### All in the Mind

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. 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?

# Noughts and Crosses

##### Stage: 3 Challenge Level:

Imagine a $3 \times3 \times3$ cube, made up from 27 unit cubes, all of which are made from clear plastic that can be filled with ease.

The location of a unit cube is described according to the following positions with respect to the three axes or directions:
• left, middle, right;
• front, middle, back;
• top, middle, bottom.

A marble is placed in the unit cube at left-middle-bottom.
Another is placed at middle-middle-middle.
Where should the third marble be placed to make a winning line of three marbles?

How many winning lines go through middle-middle-middle?

How many different types of winning lines are there?

How many winning lines are there altogether?

How many winning lines of four are there altogether in a $4 \times 4 \times 4$ cube?

How many winning lines of $n$ are there altogether in an $n \times n \times n$ cube?

This problem will feature in Maths Trails - Visualising, one of the books in the Maths Trails series written by members of the NRICH Team and published by Cambridge University Press. Maths Trails - Visualising is due to be published later this year, but for more details about the other books in the series, please see our publications page .