another logic puzzle

[desh]

This came up today. I'm not 100% sure the answer I have in mind is unique, but I think it is.

You and your spouse invite 4 other couples over for a dinner party. During the course of the party, many people shake hands with other people, except that no one shakes hands with their partner or themselves. At the end of the party, you realize that each of the other 9 people (that is, everyone but you) shook a different number of hands from each other. How many different hands did you shake?

Again, the answer doesn't involve tricks, like some people shaking with both their left and right hands or shaking the hand of someone not at the party. All comments screened for now, like before. Questions and bad guesses will be unscreened sooner; good guesses and right answers will be unscreened later. No googling allowed! Comments now contain spoilers.

Comments

[conana.livejournal.com]

Since no one shakes hands with eir spouse, the counts represented will be the integers between 0 and 8, with one number appearing twice. I filled out a binary matrix, which obviously must be symmetric (A equals A-transpose), by filling the lower-right triangle, with the result that 4 appears twice. I would like to have a proof of uniqueness, but do not. Any single handshake added either removes two numbers from the counts leaving three duplicate pairs (or three 4s and a pair), or removes the 4 pair, leaving three 5s, but I have not gone further. Thanks to deled for catching me in several appalling errors.

[desh]

Nice method! Not one I would've thought of.