A finite conflict with given payoff matrix may have many ESS's (evolutionarily stable strategies). For a given set of pure strategies { 1, 2, ...,n} a set of subsets of these is called a pattern, and if there exists ann ×n matrix which has ESS's whose supports (i.e. the playable strategies) precisely match the elements of the pattern, then the pattern is said to be attainable. In [5] and [10] some methods were developed to specify when a pattern was, or was not, attainable. The object here is to present a somewhat different method which is essentially recursive. We derive certain results which allow one to deduce from the attainability of a pattern for givenn the attainability of other patterns forn+1, and by induction for anyn+r.