Self-Checks That Catch Errors

A tally you cannot check is a guess with extra steps. Four checks sit on top of the counting, they cost a few seconds between them, and each one catches a different kind of wrong. Run them in this order and stop at the first failure.

Check one: the books balance

Add the height map. Add your six counters. The two totals have to agree.

The columns here stand 3, 2, 2, 2, 2, 1, 1, 2 and 1 cubes tall, so the pile holds sixteen cubes. Now count the numbers printed on the drawing: there are eleven. The five that are missing belong to cubes that show no face at all, and a tally that stops at eleven is not slightly short — it is missing whichever classes those five cubes happen to fall into.

You already know why it works: every cube lands in exactly one counter, so the counters have to add back to the pile. What is worth knowing here is where it sits among the four. It is the only check that touches every cube, which makes it the one to keep when the clock is against you — and the one to run unprompted, before anything has gone visibly wrong.

Check two: the answer is on the scale

The choices never change — 1 cube through 5 cubes — and the items only ask about classes whose answer fits that scale. So the counter for the class you were asked about has to land between one and five. Only that counter. The other five are free to run wherever the pile takes them: the 2-class on the pile above holds six cubes, as the tally at the foot of this page confirms — perfectly ordinary, and for that very reason never the class you are asked about.

When the counter you are about to report does fall off the scale, its direction tells you where to look. A zero means cubes are missing from that class. A six or more means the class has collected cubes that are not its own. Either way the fault is in the ledger, not in the figure.

Check three: every column has a top

A column's top cube always scores at least one: nothing stands on it, so its top face is painted. Nine columns in the pile above, nine painted tops. Read backwards, that becomes a check on your route rather than on your arithmetic — if a column put nothing at all into the ledger, you walked past it. Spend the glance on the sparse edges of a footprint, where a lone one-cube column is easiest to skip.

Check four: nothing scores six, and every zero is buried

Six is impossible, and a six in your ledger is a bottom you painted. That half you have been told already. The floor is the half worth a second look.

Zero is real, but expensive to earn: covered on top and walled in on all four sides. A cube in that position cannot be showing you anything, because the drawing draws a face only where a neighbor is missing — the same condition that paints it. So every cube you can see scores at least 1, and every zero belongs to a cube you cannot see. Give a zero to a cube that is plainly in view and the zero is wrong before you check anything else.

Peel the pile down to its bottom level, the way the previous topic does, and two of the five cubes the drawing never showed are sitting there shaded. The generator's own tally, printed further down, names them: one scores 1, and the other is this pile's only zero.

What a failed check actually costs

A failed check says the ledger is wrong. It does not say start again. Rebuild the single column you trust least and reconcile it against what you committed — a column's numbers depend on nothing but itself and the columns touching it, so nothing else in the ledger has to move.

And when the check that failed is the one on the asked class, you do not need the tally back at all. You need one number: how many cubes carry exactly that score. Sweep for that class alone and let every other cube go by unjudged.

Asked for the cubes with three painted sides, that is the entire job — five cubes, and on this pile all five are in plain view. A class you were not asked about can stay wrong all day without costing you the point.

How many cubes have three of their exposed sides painted?

A

1 cube

B

2 cubes

C

3 cubes

D

4 cubes

E ✓

5 cubes

The pile contains 16 cubes (5 hidden but required as support). Painted-face tally: 1 cube with 0, 1 cube with 1, 6 cubes with 2, 5 cubes with 3, 2 cubes with 4, 1 cube with 5. Bottoms and cemented faces are never painted.

The explanation prints the tally the generator built: one cube with 0, one with 1, six with 2, five with 3, two with 4, one with 5 — sixteen cubes. That is check one done for you. Hold your own counters against it after every item in the cube counting practice generator, and you will learn fast which of the four checks you personally need.

The takeaway

Four checks: the counters add to the height map's total; the answer for the asked class lands between one and five; every column contributed a painted top; and nothing scored six while every zero sits out of sight. When one fails, rebuild a single column or re-sweep the single class you were asked about — never the whole pile. If there is time for only one check, run the sum, because it is the only one that touches every cube.