Noughts and crosses

Ever thought of playing three dimensional Noughts and Crosses? This problem might help you visualise what's involved.
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem

 

The ideas in this problem have been expanded and made into another, more detailed problem. See Marbles in a Box.

 

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 .