The big cube shown is dipped in blue paint so every outer face is coloured, and then cut along all the lines. How many small cubes carry blue on exactly one face?

One painted face means a cube in the middle part of a face, not on any edge. Each face is a 6 by 6 square, and removing its border leaves $4\times 4=16$ squares. Six faces give $6\times 16=96$. (150 removes only one row and one column from each face: $6\times 5^2.$)