Cans in boxes
How many cans can be packed in a 50cm x 80cm box, if each can has a diameter of 10 cm? At first glance one would say forty -- if the packing is made very tight.
There is, however, a smarter way of placing the cans in the box which allows the packing of 5 x 5 + 4 x 4= 41 cans.
It may amuse you to know that there is a simple way of transforming the first packing into the second. Indeed, remove two cans, the second and the fourth in the uppermost row, shove the remaining cans (you might wish to find yourself the correct way to do it) and then add the two previously removed together with a new one.
What is more, now the packing is not even as tight as in the first case, since
10 + 8 x 5 x 31/2 = 79.28... < 80,
which means that you have enough room to move the cans slightly!