Instructions: A cube of 7 cm x 7 cm x 7 cm is kept in the corner of a room and painted in three different colours, one colour on each of the three faces that can be seen. The cube is then cut into 343 smaller, identical unit cubes.
How many of the smaller cubes have exactly one colour on them?
Show Hint
A cube with exactly one painted face has exactly one coordinate equal to 7 and the other two between 1 and 6.
Each of the three painted faces of the big cube is a 7 by 7 grid of unit-cube faces, but some of those unit cubes also touch a second painted face along a shared edge, and those must be excluded from the "exactly one colour" count.
Each painted face has $7 \times 7 = 49$ unit cubes touching it.
Along each painted face, one edge (7 cubes) is shared with another painted face, so those 7 cubes on that edge actually have two colours, not one, and must be removed: $49 - 7 = 42$.
The corner cube, where all three faces meet, was already excluded once as part of that edge, but it also lies on the edge shared with the third face, so one more cube (the corner itself) needs to be removed from this face's count: $42 - 1 = 36$.
Wait, checking this more carefully by direct coordinates is safer than the edge-count method: fixing one coordinate at 7 and requiring the other two strictly between 1 and 6 already avoids both shared edges of that face by construction, so each face directly gives $6 \times 6 = 36$ single-coloured cubes without extra adjustment.
