In psychology we sometimes use balanced latin squares for the order of our tests to counterbalance first-order carry-over effects (fatigue, learning, etc.) .
For our current study we want to pretest 12 stimuli (let's call them A-F) to see whether they're useful for a later study. We don't want to bore our participants, so we wanted to give them only half of all the material we need to test. We're indifferent about the size of the subset of 12 as long as it is anything between 4-8 stimuli per participant.
For a different reason (to achieve sufficient statistical power) we need at least 132 participants (at least 11 runs where each stimulus occurs first), we don't want to exceed this too heavily.
A B F C E D B C A D F E C D B E A F D E C F B A E F D A C B F A E B D C
But is it also possible to construct a balanced (latin) rectangle (6 columns wide), where each letter is followed by another letter an equal amount of times? How many rows (participants) would this yield?
Maybe somebody with a bit more handle on this problem will enjoy the puzzle!
Sorry if my language is too idiosyncratic, if I can clarify with the appropriate jargon I'll duly comply, this is quite outside my field.
Splitting it in the middle and then adding the broken-up orders seemed the wrong approach to me.
Edit: Can I find one computationally? I have no idea how ridiculous that question is, but the sheer number of permutations (479 001 600) does seem daunting.
Edit 2: I didn't want to make this question too much about our experiment, but apparently that made it less clear. I'm sorry. I edited the clarifications into the question.