Telling apart

Two Listings In One Cell

Two listings that agree on every filter become one listing to every question the board permits. The difference is invisible by construction.

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Take a board with two filters, one offering six choices and one offering four. Twenty-four cells.

Now take two listings that happen to carry the same value in both filters. They sit in the same cell. There is no query anybody can construct — not a clever one, not a narrow one, not a combination nobody has tried — that returns the first without also returning the second.

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They have become one listing, and the board did it.

One schema, three queries

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

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

Two facets of six and four values. Nothing set returns the whole grid; one facet set returns a column of four; both set reaches a single cell. The grid is identical in all three — only the outline moves.

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.

Not a failure of searching

This is worth separating from the ordinary frustration of not finding things, because they feel identical and have nothing in common.

Ordinary failure is a mismatch: the listing is there, the query was wrong, a better query would have found it. That is fixable by the person searching, and most advice about searching is about fixing it.

What happens inside a cell is not fixable by anybody searching. The board holds no apparatus for the distinction. Trying harder produces the same result, because the result is the full and correct answer to every question the board can be asked.

A query returns a cell or a union of cells. It cannot return part of one, because there is no "part of one" for it to address.

What falls inside a cell in practice

Whatever the filters do not cover.

A board carrying filters for category and location distinguishes listings by category and location. Two listings in the same category and the same location are identical to it, however much they differ in everything the board has no filter for.

The listings themselves may describe those differences at length. The descriptions sit inside the cell along with everything else, and descriptions are not filters.

The arithmetic of the invisible part

Here is the part that can actually be computed.

Suppose the listings on a board genuinely differ along more axes than the board carries filters for. Call the real number of distinguishable kinds R and the board's cell count C. If R is larger than C, then at least R minus C of those kinds are being merged, and no rearrangement helps: the merging is forced by the shortfall.

The record cannot tell anybody what R is, because R is a property of the world. It can state C exactly, from the page, in under a minute. Half of a comparison is still worth more than none.

What	Where it comes from	Can this record state it?
cells, C	the filters on the page	yes, exactly
real kinds, R	the listings and the field	no
forced merging	R minus C, when R exceeds C	only once R is known

Adding a filter, and what it does to the merging

Every filter added multiplies the cell count by its number of choices. A filter of three choices triples the resolution; a filter of six multiplies it by six.

That is a large effect and it is often the wrong one. The filter has to cut along an axis that listings actually differ on, or it multiplies the compartments without separating anything: the listings all take the same value and the extra cells stand empty.

A board with twenty-four cells that adds a filter of six choices has one hundred and forty-four cells. Whether it can now tell more listings apart depends entirely on whether the new filter asks about something the listings vary in, and that is not an arithmetical question.

What this looks like from the reader's chair

Nothing unusual, which is the difficulty.

Somebody sets both filters, gets a list of eleven listings, and works down it. Some of those eleven are what they wanted and some are not. They assume they have narrowed as far as the board allows and that the remaining sorting is theirs to do by reading.

That assumption is correct. They have narrowed as far as the board allows. What they have no way of knowing is whether the thing separating the useful eleven from the useless ones is a dimension the board could have carried and does not.

So the experience of hitting the ceiling and the experience of using a board properly are the same experience, and the figure is the only way to tell them apart.

Why the ceiling is worth knowing anyway

Because it bounds the argument.

If a board carries twenty-four cells and somebody is unhappy that two particular listings cannot be separated, the conversation about better search terms is over before it starts. The board resolves twenty-four kinds of listing. If those two are the same kind, that is the end of it, and the only remedy is a different filter.

Knowing which of those two conversations to have saves everybody involved a good deal of time, and the figure that decides it is a product of the numbers printed on the page.

The symmetric case, which is rarer and worth naming

A cell can also be empty.

Nothing requires every combination of values to occur. A board with one hundred and forty-four cells may have forty of them that no listing has ever fallen into, either because the combination is unusual or because it is impossible in that field.

Empty cells cost nothing and mislead nobody: a query reaching one returns nothing, which is a true answer. They matter only for the count of what the board can express, which the next section takes up, because a question with no answer is still a question the board allowed somebody to ask.