adjacent letters modulo the word’s length. We present a closed-form solution for the enumeration of necklaces in n beads, k colours and t transitions. We show that our result provides a more general solution to the problem of counting alternating (proper) colourings of the vertices of a regular n-gon.

Suppose that A is a k × d matrix of integers and write $\Re _A:{\mathbb N}\to {\mathbb N}\cup \{ \infty \} $ for the function taking r to the largest N such that there is an r-colouring $\mathcal {C}$ of [N] with $\bigcup _{C \in \mathcal {C}}{C^d}\cap \ker A =\emptyset $. We show that if ℜA(r) < ∞ for all $r\in {\mathbb N}$ then $\mathfrak {R}_A(r) \leqslant \exp (\exp (r^{O_{A}(1)}))$ for all r ⩾ 2. When the kernel of A consists only of Brauer configurations – that is, vectors of the form (y, x, x + y, …, x + (d − 2)y) – the above statement has been proved by Chapman and Prendiville with good bounds on the OA(1) term.

We show that for all $m,k,r\in \mathbb{N}$, there is an $n\in \mathbb{N}$ such that whenever $L$ is a Latin square of order $m$ and the Cartesian product $L^{n}$ of $n$ copies of $L$ is $r$-coloured, there is a monochrome Latin subsquare of $L^{n}$, isotopic to $L^{k}$. In particular, for every prime $p$ and for all $k,r\in \mathbb{N}$, there is an $n\in \mathbb{N}$ such that whenever the multiplication table $L({\mathbb{Z}_{p}}^{n})$ of the group ${\mathbb{Z}_{p}}^{n}$ is $r$-coloured, there is a monochrome Latin subsquare of order $p^{k}$. On the other hand, we show that for every group $G$ of order $\leq 15$, there is a 2-colouring of $L(G)$ without a nontrivial monochrome Latin subsquare.

Let $n,r,k\in \mathbb{N}$. An $r$-colouring of the vertices of a regular $n$-gon is any mapping $\unicode[STIX]{x1D712}:\mathbb{Z}_{n}\rightarrow \{1,2,\ldots ,r\}$. Two colourings are equivalent if one of them can be obtained from another by a rotation of the polygon. An $r$-ary necklace of length $n$ is an equivalence class of $r$-colourings of $\mathbb{Z}_{n}$. We say that a colouring is $k$-alternating if all $k$ consecutive vertices have pairwise distinct colours. We compute the smallest number $r$ for which there exists a $k$-alternating $r$-colouring of $\mathbb{Z}_{n}$ and we count, for any $r$, 2-alternating $r$-colourings of $\mathbb{Z}_{n}$ and 2-alternating $r$-ary necklaces of length $n$.

An $r$-ary necklace (bracelet) of length $n$ is an equivalence class of $r$-colourings of vertices of a regular $n$-gon, taking all rotations (rotations and reflections) as equivalent. A necklace (bracelet) is symmetric if a corresponding colouring is invariant under some reflection. We show that the number of symmetric $r$-ary necklaces (bracelets) of length $n$ is $\frac{1}{2} (r+ 1){r}^{n/ 2} $ if $n$ is even, and ${r}^{(n+ 1)/ 2} $ if $n$ is odd.

The polythene films used in manufacturing bipyramidal tsetse traps in the Central African Republic can exhibit slight variations in colours. Experiments using a new protocol that combines a Latin square design and two competing traps in the same experiment conducted on Glossina fuscipes fuscipes show that a iess intensely blue colouring decreases the attractiveness of the trap. Polythene is useful material for tsetse traps, but is necessary to verify that its colouring is homogeneous.