Painting the Bottom

The error

A cube sits on the table at the corner of a pile, standing proud with nothing crowding it. It looks like it is out in the open on every side, so you give it six. Or you think about the pile as a whole, picture the painter working around it, and count its underside as one more outside surface the brush would have reached. Both readings feel reasonable. Both are wrong, and they are the same mistake.

The rule that kills it

The instructions are explicit about the one surface the paint never touches: the group was painted on all sides except for the bottom on which it is resting. The floor the pile stands on gets nothing. And a cube higher up is no better off, because its underside is glued to the top of the cube below it, and cemented faces are not painted either.

These piles are built as full columns standing on one flat surface, so a cube's underside is always one of exactly those two things: on the table, or cemented to the cube beneath it. No cube's underside is ever painted. Five faces are in play for every cube in the pile — the top and the four sides — and never a sixth.

Five is the ceiling

That gives you a free error detector that costs nothing to run. The largest number any cube can carry is five, so any cube you scored six is a receipt for this error. You do not have to find the face you double-counted; the six already told you it was the bottom one.

Ten cubes, ten numbers, and not a six among them. The largest is the five on the cube at the top of the tower: nothing stands beside it at that level, nothing sits on it, and its underside is cemented to the cube below. A lone cube on top of a stack is the only thing that reaches five, and five is as high as any cube in any pile goes.

The corner cube on the table

Now look at the four cubes at the corners of the ground row. Each one carries a four. They are the cubes the error is aimed at — each has three sides in open air and a top nobody is standing on, which is exactly the look of a cube that ought to be worth more. A fifth four sits higher up, on the top cube of the two-high column in the nearest row; that one rests on a cemented joint rather than on the table, and the difference is about to matter.

Four is all they get, and not by accident. A pile is cemented into one piece, so a cube resting on the table has to be touching at least one other cube, and that contact costs it a face. Its underside costs it nothing further, because the underside was never in play. On the table the ceiling is four; five belongs only to a cube perched on top of something.

What the error costs

Suppose the question asks how many cubes have five painted sides. Here is the answer:

One cube. That is choice A. Now here is the set you would hand in if the faces resting on the table counted as painted:

Five cubes — the four corner cubes promoted from four, plus the one that genuinely has five. That is choice E. One mistaken face per ground-row cube moved the answer from one end of the ladder to the other.

The error comes in two strengths. Paint only the pile's underside, as above, and just the ground row inflates; nothing exceeds five, so nothing looks wrong. Paint every underside, treating each cemented joint as an outside surface, and every cube in the pile gains one: the whole tally shifts a rung, and the cube at the top of the tower reads six. The second version announces itself. The first one does not, which is why the rule has to be automatic rather than checked.

Try one

This one asks for the top of the ladder, where the error bites hardest. Score each cube on five faces only, then check yourself in the cube counting practice generator.

How many cubes have five of their exposed sides painted?

A

1 cube

B

2 cubes

C

3 cubes

D

4 cubes

E

5 cubes

The takeaway

A cube's underside is either standing on the table or cemented to the cube below, and neither one takes paint. Five faces are in play for every cube, so five is the ceiling and a six is proof you painted a bottom. On the table the ceiling drops to four, because a cube in a one-piece pile always touches a neighbour — the corner cube that looks free is worth four, not five and never six.