The kissing number problem asks how many n dimensional unit spheres can fit around a central one with no overlapping; a natural question is in what dimensions can this be done so that there is no extra space to move any outer sphere around the central one, as in the following picture:
There are such configurations in dimensions 1, 2, 8, and 24, and they tie into some exceptional mathematics. My question is, is it possible there are other dimensions that also exhibit this property? Which dimensions have been ruled out, and if little is known, are there natural conjectures or heuristics?