A packing carton measures 60 cm by 50 cm by 40 cm inside. Solid cubical blocks of edge 12 cm are to be stacked in it in upright rows, without tilting or cutting any block. What is the greatest number of blocks that will go in?

Blocks can only be counted whole along each direction: 60/12 = 5 exactly, 50/12 gives 4 with 2 cm to spare, and 40/12 gives 3 with 4 cm to spare. So $5\times 4\times$ 3 = 60 blocks fit, and the leftover strips stay empty. The trap 69 divides volume by volume, 120000/1728 = 69.4, which pretends the wasted 2 cm and 4 cm strips can be filled with fragments of a block; 80 rounds 40/12 UP to 4 rather than down; $125=5^3$ applies the exact fit of the 60 cm side to all three directions.