What can be asked

The One That Stands For Leaving It Alone

Questions are the product of each facet size plus one. That extra one is the option of not setting the filter, and it is the second ceiling.

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A query on a board does one of two things to each filter: it picks a value, or it leaves the filter alone.

That is the convention this record works under, and the whole of the second ceiling follows from it. A filter with four choices offers five states to a query — four values, plus untouched. A filter with six choices offers seven.

A practical use of employee activity dashboards is tracing time to projects and tasks while keeping the limits of every aggregate visible to reviewers. For a separate technical reference, see Schema.org.

Multiply those together and the product is how many distinct questions the board permits.

The same grid, the four kinds of query

one cell: one combination of values
the heavy outline with corner ticks: what the query returns

Four grids drawn; one of them sets every facet and so reaches a single cell; one sets none, and so returns everything.

Two facets of four and three values, twelve cells. Neither set, the first set, the second set, both set. Four of the twenty available questions, and the outline shows what each one returns.

Computed from the two facet sizes. A query sets a facet to one value or leaves it unset: that is the convention, not a claim about how anybody searches.

The two products side by side

Facets	Cells	Questions
4·3	12	20
4·6·6	144	245
2·2·2·2·3·3	144	1296
6·6·6·6·6·6	46656	117649

The first product is sizes multiplied. The second is each size plus one, multiplied. They are built from the same inputs and they do not track each other, as the third row makes plain against the second.

What the extra one is doing

It is a real option and it needs saying out loud, because it is easy to read "plus one" as an arithmetical convenience.

Leaving a filter alone is a choice a reader makes constantly. Most queries anybody actually performs leave most filters untouched — somebody sets the one or two things they care about and ignores the rest. The record counts those as queries because they are queries: they return a definite set of cells, and a different set from what any other combination returns.

Setting no filters at all is a question. It asks for everything, it returns everything, and it is one of the states the arithmetic counts.

Why the product is bigger than the cell count, always

Because every size is smaller than that size plus one, so every factor in the second product beats its partner in the first.

More precisely, the ratio between them is the product of the fractions k over k plus one, taken across the facets. Each fraction is less than one, so the cell count is always the smaller figure — and the more facets a schema has, the further apart the two products drift.

A schema of one facet with six values: six cells, seven questions, and the cell count is six sevenths of the question count. A schema of six facets with two values each: sixty-four cells, seven hundred and twenty-nine questions, and the cell count is sixty-four seven hundred and twenty-ninths of it — under a tenth.

The section on reaching a single answer is about that ratio, which turns out to be the most useful derived figure in the record.

The questions that lead nowhere

A board permits a question whether or not any listing answers it.

Of the twenty questions available on a four-by-three board, every one returns a well-defined set of cells. Whether those cells contain listings is a separate matter entirely, and not one a schema can speak to.

This is the commonest objection to counting questions, and it is a fair one: who cares how many questions can be asked if most of them come back empty? The answer is that the two things fail differently. A question returning nothing has told the reader something true — there is nothing of that kind here. A distinction the board cannot express has told them nothing at all, and they will not know it happened.

The page on empty results takes this apart at length.

Why this figure is the one that identifies a board

Cell counts collide: three hundred and eighty-one schemas share theirs with another. Question counts collide far less — the largest group of schemas sharing a question count holds three, against seven for cells.

And the two together collide not at all. Across the four hundred and sixty-one schemas, every pair of figures is distinct.

The reason is visible in the arithmetic. Cell counts are products of numbers drawn from two to six, a range full of shared factors. Question counts are products of numbers drawn from three to seven, which includes two primes near the top — and a product containing a seven cannot be rearranged into one that does not.

A board where the two figures nearly agree

One facet of six: six cells, seven questions. The cell count is six sevenths of the question count — the closest the two figures ever get in this record.

That closeness is a property of short schemas. With one facet there is one fraction to apply; with six there are six, and they compound.

Schema	Cells over questions
6	6/7
6·6	36/49
6·6·6·6·6·6	46656/117649

Even six large facets only bring the ratio down to about two fifths. Six small ones take it under a tenth, which is the lowest in the record and belongs to six toggles.

The figure to take away

Count the choices in each filter, add one to each count, and multiply. That is the second of the two numbers worth knowing about any board, it takes the same minute as the first, and it is the one that turns a group of seven candidate structures into a single identified one.