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A Christmas packing puzzle

By David Shaw, 24 December 2019

Merry Christmas! Here’s our maths gift to you – a puzzle.

You will need

Stick the middle of piece A to the end of piece B.

• Scissors
• Glue
• Colouring in pencils or textas
• Puzzle piece printout (three pages)
• Big box printout (two pages)

What to do

1. Print out and colour the two pieces for the big box. Then, cut them both out.
2. Stick the middle of piece A to the end of piece B, to make a ‘T’. There are two triangles to show you where to match them.
3. Fold along each dotted line and make a box. Glue all the tabs, except the ones on the top of the box (they are labelled).
4. Print out all the puzzle pieces and colour them in.
5. Cut out each of the pieces along the thick black lines.
   

   Don’t glue the tabs on the top of the big box.

6. Fold along the dotted lines so your colouring is on the outside.
7. When the shape looks like a box, glue the tabs to hold it together.
8. You should end up with six larger boxes, and three small ones.

The puzzle

Use the nine presents to make a cube. If you have made the big box, you should be able to fit all the small presents into the big box.

Some hints

• The bigger presents are the same size (volume) as four little presents. Work out how many little presents it would take to make the big cube.
• Look at the bottom layer. How many little presents would it take to fill the bottom layer? Can you fill the bottom layer with just the larger boxes? What about the middle layer, or the top one?
  

  You need six larger boxes and three smaller ones.

• If you’re still having a hard time, make sure you haven’t put two small boxes on top of each other.

What’s happening?

The secret to this puzzle is where you put the little boxes. Working out where to put them is all about odd and even numbers.

Firstly, it’s useful to imagine the big box being made up of little cubes, each as big as one of the small presents. The big box is three cubes wide, three cubes tall and three cubes deep. Looking at the bottom layer of the big box, we find it can fit nine cubes. No matter how you arrange the larger presents, they take up an even number of cubes in the bottom layer, so to completely fill the bottom layer, you’ll need to put at least one little present in there, too. For exactly the same reason, you’ll need to put a little cube in the middle layer, and one in the top layer.

A large paper box filled with smaller rectangular and cubic paper boxes.

If you slice off all the cubes on the left hand side, you’ll find another square of nine cubes. This means you’ll need a small present on the left hand side, one on the right hand side, and one in the middle. You also need one on the front face, and one on the back face, and one in the middle. If you put the three little cubes on a diagonal going from top front left to bottom back right, then you should be able to fit the other presents around them.

Real-life maths

This sort of puzzle is called a packing problem. Other examples of packing problems include trying to fit bags into a car when you go on holidays, or trying to get all your stuff into a cupboard when you are tidying your room. A good strategy for these sorts of problems is to make big, closely packed groups of things, and then try to fit all those groups into the container. This will often give you a good packing, but it won’t necessarily give you the best packing. In this puzzle, making a big block out of three big presents won’t lead to the correct solution.

If you’re having difficulty try making this shape and then see if you can fit the rest of the pieces in.

Finding the best solution to a packing problem is very hard. Mathematicians have spent many years investigating these sorts of problems, and they still don’t have a quick method for solving all of them. In fact, many mathematicians think there is no easy solution, and most packing problems will always be hard.

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