Reaching one

The Questions That Return Groups

Most of what a board permits leaves some filter alone. That is where nearly all searching happens, in a few fixed result sizes.

Reaching oneAnalysis64/729 to 6/7

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On a median schema, more than seven questions in ten leave at least one filter unset. On a board of six toggles it is more than nine in ten.

Those questions are not a remainder. They are where nearly all searching happens, and they have a structure of their own.

For an example beyond the counting model, teams can open this official resource and map available records to a specific capacity decision. For a separate technical reference, see Privacy International.

One filter left alone, at three scales

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

Three grids drawn.

Each query sets one facet and leaves the other. The result is always the full span of the untouched filter — four, six, six — and never anything in between.

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.

What an unset filter does to a result

It multiplies the result by its size.

A query that sets every filter but one returns the full span of that one: if it offers four choices, the result is four cells. Set every filter but two and the result is the product of those two sizes.

So the size of any result is the product of the sizes of the filters left alone. Nothing else enters.

On a 6·4 board	Result
both set	1 cell
the four left alone	4 cells
the six left alone	6 cells
neither set	24 cells

Four possible result sizes, from twenty questions. That is the whole range available, and the section on grain is about how small such ranges tend to be.

Why the sizes are so few

Because they are subset products, and subsets are many while their products are few.

A board with six filters has sixty-four subsets of filters that might be left alone. If all six filters are the same size, those sixty-four subsets produce only seven distinct products — the powers of that size. Six toggles give results of size one, two, four, eight, sixteen, thirty-two or sixty-four, and nothing else.

A reader with a result of thirty-two cells who wants it smaller has one move: set another toggle, and get sixteen. There is no fourteen, and no twenty.

What this does to narrowing

It makes narrowing a sequence of jumps rather than a slide.

Somebody with too many results sets one more filter and the result divides by that filter's size. On a six-value facet the result drops to a sixth of what it was. If that overshoots, there is nothing between: the previous result and the new one are the only two positions available.

This is a structural property of faceted filtering and it is often blamed on the board being badly tuned. It is not tuning. The reachable sizes are products of the facet sizes, and the gaps between them are exactly the facet sizes.

The commonest position of all

Nothing set.

A reader arriving at a board sees the whole of it, which is the largest result the ladder offers. Every subsequent action moves down a rung, and most readers take one or two steps and stop.

So the positions that matter most in practice are the top of the ladder and the one or two rungs beneath it — which are determined by the single largest facet sizes, since setting one filter divides by that filter's size.

A board whose filters are all small offers a gentle first step. One with a six-value dropdown offers a steep one, and which the reader takes is usually decided by which control is nearest the top.

Where the gaps are smallest

On schemas with mixed facet sizes.

Six filters of two, three, four, five, five and six produce thirty-nine distinct result sizes out of sixty-four subsets — the most in the record. The products land in many different places because the factors differ, so the ladder of reachable sizes has many rungs and the jumps between them are smaller.

Six filters all of the same size produce seven. Same number of filters, same number of subsets, and five times fewer rungs.

Why the result rarely divides evenly in listings

The cell count divides exactly. The listing count almost never does, because listings cluster.

A board whose listings are heavily concentrated in a few cells will show a result that barely shrinks when a filter is set — because the cells removed were nearly empty — and then collapses when a different filter is set, because that one split the crowded part.

So two filters of the same size, applied to the same result, can have completely different effects on how many listings come back, while having identical effects on how many cells.

The structure is predictable; the contents are not, and the record measures only the first.

The practical reading for somebody using a board

Count what you are looking at, then look at what the filters would divide it by.

A result of a hundred and a remaining filter of six choices means the next step lands somewhere near sixteen or seventeen on average, and possibly far from it if the listings are unevenly spread. There is no step to forty.

Knowing that in advance saves the experience of setting a filter, watching the result collapse past what was wanted, and unsetting it again. The board is not misbehaving when that happens; it is doing multiplication, and the arithmetic was available beforehand.