## Task 2 - 320622

A wooden cube, whose six side surfaces have been painted red, is subsequently sawn into a number of equally sized smaller cubes.

a) How large is this number if it is known that exactly 72 of the resulting smaller cubes have exactly two red side faces?

b) How many of the remaining resulting smaller cubes have exactly one red side face,

c) How many have no red side face?
a) Exactly two red side faces are located on those sub-cubes that, before being cut, were adjacent to an edge of the original cube but did not contain any of its corners.

Since the original cube had exactly 12 edges, there were thus 72 / 12 = 6 such sub-cubes along each of its edges. Extending from both sides of a row of such sub-cubes, there was exactly one more cube on each side that contained a corner of the original cube. Including these two cubes, such a row thus consisted of exactly 8 sub-cubes. Therefore, the original cube was cut into 8^3 = 512 sub-cubes.

b) Exactly one red side face is located on those sub-cubes that were adjacent to a side face but not to an edge of the original cube. For each of the 6 side faces of the original cube, these sub-cubes formed a square arrangement of 6 x 6 = 36 pieces; hence, there are a total of 6 x 36 = 216 such sub-cubes.

c) The sub-cubes without red side faces formed a cubic arrangement that lay entirely inside the original cube and consisted of exactly 6^3 = 216 sub-cubes.