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Well done to all of you who sent in solutions to this problem - there are too many to mention you all, I'm afraid. Not so many of you answered both parts of the question, though! Mrs Alford's Maths Challenge Group at West Lodge Middle School, Mathew and Sam from Woodfall Primary and Linden from Comberton Village College explained their thinking clearly. West Lodge pupils say:

40 cubes were not drilled.
By drilling a hole in each corner of the face that is facing you, you get 16 cubes that have been drilled through.
When the friend drills through, 2 cubes on each edge have already been drilled so they don't count.
A total of 8 extra cubes have been drilled.
So 8 + 16 = 24 (cubes drilled.)
So 64 - 24 = 40 not drilled.    

Sakib from Swanlea School sent us a picture which really helps the visualisation:

picture of cube being drilled

William from St Patrick's School found a solution to the second part of the problem too:

If you were to drill through the centre 4 cubes (that are also 4 cubes deep) then 4 cubes x 4 cubes = 16.
Than if you drilled down from the top, only 2 cubes would have new holes
4 x 2 = 8, 8 + 16 = 24, 64 - 24 = 40.

Another William, from Dane Court Grammar, agreed with this.