optimization - Optimize pattern of rotating holes for all combinations -

sort of programming question, sort of general logic question. imagine circular base pattern of circles:

and circle, mounted above , able rotate, holes expose colored circles below:

there must optimal pattern of either colored circles or openings (or both) allow n possible combinations of colors... have no idea how attack problem! @ point, combinations of 2 seem easiest , fine starting point (red/blue, red/green, red/white, etc).

i imagine there need gaps in colors, unlike example above. suggestions welcome!

edit: clarified question (hopefully!) feedback robert harvey

for 2 holes, perfect matching in bipartite graph, each permutation described 2 nodes, 1 in each partition. nodes connected if share 1 element, i.e. (blue,red) node first partition connected (red,green) node of second. circles arranged in same distance allow both of these patterns. perfect matching in graph correspond chains or cycles of permutations 2 of them share single color. bit dominoes. if had set of cycles of same length, interleave them form pattern on lower disk. i'm not sure how easy obtain these same length cycles, though, , don't know how generalize more 2 elements in each permutation.