A Cube's Painted Count Is Its Exposed Faces

One cube, five faces

Stand a single cube on the table and paint everything the brush can reach. Five faces take paint: the top and the four sides. The sixth is underneath, pressed against the table, and the brush never gets there.

A cube's painted count is the number of its faces that meet open air, counting only the top and the four sides. That sentence is the whole of cube counting. Everything else in this unit is bookkeeping built on top of it.

The underside is never in play

The section's instructions say each group was painted on all sides except for the bottom on which it is resting, and that faces where two cubes are cemented together are not painted. For a cube's underside those two clauses reach the same verdict by different routes. A cube on the table has its underside against the table. A cube higher up has its underside cemented to the cube below it. Neither takes paint.

That holds here because of how these piles are built: every column stands on the table and is filled solid from the table upward, so nothing overhangs and no cube floats. Read the rule as a fact about these piles rather than a law about undersides in general — a cube hanging out over open air would be a different problem, and you will not meet one.

Each neighbour costs exactly one face

Cement a second cube against one side of the first and that side stops being a surface. It is now an interior seam between two cubes, and the instructions rule it unpainted. The face is gone from the count.

Each neighbouring cube removes exactly one painted face — no more, no less. A cube presents exactly one face toward each of the five directions in play, so one neighbour cancels one face. Start from five and subtract one per neighbour.

Every cube in that pile wears its own painted count. Read a few of them against their surroundings and the arithmetic is visible: the cube crowned by nothing and hemmed in on one side reads 4, the cube with something above it and neighbours on two sides reads 2. Ten badges for eleven cubes: one cube of this pile is covered from every direction the drawing shows, so it draws nothing at all. It still has a painted count, and finding such cubes was the previous lesson's business.

Zero through five, and never six

Only five directions are ever in question, so a cube's painted count runs from zero to five. Six is impossible for every cube in every pile, because the underside was out of the running before the counting started. If you ever write down a six, you have counted the bottom.

Five is the other end: it means all five directions are open. Nothing above, nothing on any side.

Add neighbours and the count falls one face at a time. Both cubes marked below carry a cube overhead and a neighbour on two sides, which is three faces gone and two left.

Zero is the floor, and it is reachable. A cube with all four sides packed and another cube sitting on top has no face left that meets air.

Six cubes in that pile, five numbers on it. The missing one is the cube at the middle of the cross: four neighbours packed around it, one cube on top, its underside on the table. It carries zero painted faces, and no drawing of the pile can ever show you that, because a cube with nothing exposed has nothing to draw.

Judge each cube by its own neighbours

The temptation is to grade cubes by how the pile looks. An airy, sprawling pile feels like it should be full of fives; a squat, packed one feels like it should be full of ones. Neither impression is worth anything.

A cube's count depends only on its own five directions. A sprawling pile can hide a cube with one painted face at the bottom of its tallest stack, and a compact pile can carry a five on top. The pile's overall shape tells you nothing about any particular cube; the five slots around that cube tell you everything.

That is also why the count is worth trusting. You never have to see a cube to know its number. You only have to know what surrounds it.

The takeaway

A cube's painted count is the number of its faces touching open air among the top and the four sides. Its underside never counts, because in these piles it is either on the table or cemented to the cube below. Start at five and subtract one for each neighbour, which puts every cube somewhere between zero and five and makes six impossible. Decide each cube by its own five neighbours, never by how exposed the pile looks.