# Sunday Times Teaser 2673 – Footprints

*by Nick MacKinnon*

Each of the faces of a six sided cubical die are numbered one to six as usual and have dimensions that exactly match the dimensions of each of the nine cells in a three by three grid.

The die is placed on one of these cells and is then ‘rolled’ on one of its lower edges in such a way that it moves onto another grid cell. This move is repeated a total of eight times and in such a way that the die occupies every grid cell exactly once.

The score for such a sequence of moves is the sum of the nine die faces that have been in contact with the grid (some added more than once).

What are the minimum and maximum possible scores and what scores between these two values are impossible?

This is my first attempt at this, one which I may improve later.

3×3 grid gives us the central and the four corners as starting points as a result of which 21 and 42 being the minimum and the maximum scores which can be scored respectively with 29 and 34 being impossible scores to reach.

There are 40 distinct ways of letting the die to touch each cell only once and each way having 24 different possibilities to apply.