Here is more strange data on Grabbing Cubes. It looks like Jim Propp was right to be suspicious of the proof in Math Horizons, because it does not say why 5 x 13 x 31 has no shortcut, as several other box sizes do. Consider all 120 boxes with distinct odd prime dimensions p1<p2<p3<32. For each box size, I ran 1000 games. Whenever more than one cube had maximal yield, a random choice among those was made. 87 box sizes had the same game length in all 1000 games, namely p1*p2. But the other 33 box sizes have a distribution of game lengths. The maximum game length seen is p1*p2. In general, for larger boxes, the game lengths seen becomes farther and farther below p1*p2. It seems that there are shortcut strategies available for these boxes, and the larger the box, the more the shortcuts become unavoidable. What distinguishes the box sizes that have shorter games??? The 33 are: box 5 11 13, length 49 to 55 no gaps box 5 17 19, length 78 to 85 no gaps box 5 29 31, length 138 to 145 no gaps box 7 11 13, length 63 to 75 with gaps box 7 13 17, length 85 to 91 no gaps box 7 17 19, length 107 to 118 no gaps box 7 19 23, length 126 to 133 no gaps box 7 29 31, length 190 to 201 no gaps box 11 13 17, length 123 to 140 no gaps box 11 13 19, length 131 to 142 no gaps box 11 17 19, length 157 to 177 with gaps box 11 17 23, length 175 to 187 no gaps box 11 19 23, length 188 to 205 no gaps box 11 23 29, length 242 to 253 no gaps box 11 23 31, length 248 to 253 no gaps box 11 29 31, length 288 to 309 with gaps box 13 17 19, length 178 to 201 no gaps box 13 17 23, length 202 to 218 with gaps box 13 19 23, length 218 to 235 no gaps box 13 19 29, length 242 to 247 no gaps box 13 23 29, length 278 to 295 no gaps box 13 23 31, length 288 to 299 no gaps box 13 29 31, length 331 to 357 no gaps box 17 19 23, length 263 to 294 no gaps box 17 19 29, length 305 to 319 no gaps box 17 19 31, length 312 to 323 no gaps box 17 23 29, length 348 to 373 no gaps box 17 23 31, length 360 to 381 with gaps box 17 29 31, length 407 to 451 with gaps box 19 23 29, length 379 to 410 with gaps box 19 23 31, length 393 to 419 with gaps box 19 29 31, length 454 to 497 with gaps box 23 29 31, length 527 to 584 with gaps The range of game lengths is those seen in 1000 games. "With gaps" means some value(s) in the range were not seen. For instance, box 7 11 13, game length 64 was not seen. But running 10^4 games not only saw 64, but also saw 76 and 77, extending the range to p1*p2 = 7*11 = 77: box 7 11 13, 10000 games, length 63 to 77 no gaps But running box 7 17 19 for 10^4 games saw only one smaller game length (106), the max seen (118) still falling short of p1*p2 = 119. The high end tail for 7 17 19 might be very thin, so running many more games might encounter length 119. That seems unlikely for 11 17 19, where 10^4 games saw only a new low game length (156). Its high game length seen (177) is far short of p1*p2 = 187. -- Mike