I know that does not sound like a math question, but I believe there should be a really simple solution to this somewhere in combinatorics (I am sorry in advance if I am wrong). There is a group of 6 people, whom we have to divide into two equal groups of 3 according to their stated preferences. How this mapping of preferences to subgroups should like so the groups are always of the same size of 3?
For example, in this famous experiment by Tejfel the participants had to choose between Klee and Kandinsky and were assigned into two different groups based on their choice (Tajfel, Henri. "Experiments in intergroup discrimination." Scientific American 223.5 (1970): 96-103.)
But there is no guarantee that the preferences will split the group into two equal parts.
If for instance, we would offer them to order N pictures from the most to the least preferred, and then assign the group membership based on the preferences, that would allow us to divide them. Unless all 6 of them would choose the same order (unlikely but possible).
Does such ordering exist?