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?