Worked Run Walkthroughs

One large pile, three questions, one analysis. Nothing new is introduced here — the sweep and the self-checks are the ones you already have. What this page shows is the order they go in, and how far a single tally carries you.

The run arrives

How many cubes have one of their exposed sides painted?

1 cube

2 cubes

3 cubes

4 cubes

5 cubes

Question one of a three-question run. The same pile serves all three.

This is question one of three. The same drawing serves all three, and the other two are not visible yet — which is the trap in disguise. If you count only what this question asks, you will pay for the pile again in a moment.

Step 1 — read the heights, get the total

Read the height of every column off the drawing and lay the footprint out as a grid. It does not matter which way you turn that grid, and it does not matter which row you call the first one: a cube's painted count depends only on which of its neighbors are present, so any labeling that keeps neighbors next to each other gives the same answer. This pile's footprint is four columns by four, and two of the columns stand four cubes high.

	C1	C2	C3	C4	Row
R1	2	1	3	4	10
R2	2	1	1	—	4
R3	—	2	2	1	5
R4	—	1	3	4	8

Thirteen columns, 27 cubes. Hold that number; it is the only thing standing between you and an undetected miscount.

Step 2 — sweep the columns, fill the tally

Take the columns in a fixed order and score each cube against its neighbors. Here is the whole sweep, one row per column, listing that column's cubes from the bottom up:

Column	Height	Painted faces, bottom to top
R1C1	2	2, 4
R1C2	1	2
R1C3	3	1, 3, 4
R1C4	4	3, 3, 3, 5
R2C1	2	2, 4
R2C2	1	1
R2C3	1	2
R3C2	2	1, 4
R3C3	2	0, 3
R3C4	1	3
R4C2	1	3
R4C3	3	1, 2, 4
R4C4	4	2, 3, 3, 5

The figure below badges every cube you can actually see with its count, so you can check the sweep against the drawing:

The same pile with every visible cube badged; twenty-three badges for twenty-seven cubes.

Twenty-three badges for twenty-seven cubes. The four without badges are the ones hidden behind their neighbors, and in this pile all four rest on the table underneath something. They are the bottom cubes of R1C1, R1C3, R3C2 and R3C3, and they contribute 2, 1, 1 and 0 painted faces. Nothing on the drawing shows them to you. You get them from the height map or you do not get them at all.

Step 3 — check the sum before answering anything

Painted faces	Cubes
0	1
1	4
2	6
3	9
4	5
5	2
Total	27

1 + 4 + 6 + 9 + 5 + 2 = 27, and the height map said 27. The tally is closed. Now — and only now — read the questions.

Step 4 — answer all three off the table

Question one asked for one painted face. The tally says 4 cubes.

How many cubes have one of their exposed sides painted?

1 cube

2 cubes

3 cubes

D ✓

4 cubes

the tally's one-face row

5 cubes

Question one answered from the tally row for one painted face.

Question two asks for four. The tally says 5 cubes. You do not look at the pile again.

How many cubes have four of their exposed sides painted?

1 cube

2 cubes

3 cubes

4 cubes

E ✓

5 cubes

Question two, answered without looking at the pile again.

Question three asks for five. The tally says 2 cubes.

How many cubes have five of their exposed sides painted?

1 cube

B ✓

2 cubes

3 cubes

4 cubes

5 cubes

The pile contains 27 cubes (4 hidden but required as support). Painted-face tally: 1 cube with 0, 4 cubes with 1, 6 cubes with 2, 9 cubes with 3, 5 cubes with 4, 2 cubes with 5. Bottoms and cemented faces are never painted.

Question three, and the item's own explanation printing the full tally.

Three answers, one analysis. The second and third questions cost you a glance at a table you had already built.

Notice what was never asked. The tally's two biggest rows — nine cubes at three painted faces, six at two — did not appear in any question, and neither did the single fully buried cube. The choices only run from one cube to five, so a row holding six or more is unanswerable, and zero is not on the ladder either. On large piles that removes the crowded middle, which is why this run asked for one, four and five.

Where a wrong answer would have come from

The five choices in this section are just counts. There is no clever wrong shape to fall for — a wrong answer here means a wrong number, and the number came from one of the steps above. Two slips account for most of them, and they leave different fingerprints.

Never inferring the hidden cubes. Drop the four cubes you cannot see and the tally becomes 2 at one face, 5 at two, 9 at three, 5 at four, 2 at five — 23 cubes. Question one now reads 2 and you answer B instead of D. Questions two and three are untouched, because hidden cubes can never carry more than two painted faces. The fingerprint is the sum: 23 against a height map that says 27.

Painting the underside that rests on the table. Add one face to each of the thirteen ground-level cubes and the tally becomes 1 at one face, 5 at two, 11 at three, 8 at four, 2 at five. The total is still 27, so the sum check passes and tells you nothing. But question two now reads 8 cubes — and there is no choice for 8. An answer that will not fit on the ladder is itself the error message. When that happens, re-derive the columns you touched rather than the whole pile.

Now predict one yourself

Same routine, one third the size. Work this run before you open the answer: read the heights, get the total, sweep, close the tally, then read off all three answers.

How many cubes have one of their exposed sides painted?

1 cube

2 cubes

3 cubes

4 cubes

5 cubes

A small pile, and the first question of its run.

The tally for that pile

Ten cubes, none hidden — every cube in this pile shows itself.

Ten cubes, ten badges: nothing in this pile has to be inferred.

Painted faces	Cubes
1	1
2	3
3	3
4	2
5	1
Total	10

1 + 3 + 3 + 2 + 1 = 10, which matches the height map. The run asks for one painted face (1 cube, choice A), then two (3 cubes, choice C), then three (3 cubes, choice C again — repeats are ordinary, and a repeated answer is not evidence you erred).

How many cubes have two of their exposed sides painted?

1 cube

2 cubes

C ✓

3 cubes

4 cubes

5 cubes

The pile contains 10 cubes (none hidden). Painted-face tally: 1 cube with 1, 3 cubes with 2, 3 cubes with 3, 2 cubes with 4, 1 cube with 5. Bottoms and cemented faces are never painted.

The run's second question, marked, with the generator's own tally printed underneath.

Every class in this small pile holds between one and five cubes, so all five were legal to ask about — and the questions landed in the middle of the ladder, where a small pile keeps most of its cubes.

Say the sequence back

Both runs are the same four moves in the same order: heights and total, sweep, close the tally against the total, read every question off the table. The order is what does the work, because reading a question first tempts you into scoring one class and moving on.

So drill it out loud. After a run — the ones you got right too — say the total, say the tally, and say which row each question took. Where you hesitate is the step that is not automatic yet. And when you miss, name the step rather than the letter: a total that disagreed with your height map is a different disease from an answer that would not fit on the ladder.

The takeaway

Count the pile once, completely, before you read what is being asked. Close the tally against the cube total, then answer every question in the run off that table without looking at the drawing again. When a count comes out wrong, the tally itself names the step: a total that misses the height map means you skipped cubes you cannot see, and an answer with no choice to hold it means you painted faces that were never exposed.

Worked Run Walkthroughs

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