322. Cigarette Packaging. Cigarettes are shipped from the factory in boxes of 160. They are arranged in 8 rows of 20 each, completely filling the box.

Can more than 160 cigarettes be placed in the same box with a different packaging method? If so, how many more cigarettes can be added?

At first glance, it seems absurd to think that extra cigarettes can be added to a fully filled box, but after a moment's thought, you might find the key to this paradox.
322. Let the diameter of a cigarette be 2 units, and let 8 rows of 20 cigarettes each (see case \(A\)) completely fill a box. In this case, the internal length of the box is 40, and the depth is 16 units. Now, if we place 20 cigarettes in the bottom row and if instead of 20 in the next row we place 19, as shown in case \(B\), we save 0.268 (more precisely \(2-\sqrt{3}\)) in height. This second row and each additional row of 20 or 19 (alternating) cigarettes increases the height by 1.732. Therefore, we will get nine rows with a total height of \(2 + 8 \times 1.732 = 15.856\) units, which is less than our depth of 16 units. Thus, we will increase the number of cigarettes by 20 (thanks to the additional

ANSWER

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row) and decrease it by 4 (1 unit in each row of 19), resulting in a net increase of 16 cigarettes.