From Drawing to Height Map

Three faces, and often fewer

A cube has six faces. The drawing gives you three of them: the top, and the two sides that turn toward you. The underside and the two sides that face away are never drawn, in any pile, ever.

Even those three appear only when the space beside them is empty. Set a second cube on top and the lower cube's top face disappears from the picture — the two are cemented there, and no line marks the joint. So most cubes in a pile show one face or two, and a cube in the middle of a crowd can show none at all.

Accept that early: the picture is not an inventory of cubes. No amount of counting faces, or counting the cube shapes you can pick out, will hand you the pile.

A pile is a set of columns

Every pile in this section is built one way. Cubes stack into columns, every column stands on the same flat surface, and every column is solid from that surface up to its top. Nothing floats and nothing hangs over an edge.

That single fact does all the work. A column that is solid and starts at the surface is described completely by one number — its height. A column of three is three cubes, bottom to top, with no gap to wonder about. So the whole pile is a small grid of numbers, one per column. That grid is the height map.

The map is the pile's footprint seen from above, turned to the same orientation as the drawing, with each column's height written into its cell. A dashed cell is empty: no column stands there.

Reading the heights

Every column ends in a cube with nothing above it, so the drawing always draws that cube's top face. The upper surface of the pile is a staircase of top faces — exactly one per column, never more.

Start at the column nearest you, the one at the bottom of the drawing. Here it is one cube tall. Then step backward, and notice that backward splits into two directions: up and to the left, or up and to the right. Up-left from that near column stands a column of two; up-right stands another of one. Each step lands on the next top face — level with the one you left, or one cube above or below it — and each step fills one more cell of the map.

Seven columns, seven top faces, seven numbers. From here on you count from the map, not from the picture. Every question the rest of this unit asks — how many cubes altogether, how many with four painted sides, how many buried — is a question about those numbers. The drawing's job ends the moment the map is written.

When the picture holds back

An isometric view looks along one exact diagonal, and two columns can line up along it. A column that sits one step farther away, one step to the side and one cube lower than its neighbor is covered by that neighbor completely: its top face is drawn, and then painted over.

Six cubes stand in that pile. The far corner column is one of them, and not a pixel of it reaches your eye.

The test's own instructions are what keep real figures out of this trap: the only hidden cubes are those required to support other cubes. Read literally, that forbids exactly this arrangement — a cube you cannot see that is holding nothing up. Generated practice piles do not always honor it, so if a pile refuses to add up while you drill, look along the diagonals before you doubt your arithmetic.

Read one yourself

Here is a pile straight from the practice generator. Walk it from the near corner and write its map before you look at the answer.

Eight columns: a far row of 3, 3, 3 and 2, a middle row holding 1 and 2, and a near row holding 1 and 1. Whether those numbers add to a sensible total is the next topic's business.

The takeaway

The drawing shows each cube's top face and the two sides that turn toward you, and nothing else. Because every pile is solid columns standing on one flat surface, one number per column describes it exactly, and every column's top face is drawn for you to find. Convert the picture into that grid of numbers first. Everything after this counts numbers, not cubes in a drawing.