A figure composed of congruent cubic blocks has four layers. On the lowest layer are 7 rows of 7 blocks each. Centered on the bottom layer are 5 rows of 5 blocks each. Centered on the top of that are 3 rows of 3 blocks each. Finally, a central block is placed on top of the entire structure. Then, the figure is painted, except for the bottom. How many blocks have 6 painted faces? 5? 4? 3? 2? 1? 0?
Suppose a similar figure has 99 rows of 99 blocks each on the bottom layer. How many blocks have 5 painted faces? 4? 3? 2?
Alternate Problem 18
A rectangular block measuring 10 units by 8 units by 6 units is made up of cubes measuring 1 unit on a side. The base of the block is 10 units by 8 units. The outside of the block other than the base is painted red. How many of the unit cubes have exactly 1 face painted red?
Extension Problem 18
Three sets of parallel planes are mutually perpendicular in pairs. There are 5 planes in the first set, 6 planes in the second set, and 7 planes in the third set. The planes separate three-dimensional space into several regions, some of which are bounded on all sides and some of which are not. How many unbounded regions are there?