## Task 4 - 050614

In a company, 1600 packages, each 1.6 dm long, 7 cm wide, and 45 mm high (external dimensions), are to be shipped.

Workers want to stack them in boxes with internal dimensions of 64 cm in length, 0.28 m in width, and 1.8 dm in height.

What is the smallest number of boxes needed to ship all these packages at the same time?
We can pack 64 packages into each of the boxes if we place the packages so that their longest edges are parallel to the longest edge of the box and their second-longest edges are parallel to the second-longest edge of the box. In this case, we get 4 overlapping layers of 16 packages each.

With 25 such packed boxes, we are sufficient; because the number of packages in them is 25 × 64 = 1600.

The sum of the volumes of the inner spaces of all 25 boxes is exactly the same as the sum of the volumes of the packages. Since fewer than 25 boxes would have a smaller volume than the one calculated here, 25 is the smallest number of boxes that is sufficient to ship 1600 packages of the size specified in the problem simultaneously.

Solutions of the I. Round 1965 adopted from [5]

### 3.6.2 II. Round 1965, Class 6