### All in the Mind

Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?

### Painting Cubes

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

### Tic Tac Toe

In the game of Noughts and Crosses there are 8 distinct winning lines. How many distinct winning lines are there in a game played on a 3 by 3 by 3 board, with 27 cells?

# Nicola's Jigsaw

### Why do this problem?

This problem offers students an opportunity to reason spatially in three dimensions, practise interpreting and drawing two dimensional representations of 3D shapes, and work systematically to find all the possible solutions to a problem.

### Possible approach

The problem is much more manageable if students have cubes available to work with.
Introduce the problem and show students the diagrams of the five pieces:

Some students might want to start by making the five pictured pieces.

In order to find what the missing piece might look like, encourage students to consider what they know about volume to deduce how many cubes make up the missing piece.

They could then make each possible missing piece to see if it can be used to complete the puzzle, and use isometric paper to record their findings.

### Key questions

What is the volume of each piece?
What is the volume of the finished puzzle?
What is the volume of the missing piece?
What possible shapes could the missing piece be?

### Possible extension

Nine Colours and Marbles in a Box offer suitable follow up challenges requiring reasoning in three dimensions.

### Possible support

Students could work on The Third Dimension to practise working systematically and recording their work on isometric paper.