First day of school today. In biology, we had to do a metric measurement worksheet, and some of the problems involved a rubber stopper shaped like a clipped cone. The top diameter was 2.2 cm, the bottom diameter was 2.7 cm, and the non-lateral height was 2.5 cm. One question asked the volume. Another question asked how many could fit in a box the size of the biology book, which is 20.65 x 25.5 x 4.5 cm. The instructor simply divided volume and got 200-some. But this raises an interesting question: exactly how many clipped cones can fit into a box? I got 110, my method was a hexagonally-tiled row along the 20.65 cm axis. By having the upper row flipped from the bottom row, I figured 11 could fit together without exceeding the height of the book (6 on bottom, 5 on top). Then 10 of these rows could fit in the 25.5 cm length. That gives 110, but can anybody do more?

For a large enough box, I think the most efficient way is to tile them in hexagonal layers like cylinders, but have every other row "upsode-down". Then stack layer upon layer until the box is full. The top or bottom of each layer would be like so: