This paper highlights the essential need for appropriate statistical design and randomisation in laboratory animal studies. Using an example of a 21 day weight gain study in mice, we show that without the use of an appropriate statistical design and randomisation, incorrect conclusions may have been drawn. We used an experimental design that allowed comparisons to be made between five treatments that were free from systematic error. Two alternative designs that are practically attractive, yet had no statistical basis, are also described in this paper and the potentially incorrect conclusions highlighted. The use of appropriate statistical design is ethical because it results in clear, unambiguous conclusions. Conclusions that may be biased or ambiguous will require verification by further research and this, in the long term, is contrary to the reduction element of the Three Rs.

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.

A Latin square is reduced if its first row and first column are in natural order. For Latin squares of a particular order $n$, there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order $n\rightarrow \infty$.