Problem of the Week 18

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?