Cemented Joints and Double Counting

A joint costs two faces, not one

Cementing two cubes together destroys two painted faces, one on each cube. The joint is shared and both owners pay for it. Start with a cube that nothing touches:

Five. Now cement a second cube to its side:

Both read four. One joint, two faces gone. The mistake is to deduct once — to spot the contact, take a face off the cube you happen to be looking at, and move on without taking one off its partner. Working column by column removes the need to remember: each cube is scored on its own five faces in its own turn, so the joint is deducted twice because you arrive at it twice, once from each side.

Meeting a cube twice

The opposite bookkeeping failure is scoring the same cube two times. It happens when you hunt for cubes visually — you work along the top of the pile, then down one exposed face, then the other, and a cube standing where those runs meet gets counted in both. A fixed traversal is the whole cure. Take the columns in a settled order, commit each column's contribution before starting the next, and every cube is visited exactly once no matter how many times it catches your eye.

Covered is not the same as unseen

Now the harder half of this topic. The drawing gives each cube at most three faces: its top, and the two that turn toward you. Every other face of every cube in the pile is invisible — and invisible covers two situations that have nothing to do with each other.

• A face pressed against a neighbour is covered. It is a cemented joint and it is unpainted.
• A face turned away from you with nothing behind it is exposed. It is painted, in full, and you simply cannot see it.

A face you cannot see is not the same as a face that is covered. The drawing hides both kinds equally well, which is why judging a cube by how exposed it looks fails in both directions at once.

Here is a twelve-cube pile with every count on show, and the four cubes that have exactly three painted sides tinted:

Two of those four make the point by themselves. The cube at the far corner shows you exactly one face, its top, and looks thoroughly walled in — yet two more of its sides are turned away with nothing behind them, so it has three painted sides. The cube nearest you shows three faces and looks completely out in the open, and it also has three: the two sides it turns away from you are both pressed against neighbours. Same true count, opposite reasons, opposite appearances.

The two errors this produces

Read the pile by eye and you make both mistakes in the same pass. Cubes along the far edge look enclosed, so you score them low; their away-facing sides are open air and painted, and you have undercounted. Cubes toward you look free on every side, so you score them high; their away-facing sides are cemented to the cubes behind them, and you have overcounted. Appearance is not evidence: a cube is judged by its own five neighbouring slots or it is not judged at all.

Scoring each cube by the faces the drawing shows has a signature you can see coming. The most any cube can show is three, so that student's tally holds nothing above three at all. This pile contains a cube with four painted sides and a cube with five, and neither can be reached that way. Ask that student for the cubes with three painted sides and the tint lands here instead:

Two cubes, where the true answer is four: a slide of two rungs down the ladder. And only one of the two belongs in the true set at all — the cube nearest you, which lands in both because the three faces it shows happen to be the only three it has. The other is there by accident.

Checking a cube you do not trust

When a cube looks wrong, do not squint harder at the drawing. Name its neighbours instead: the four cubes that would stand beside it and the one that would sit on top. Count how many of those five slots are empty. That is its painted count, and it does not depend on where you are standing or on which faces happened to be drawn.

Try one

This one asks for a class no picture-reading can reach. Score each cube from its neighbours, then check yourself in the cube counting practice generator.

How many cubes have four of their exposed sides painted?

A

1 cube

B

2 cubes

C

3 cubes

D

4 cubes

E

5 cubes

The takeaway

Every cemented contact removes two painted faces, one from each cube, so deduct on both sides of every joint and let a fixed column order stop you meeting a cube twice. The drawing shows a cube three faces at most, and the faces it withholds are a mixture of covered ones and exposed ones — far-edge cubes have painted sides you cannot see, near cubes have covered sides you cannot see, and neither is readable from appearance. Score every cube by naming its neighbours.