Quote:
Originally Posted by THILLIAR
What about the additional space found within the 1meter cube? Each layer has some additional space that the balls will, like a liquid, "fill". The first layer has 3.4 cm extra.

Yes, what about it?
This is what I've called the wiggle room. I've already offered my opinion about how many additional balls can be fit into the box because of this. The space you mention on the bottom layer is mostly filled by the balls on the second layer, but that last row could roll 5 mm further and drop slightly, opening a tiny gap between the last two rows on the second layer. The the last row on the third layer could drop into this gap. Which would make it possible for the last two rows of the fourth layer to drop slightly. This process could be continued until the top layer, letting gravity packing slide the ends down slightly.
There are other possibilities that you haven't mentioned. The edge row of the top layer doesn't have to nestle down into the tetrahedral packing. It could be rolled up slightly to hit top of the box. The the next rows over could be rolled up slightly to touch the displaced row. This process could be continued across the top of the box. Continuing all the way across the top, this extra space could join the extra space from the gravity packing.
It's possible that these moves in combination would create enough space to squeeze in one more row of balls. This possibility is why I said "
at least 1254 balls.
With only 5 mm spare on each layer and less than 2 mm spare at the top, I think it's unlikely that these manuevers will gain enough room to fit more 50mm balls. I judge the likelyhood to be low enough that I'm not going to do the calculations to see if they fit. But if you think more can fit, go for it. If you're right, then we will all applaud your superior solution. Of course, be sure to show all the math so we can confirm the answer.