The photo above is from one of two packing puzzles in Leonardo's Mirror & Other Puzzles by Ivan Moscovich, about whom I have written lately. I just noticed that you can actually get new copies of this book in paperback at Amazon for under $1. (I paid more, and found it well worth the price.)

So on to the problem. The image shows a crate of 48 bottles of your favorite beverage, and they are compactly packed in a rectangular array so as not to rattle in the case. But is this the best you can do? Is it possible to alter the packing and squeeze in one or more extra bottles.

I will cleverly divert your eye with the link to the book, but if you need a hint, I have placed a key word below the link.

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"Flag"

If you've figured it out, or just want to go on, I give the answer down the page a bit.

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Yep, if you look at the stars of the US Flag, you can see a tighter packing for the bottles. Alternating rows of five between rows of six allows you to get 50 bottles in the same case. Actually they will fit in a somewhat smaller case. If for ease of computation, we put the diameter of the bottles as 2 inches, it is simple to figure out the original case had to be 12 inches by 16 inches to pack the bottles in the rectangular style. But if you uses the six-five-six hexagonal packing, the distance between the centers of bottles from one six row to the next is only the square root of 3 times the diameter of the bottles. so since there are five rows of six bottles, there will be five distances of 1.732 times the diameter, plus 1 additional diameter for the two outside rows to reach the edge. My calculator gives a little under 15.9 inches with the fifty bottles packed in patriotic style.

Of course having the answer, I'm sure you are wondering if it is possible to do the same thing with smaller rectangular packs. Could we do it with a 12 pack case, or 24, or..... well, what are you waiting for. Get the pencil and paper out and get to figuring. Send me your record beating packing, or proof that there can't be one.....

And as long as we are looking at hexagonal packings, here is another nice problem from the same book.

If you look at the 23 ping-pong balls packed in the rectangular crate, you can remove one and the pack is still rigid; none of the ping-pong balls can slide around. But can you take out two, or three, or ????

Enjoy.