What’s the largest number of pennies that you can pack inside a circular tray to form a carpet of non-overlapping coins? What about inside a square or triangular tray? What if you could expand or contract the size of all your pennies to fit a required number of them snugly in a given tray?
Mathematicians have long pondered the problem of packing identical circles inside a variety of geometric shapes. Indeed, “the optimal packing of equal disks . . . is an ancient and extremely difficult problem,” says mathematician Ronald L. Graham of the University of California, San Diego. “Some of these very simple problems—like how you pack 27 disks in a triangle, square, or circle—are very stubborn.”