Desirable and undesirable properties of Latin squares in experiments? A cursory search reveals that Latin squares are fairly extensively used in the design of experiments.  During my PhD, I have studied various theoretical properties of Latin squares (from a combinatorics point-of-view), but do not have a deep understanding of what is it about Latin squares that make them particularly well-suited to experimental design.
I understand that Latin squares are good at allowing statisticians to efficiently study situations where there are two factors which vary in different "directions".  But, I'm also fairly confident there would be many other techniques that could be used.

What is it, in particular, about Latin squares that make them so well suited for the design of experiments, that other designs do not have?

Moreover, there are zillions of Latin squares to choose from, so which Latin square do you choose?  I understand that choosing one at random is important, but there would still be some Latin squares that would be less suited to running experiments than others (e.g. the Cayley table of a cyclic group).  This raises the following question.

Which properties of Latin squares are desirable and which properties of Latin squares are undesirable for experimental design?

 A: Imagine:


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*you were interested in the effect of word type (nouns, adjectives, adverbs, and verbs) on recall. 

*you wanted to include word type as a within-subjects factor (i.e., all participants were exposed to all conditions)


Such a design would raise the issue of carry over effects. I.e., the order of the conditions may effect the dependent variable recall. For example, participants might get better at recalling words with practice. Thus, if the conditions were always presented in the same order, then the effect or order would be confounded with the effect of condition (i.e., word type).
Latin Squares is one of several strategies for dealing with order effects.
A Latin Squares design could involve assigning participants to one of four separate orderings (i.e., a between subjects condition called order):


*

*nouns adjectives adverbs verbs

*adjectives adverbs verbs nouns 

*adverbs verbs nouns adjectives 

*verbs nouns adjectives adverbs 


Thus, the Latin Squares design only entails a subset of possible orderings, and to some extent the effect of order can be estimated.
In a blog post I suggest the following simple rules of thumb:


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*"If order is the focus of the analysis (e.g., skill acquisition looking at effects of practice), then don't worry about order effects

*If order effects are very strong, it may be better to stick to between subjects designs

*if order effects are small or moderate or unknown, typical design strategy depends on the number of levels of the within-subjects factor of interest.

*

*If there are few levels (e.g., 2,3,4 perhaps), present all orders (counterbalance)

*If there are more levels (e.g., 4+ perhaps), adopt a latin squares approach or randomise ordering"



To specifically answer your question, Latin Squares designs allow you to get the statistical power benefits of a within-subjects design while, potentially at least, minimising the main problem of within subjects designs: i.e., order effects.
