In helping my aging parents move from one apartment to another, I faced the problem of bundling rolls of wrapping paper together. I laid a bungee cord on a flat surface, stacked the rolls athwart the cord in what seemed to be a reasonably compact formation, fastened the cord (turning the triangular stack into something more circular), and picked up the bundle of rolls. Unfortunately, when I turned the bundle on its side, one of the middle rolls fell out. In terms of its cross section, that roll was what in disk-packing parlance is called a “rattler”, and friction wasn’t holding it in place. Are there good heuristics for avoiding the rattler problem when packing cylinders? For instance, would my chances of success have been greater if I’d put the thicker rolls in the middle of the stack and the thinner rolls near the periphery? (As it happens, the rattler was a thin roll near the middle.) And should I have tried to ensure that adjacent rolls had roughly the same diameter? If that’s too loosey-goosey for you, here’s a precise math question (though perhaps in its current form it’s merely pre-precise): Is there a set of rolls (i.e., a multiset of diameters) that cannot be bungee-bundled, in the sense that for any stable way of wrapping them circumferentially with an elastic cord, there’ll be a rattler? My notion of stability is based on the assumption that the (potential) energy of the system is the length of the cord; I assume that energy cannot increase, that there is no slippage between rolls, and that there is no slippage between the bungee cord and the rolls that it touches. At first I thought that if one diameter is tiny relative to all the others, it must be a rattler, but when I thought about it harder and remembered the no-slip condition, it no longer seemed obvious to me. Indeed, when the diameters are epsilon, 1, and 1, I think the 1,1,epsilon “shishkebab arrangement” is stable. Jim Propp PS: Maybe my attempted formalization of the notion of stability needs some improvement. I think that the 1,epsilon,1 shishkebab is technically stable because of the no-slip conditions, but this doesn’t correspond to physical reality.