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The Counting Method
The Tally Table
Summary
Six counters, one pass
The sweep hands you a number for every cube in the pile. Those numbers arrive in column order, which is the wrong order for answering anything, so they need somewhere to go.
Keep six counters, labeled zero through five, and add one mark to a counter as each cube is finished. Nothing is written down twice and nothing is held in your head. When the last column is done the six counters describe the pile completely.
The counters are the whole product. You are not counting toward an answer while you sweep; you are building a table, and the answers come out of it afterward.
The table
Here is the seventeen-cube pile from the end of the last topic, badged with each visible cube's painted count.
Sweep its ten columns, dropping each cube into a counter as you go, and the table comes out like this.
| Painted sides | Cubes |
|---|---|
| 0 | 1 |
| 1 | 3 |
| 2 | 4 |
| 3 | 3 |
| 4 | 6 |
| 5 | 0 |
| Total | 17 |
Two entries are worth a glance. The five-counter finished empty: no cube in this pile has all five faces open, which is a fact about the pile and not a mistake. And the four-counter finished at six, past the top of the answer scale — counters are free to run wherever the pile takes them, and which of them a question names is a separate matter taken up later in this unit.
The total is your checksum
Before any of this you added the height map's numbers and got the pile's total. That total now has a job.
The six counters must sum to the pile's cube count. Here 1 + 3 + 4 + 3 + 6 + 0 comes to 17, and 17 is what the height map promised. Every cube in the pile entered exactly one counter, so any other sum means the bookkeeping broke somewhere.
A sum that comes up short means you dropped a cube — most often the buried ones, whose columns are easy to leave half-swept. A sum that runs over means you counted one twice. Either way you learn it in five seconds, before the answers are committed, rather than after three questions have been built on a bad table.
Reading the answer off a counter
Once the table balances, answering is lookup. The question names a number of painted sides; you report the counter with that label.
Asked for the cubes with three painted sides, you read the three-counter and answer 3. The tinted cubes are those three. The table needed no help finding them, and it had already logged something the drawing cannot show you at all: the cube in this pile with no painted face.
One table, the whole group
This is where the work pays off. A cube counting figure does not serve one question; it serves several, and they name different counts.
How many cubes have one of their exposed sides painted?
The pile contains 9 cubes (1 hidden but required as support). Painted-face tally: 2 cubes with 1, 1 cube with 2, 3 cubes with 3, 1 cube with 4, 2 cubes with 5. Bottoms and cemented faces are never painted.
Its explanation reports a nine-cube pile with one hidden cube, and the tally 2 cubes with 1, 1 cube with 2, 3 cubes with 3, 1 cube with 4, 2 cubes with 5. Those six numbers are the whole table for this figure, and they sum to 9. Now watch what the rest of the group costs.
How many cubes have two of their exposed sides painted?
How many cubes have three of their exposed sides painted?
Three questions, one pile, three different counters read. The second and third cost nothing but a glance at the table, because the pile was never the question — it was the input, and it was consumed once.
The alternative is what makes people slow. Counting only the number the first question names means starting over when the second arrives, on a pile you have already stopped concentrating on, under a clock that has been running the whole time. Sweep once, tally everything, answer from the table.
The counter you never report
The zero counter is the odd one. Cubes with no painted face are real and this pile has one, but the questions in this section name counts from one to five, so the zero counter is never the thing you report.
Keep it anyway. It is part of the checksum, and dropping it is a silent way to break the check: the sum comes up one short, you go hunting for a cube you never lost, and the table you were about to trust starts to look wrong. Log the zeros, sum all six, report from the other five.
Build tables on fresh figures in the cube counting practice generator, where each figure comes with a run of questions to spend it on.
The takeaway
Run six counters, zero through five, and mark one per cube as the sweep finishes it. Check the six against the pile total before you answer anything, because a table that does not balance is a table that will produce wrong answers confidently. Then read each question's answer straight off the counter it names. The zero counter never gets reported but always gets counted, and one table answers every question the figure is going to ask.
Practice Questions
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