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 three faces meeting at the corner facing into the room). The cube is then cut into 343 smaller, identical unit cubes.
How many of the smaller cubes do not have any face painted?
Show Hint
A cube has zero painted faces only if none of its three coordinates lie on the outer painted layer; that gives a 6x6x6 block.
Instead of counting the unpainted cubes directly, this method counts the painted ones first and subtracts from the total, which is a useful cross-check.
The total number of unit cubes is $7^3 = 343$.
Consider the three painted outer layers, each of size $7 \times 7 = 49$ unit cubes, one for each painted face: $3 \times 49 = 147$.
This counts the cubes lying on two painted faces at once (the three edges where two painted faces meet) twice, so those must be subtracted once. Each such edge has 7 cubes, and there are 3 such edges: $3 \times 7 = 21$, so subtract 21.
The single cube at the very corner, where all three painted faces meet, was added three times and subtracted three times in the previous steps, so it needs to be added back once.
Total painted cubes $= 147 - 21 + 1 = 127$.
Unpainted cubes $= 343 - 127 = 216$.
This matches the direct count, confirming the answer is option D, 216.
