Forgetting the Buried Cubes

The error

You scan the drawing, find every cube with the asked number of painted sides, count them, and pick that number. It is the natural thing to do and it is the signature mistake of this section. The drawing is not the pile. An isometric drawing shows the cubes on the outside of the group and says nothing about the ones packed in behind them, and the question asks about every cube in the pile, drawn or not.

The pile below holds twenty-one cubes. Here is every painted count the drawing is capable of printing:

Twelve numbers. A cube can only carry a number where it shows you a face, and nine of these twenty-one cubes show none at all. Read the pile as column heights instead and nothing goes missing:

Row	Column heights
Farthest from you	3, 3, 2, 1
Middle	3, 2, 2, 1
Nearest you	2, 1, 1, 0

Rows run from the row farthest from you to the row nearest you, and inside a row the columns read left to right exactly as they do in the drawing. Nine plus eight plus four is twenty-one. The height map counts cubes; the picture only counts surfaces. That single difference is what makes the map immune and the picture not.

It only ever undercounts

Every cube you drop was a genuine member of some class, so the number you report comes out too small, never too large. That matters because of the shape of the answer choices. The choices are the fixed ladder from one cube to five cubes, so undercounting slides your answer down the ladder rather than off it. You never get the relief of finding your number missing from the list; the wrong answer is sitting right there looking perfectly reasonable.

Which questions it can corrupt

Not all of them, and the exception is worth memorizing. A cube is invisible only when it is boxed in on all three of the faces the drawing could have shown: its top, and the two sides that turn toward you. That is the test from Finding the Hidden Cubes. Boxed in means cemented to a neighbour, and cemented faces are never painted, so three of a cube's five paintable faces are gone before you start. Only the two faces turned away from you are left.

A buried cube can never have more than two painted sides. The classes of three, four and five are therefore complete on the surface: every member of them shows you a face. Tint every cube in this pile with exactly four painted sides and you see the whole class.

Three cubes, and three is the true answer. Now tint every cube with exactly two painted sides. A tint can only land on a face the drawing draws, so three cubes light up here as well:

This time three is wrong. That class holds five cubes; two of them are buried and cannot be tinted, because they cannot be drawn. The picture gives the same answer to both questions and is right only once.

Read that as a rule for the run. A question asking for three, four or five painted sides is safe from this error. A question asking for one or two is the one that can go wrong, and it can go wrong badly: on this pile the class of one painted side has five members and not one of them is visible. A picture-counter arrives at zero, which is not even on the ladder.

The diagnostic

You do not need to suspect the error to catch it. Add up your six counters, from zero painted sides through five. The sum has to be the pile's total cube count. If your counters sum to twelve and the height map says twenty-one, you did not miscount a class — you never counted nine cubes at all, and those nine are the buried ones. The gap is the error, measured.

The mirror error

The opposite mistake is rarer and just as expensive: inventing buried cubes the height map does not support. Once you know the pile hides cubes, it is tempting to add a few for safety, or to imagine a solid block behind a tall column. Do not. A column of height three contains exactly three cubes and never a fourth. The map is not an estimate you pad; it is the pile, written down.

Try one

Work this from a height map, not from the picture. Count every class as you go, then check yourself in the cube counting practice generator, which prints the pile's total, how many of its cubes are hidden, and the full tally.

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 takeaway

Counting the cubes you can see always undercounts, and it slides your answer down a ladder where every rung looks legal. Buried cubes carry at most two painted sides, so only the questions asking for one or two can be corrupted this way — and those are exactly the questions where the buried cubes may outnumber the visible ones. Count from the height map, and make your six counters sum to the total before you answer.