### Brain Teasers

# Wrapping It Up

I need to wrap eight cube-shaped boxes that are four inches on each side. I may wrap them separately, or I may wrap them together in a variety of different shapes and configurations.

In what way should I wrap all the boxes to use the least amount of wrapping paper?

In what way should I wrap all the boxes to use the least amount of wrapping paper?

### Hint

Think of different ways to arrange the boxes for wrapping. The number of box sides to be covered by the wrapping paper determines how much paper is needed.### Answer

To use the smallest amount of wrapping paper, I need to arrange the boxes as a cube.I can wrap the boxes separately; in a long line of eight; as a rectangle made up of two rows of four boxes each; or together as a cube.

Wrapping separately:

48 box sides x 16 square inches per box side = 768 square inches;

Wrapping as a line of eight:

34 box sides x 16 square inches per box side = 544 square inches;

Wrapping as a rectangle:

28 box sides x 16 square inches per box side = 448 square inches;

Wrapping as a cube:

24 box sides x 16 square inches per box side = 384 square inches.

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## Comments

Fun but very easy, just use common sense.

The answer is wrong. The best arrangement is a 2x4 rectangle of boxes.

The problem with the answer is that it does not account for the waste involved in actually wrapping the boxes. To wrap the boxes, a piece of paper long enough to go around the boxes and wide enough to go across the boxes PLUS the height of the boxes is needed.

To wrap the cube, a piece 8 sides long by 4 sides wide is needed. This is 4 x 8 = 32 box sides.

To wrap the 4x2 rectangle either a piece 6 long by 5 wide or 10 long by 3 wide is needed, depending on how it is wrapped. Either way this is 5 x 6 = 30 box sides.

The 8x1 rectangle needs a piece either 18 long and 2 wide or 4 long and 9 wide, for a total of 36.

So the least amount of paper is used for the 4x2 rectangle.

The problem with the answer is that it does not account for the waste involved in actually wrapping the boxes. To wrap the boxes, a piece of paper long enough to go around the boxes and wide enough to go across the boxes PLUS the height of the boxes is needed.

To wrap the cube, a piece 8 sides long by 4 sides wide is needed. This is 4 x 8 = 32 box sides.

To wrap the 4x2 rectangle either a piece 6 long by 5 wide or 10 long by 3 wide is needed, depending on how it is wrapped. Either way this is 5 x 6 = 30 box sides.

The 8x1 rectangle needs a piece either 18 long and 2 wide or 4 long and 9 wide, for a total of 36.

So the least amount of paper is used for the 4x2 rectangle.

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