I call to math-funsters' attention my recent MathOverflow post http://mathoverflow.net/questions/202165/transitivity-of-balanced-mass-trans... which has a math-fun-ish quality. One thing that I know, but that I don't prove in that post (though I do mention it in one of the two affiliated posts to MathOverflow) is that if mu is the measure that gives unit mass to every integer greater than or equal to 1, and nu is the measure that gives unit mass to every integer greater than or equal to 0, then nu cannot be obtained from mu by a single balanced-transport move, or by a finite sequence of such moves, or even by an infinite sequence of such moves convergent in the total variation metric. I pose this as a puzzle for math-fun, not only because it has a cute answer, but because I'm hoping it has more than one cute answer, and that one of you will find a different answer to this puzzle than the one that I know! (And it would also be nice if one of you proved or disproved transitivity, but that's a separate issue.) Jim Propp