Per-Column Bookkeeping Without Scratch Paper

The test hands you a pile and takes away your pencil. Everything you work out has to live in your head — and it has to still be there for the next question about the same figure. That is a bookkeeping problem, not a geometry problem, and bookkeeping is solved the way it always is: a fixed order, and a number committed the moment you have it.

What you are carrying

Two things, and nothing else: the height map — one number for each occupied column — and the six counters of the tally table, zero through five. Neither is new. What is new is that both have to live in your head, unwritten, while you work and while you answer.

The height map is the part people assume they cannot hold. They can. The piles you will drill here stand on footprints of nine, twelve or sixteen cells, most of them low or empty, and the map pays for itself the moment you have it: add the heights and you have the pile's total cube count, buried cubes included, before you have scored a thing.

One order, every time

A fixed traversal is already the cure for meeting the same cube twice. What matters under pressure is the part that sounds like nothing: it does not matter which route you fix on, only that it never changes. Start at the same corner of the footprint, finish a row before you begin the next, let empty cells go by without stopping — and run that same route on every practice pile until it costs no attention at all. Attention is what you need for the counting.

The other unit of work: one level at a time

Columns are not the only way to slice a pile. You can also take it one horizontal level at a time, and on a tall pile that is often the easier ledger, because a level is decided by very little: which columns reach it, and which of those keep going up.

The demo below does that to a real pile, one level per step, so you can see what your bookkeeping would otherwise reconstruct blind. One warning first, and it is the whole price of counting this way: lifting a level uncovers tops that are cemented in the intact pile. A cube shows its top face exactly when nothing stands on it — so on a freshly uncovered surface a number reads one too high unless that cube was already showing its top, which is to say unless its column ends there.

Peel the pile, one level at a time Step 1 of 4

Sixteen cubes. Six counters. Numbers on cubes are a teaching aid, not something the test prints.

1. What the test gives you

A pile, drawn once. Read the height map off it first: nine occupied columns, standing 3, 3, 1, 2, 2, 1, 1, 1 and 2 cubes tall. That adds to sixteen, so sixteen cubes have to end up in your counters — however many of them you can actually see.

The pile as the section draws it: no numbers, no cutaway, no second view.

0s 0 1s 0 2s 0 3s 0 4s 0 5s 0 committed 0 of 16

2. Score the top level

Every cube that shows a face now carries its own score — twelve numbers for sixteen cubes. Four cubes show no face at all, which is why a surface count is not a tally.

Start at the top. Two cubes stand on the highest level, both reading 4. Nothing sits on either of them, so both numbers are final. Commit them and forget them.

The same pile with each visible cube's painted-face count printed on it — twelve numbers for sixteen cubes.

0s 0 1s 0 2s 0 3s 0 4s 2 5s 0 committed 2 of 16

3. Lift the top level

Fourteen cubes left, and the middle level is the surface. Five cubes sit on it: four 3s and a 5. Two of the five were carrying a cube a moment ago — the warning in action — so that top face is cemented, not painted, and those two score 2 apiece.

The other three keep what they show, 3, 3 and 5: each was already showing its top in the previous step, so each column ends here. And three cubes are still out of sight — eleven numbers for fourteen cubes. One lift was not enough.

Top level lifted away. Five cubes are on the fresh surface; two of them were carrying a cube.

0s 0 1s 0 2s 2 3s 2 4s 2 5s 1 committed 7 of 16

4. Lift the middle level

The bottom level, one cube per column: five 2s, a 3 and three 4s. Four of these columns are a single cube tall — two of the 4s and two of the 2s, the four that were showing their tops back in step 2 — so those four are final.

The other five carry a cube, so each drops by one: the 3 becomes a 2, three 2s become 1s, and the last 4 becomes a 3. Look at what that just produced — three cubes scoring 1, and only one of the three ever appeared in the drawing at all.

Middle level lifted away, leaving one cube per column: five 2s, a 3 and three 4s.

0s 0 1s 3 2s 5 3s 3 4s 4 5s 1 committed 16 of 16

All four steps are printed below one another if this stepper does not respond.

Sixteen committed out of sixteen. Whether a finished tally is right is the next topic's business; what the ledger guarantees here is only that nothing was left on the table.

Twins are one piece of work

The rule that scores a cube reads its neighbors and nothing else — never where in the pile it sits. So two cubes in matching situations carry the same number, and two columns in matching situations contribute the same list of numbers.

Four equal columns in two situations: the inner pair contributes 2 and 3 each, the outer pair 3 and 4 each.

Four columns, all two cubes tall. The two inner ones have a neighbor on each side and contribute a 2 and a 3 apiece; the two outer ones have one open flank and contribute a 3 and a 4 apiece. Spot the repeat and it is two pieces of work doubled, not four. Real piles rarely repeat this cleanly, but they repeat — and the discount is worth taking deliberately, because a wrong twin puts two bad columns in the ledger instead of one.

Commit, then move

Never carry a partial result into the next column. "That one had a two and a three" is the thought the next column overwrites, and you will not notice it being overwritten. Add the numbers to the counters, then let the column go entirely. Counters that are always current beat a perfect memory of where you are, because counters survive an interruption and your place in the pile does not.

When you lose your place

You will lose it. A number slips, the eye jumps, and you cannot tell whether you already did that stack. The recovery is small — redo one column, not the pile — and what makes it safe is that a column's numbers depend on nothing but that column and the columns touching it. Nothing on the far side of the pile can change them. Rebuild the one you doubt, reconcile it against what you committed, and correct only the difference.

One tally, the whole run

The payoff is that the ledger outlives the question. This section asks more than one question about the same figure, and a committed tally answers all of them without a second look at the pile. Here is the run this pile actually feeds.

How many cubes have one of their exposed sides painted?

1 cube

2 cubes

C ✓

3 cubes

4 cubes

5 cubes

The pile's first question, answered from the finished tally.

Three cubes with a single painted side — and only one of the three ever appears in the drawing. Score the surface alone and you answer 1 cube: a number the scale offers, and one this pile does not support.

How many cubes have two of their exposed sides painted?

1 cube

2 cubes

3 cubes

4 cubes

E ✓

5 cubes

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

The next question about the same figure — same tally, no second look.

Five, straight off the counters, with no fresh look at the figure. The generator prints its own tally underneath; hold it against the one the demo built. Then go build your own on new piles in the cube counting practice generator.

The takeaway

You are carrying a height map and six counters, zero through five, and nothing else. Fix one route through the footprint and never vary it, or take the pile a level at a time — either way the discipline is the same: commit each unit's numbers to the counters before you move, so that losing your place costs you one column instead of the pile. Add the heights first; that total is what your counters owe.

Per-Column Bookkeeping Without Scratch Paper

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