Few people would find this to be an unsatisfactory proof that the nth triangular number is equal to the total number of pairings of n+1 objects: http://i.imgur.

Few people would find this to be an unsatisfactory proof that the nth triangular number is equal to the total number of pairings of n+1 objects: http://i.imgur.com/HCfGOYp.gif http://i.imgur.com/HCfGOYp.gif It would be interesting to test this. I forced myself to stare at that image longer than was comfortable, and I got nothing from it. Then I came back and read your words, then went back to the diagram, and still couldn't figure out what relevance the diagram had. Possibly I'm part of a small minority, possibly you are, or possibly it's 50/50. Without testing, I wouldn't wager a guess which of these is most likely. You are likely better than me at visually assembling the individual frames into a coherent whole. The sequence forms a 3-dimensional image, but with 2 dimensions of space and 1 dimension of time. I feel my brain struggling (and failing) to project it into 3 spatial dimensions. The frenetic chaos leaves me longing for the serenity of words.