Verify the Total First

Add the map before you answer anything

You have a height map. Do not start counting painted faces yet. Add the map's numbers first and get one number out of it: how many cubes are in the pile.

Seven, six and six. Nineteen cubes.

It takes about ten seconds and answers no question on the screen, which is exactly why it feels skippable. It is not. That number is the cheapest insurance in the section.

The sum counts what you cannot see

Three of those nineteen cubes are buried. The map counted all three without being asked, because a column of three contributes three cubes whether or not you can see all three of them. This is the payoff for reading columns instead of reading cubes: the total is right the first time, with no separate hunt for buried cubes.

Try to reach nineteen by counting cube shapes in the drawing instead and you cannot get there. The picture has fewer than nineteen cubes to offer.

Two passes, one number

Read the map a second time and add it again. If the two totals agree, the map is almost certainly right and you can spend the rest of the run counting rather than second-guessing. If they disagree, you misread a column — and finding out now costs you ten seconds, while finding out later costs you the entire run.

The columns most often dropped are the short ones. Four columns in this pile hold a single cube each, and a single-cube column can show as little as one small face wedged between taller neighbors. Four cubes of nineteen ride on noticing them.

A total that changes between passes is not a disaster; it is the system working. Re-derive the column you are unsure of rather than starting the whole pile again.

The total is the number everything else has to match

Hold on to it. When you later sort all nineteen cubes by how many painted sides each one has, those groups have to add back up to nineteen — and if they do not, you know something is wrong before you answer instead of after. Building that sorted count is the next unit's work; producing the number it has to match is this topic's.

What totals look like

Sizes are predictable enough to sanity-check yourself. Across two thousand generated piles of each size, small piles ran 6 to 16 cubes and averaged about 10; medium piles ran 10 to 33 and averaged about 20; large piles ran 15 to 48 and averaged about 31. A small pile that you total at 25, or a large one you total at 8, means you have misread the footprint, not the heights.

Check your number against the pile itself

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 pile contains 19 cubes (3 hidden but required as support). Painted-face tally: 3 cubes with 1, 6 cubes with 2, 3 cubes with 3, 5 cubes with 4, 2 cubes with 5. Bottoms and cemented faces are never painted.

That is the same pile, generated as a real question. Its explanation opens with the pile's total and the count of buried cubes — compare both against what you produced. The painted-face tally that follows is the next unit's business; for now, read the first number and move on.

The takeaway

Before you answer anything, add the height map and get the pile's total. The sum includes the buried cubes automatically, which is the whole reason to count from columns rather than from the picture. Add it twice: a disagreement means you misread a column, and catching that before you answer costs seconds instead of a run.