What can be asked

More Questions, The Same Sight

A factor of five in available questions buys nothing in resolution. The extra questions are unions of cells the board already had.

What can be askedAnalysis3 to 117 649

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Seven schemas resolve one hundred and forty-four listings. The questions they permit run from two hundred and forty-five to one thousand two hundred and ninety-six.

The question this page is about: what did the extra one thousand and fifty-one questions buy?

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One resolution, cut two ways

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

Four grids drawn; one sets none, and so returns everything.

The first pair is a facet of six and a facet of four; the second is two facets of two. Splitting a resolution into smaller facets multiplies the questions without adding a single cell.

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 the extra questions are

Unions of cells that were already there.

The resolution is fixed at one hundred and forty-four. Every query, on every one of the seven schemas, returns some collection of those same one hundred and forty-four cells. The boards do not disagree about what exists; they disagree about which collections a reader is allowed to name.

A schema of three filters lets a reader name two hundred and forty-five of the possible collections. A schema of six filters lets them name one thousand two hundred and ninety-six. The cells being collected are identical in both cases.

The extra questions do not reveal anything new. They are additional ways of grouping what was already visible, and that is both less and more useful than it sounds.

Why it is less useful than it sounds

Because no new distinction becomes possible.

If two listings fell in the same cell on the three-filter board, they fall in the same cell on the six-filter board, and no query on either separates them. The thing most people want from a filter sidebar — getting down to the listing they are after — is bounded by the resolution, and the resolution did not move.

A reader who could not isolate what they wanted on the first board cannot isolate it on the second. They have more routes to the same set of destinations.

Why it is more useful than it sounds

Because routes matter when the reader does not know where they are going.

A six-filter board lets somebody approach the same one hundred and forty-four cells from six directions and in combinations that suit how they happen to be thinking. They can set the two things they care about and leave four alone. On the three-filter board the same reader has three controls, each coarser, and fewer ways to express a partial idea.

Schema	Filters a reader sees	Questions
4·6·6	three	245
2·2·3·3·4	five	720
2·2·2·2·3·3	six	1296

The last row offers four simple toggles and two three-way choices. That is a different experience of searching from three dropdowns, and the difference is real even though the resolution is identical.

The arithmetic of splitting a facet

The move that produces this is splitting: replace a facet of six with a facet of two and a facet of three.

Cells: unchanged. Six equals two times three, so the product is exactly as it was.

Questions: multiplied by twelve sevenths. The factor seven becomes three times four, which is twelve.

So each split leaves the resolution alone and multiplies the question count by about 1.71. Do it to both sixes in a schema of four, six and six, and then split the four as well, and the question count climbs from two hundred and forty-five to one thousand two hundred and ninety-six while the cells sit at one hundred and forty-four throughout.

The reverse edit, and why nobody makes it

Joining two facets into one — the opposite of splitting — leaves the resolution alone and cuts the question count.

Replace facets of two and three with a single facet of six: the cells stay at one hundred and forty-four, and the questions fall from four hundred and twenty to two hundred and forty-five.

It also removes a control from the sidebar, which is the usual reason anybody considers it. The arithmetic says the cost is expressiveness and the saving is one row.

Nobody makes this edit either, for the same reason nobody makes the others: the figures that would inform it are not computed. A board simplifying its sidebar is working from how busy the page looks, which tracks neither of the two products.

What to take from it

A longer sidebar is evidence about expressiveness, not about resolution.

That is the correction this section exists to make. Counting the controls on a page gives a rough sense of how many ways there are to ask, and no sense at all of how much the board can tell apart. The two properties are independent enough that one can be multiplied by five while the other does not move.

Anybody choosing between two boards on the strength of a richer sidebar should know which of the two things they are buying. Often it is the one they wanted. Sometimes it is not, and the arithmetic takes a minute to settle it either way.