Finding the Hidden Cubes

Hidden means blocked in three directions

Three directions face you in this drawing: up, toward you on the right, and toward you on the left. They are the only three directions the drawing ever uses. So a cube is hidden when all three are occupied — a cube above it, a cube in front of it on the right, and a cube in front of it on the left.

A cube in that position draws nothing at all. Not a sliver, not a partial view: three faces were available and a neighbor blocks every one. Being hidden and drawing nothing are the same condition, not two related ones — the three directions that have to be blocked are precisely the three the drawing would otherwise have used.

A buried cube is always holding something up

Piles are solid columns standing on one flat surface, so nothing floats and nothing overhangs. A cube you cannot see is under a cube you can, or wedged behind two of them, and either way it is load-bearing. The instructions say it outright: the only hidden cubes are those required to support other cubes. You never invent a buried cube to fill a gap. You read it off the height map, which counted it without being asked.

One caution carried over from the previous topic: buried is a claim about a cube's neighbors, not about your line of sight. A column can also disappear behind a taller one that lines up with it along the viewing diagonal, and that is a different problem with a different cause.

Two painted sides, and never more

Line up the six faces of a buried cube and ask which of them the paint could have reached.

Face	Can it be painted?
Top	No — a cube sits on it. That is the first condition of being hidden.
Side toward you, right	No — the second condition.
Side toward you, left	No — the third condition.
Side facing away, right	Possibly.
Side facing away, left	Possibly.
Underside	Never — it rests on the table or on the cube below.

The three conditions that make a cube hidden are the three faces the drawing would have shown, and each of them is covered by a cemented neighbor, so none of them carries paint. The underside never counts for any cube. Two faces are left. A buried cube has at most two painted sides — a ceiling, not a tendency.

Turn that around and it becomes the most useful fact in the section: no cube with three, four or five painted sides is ever buried. Reaching three painted sides takes three free directions, and a buried cube has none of the three free.

Count the badges above: one 1, four 2s, five 4s. Ten badges for a pile of twelve cubes. The two missing badges belong to the buried pair, and the ceiling says neither of them could have carried more than a 2.

Which questions the buried cubes touch

Question asks for	Cubes in the pile	Badges you can see
One painted side	2	1
Two painted sides	5	4
Four painted sides	5	5

Asked for four painted sides, you can answer from the picture and be right, and that holds for three and five as well. Asked for one or two, the picture is short by exactly the buried cubes: you would answer one cube instead of two, and four instead of five, landing one rung down a five-rung ladder every time. That gap is the whole reason this topic exists.

How many to expect

Small piles often have none — across two thousand of them, about a third contained no buried cube at all and the average was around one. Every medium and large pile is built to contain at least one, so on those the trap is always armed: medium piles averaged under three buried cubes, large ones about five, with the biggest running to fifteen. Expect to account for a few, not for a crowd, and expect never to be able to skip the accounting on a large pile.

The same pile as a real question

How many cubes have one of their exposed sides painted?

A

1 cube

B ✓

2 cubes

C

3 cubes

D

4 cubes

E

5 cubes

The pile contains 12 cubes (2 hidden but required as support). Painted-face tally: 2 cubes with 1, 5 cubes with 2, 5 cubes with 4. Bottoms and cemented faces are never painted.

The question asks for one painted side, and the answer is two cubes — one of which you cannot see. Half of that answer is invisible, and no amount of staring at the drawing produces it.

The takeaway

A cube is hidden exactly when something sits above it and something blocks both sides that turn toward you, which is precisely why it draws no face at all. Those three blocked directions cap it at two painted sides, so questions about three, four or five painted sides never involve a cube you cannot see, and questions about one or two always might. Find the buried cubes from the height map, before the counting starts.