Reaching one

Six Sevenths, And Under A Tenth

On a board with one filter of six choices, six questions in seven reach a single kind of listing. On a board with six filters of two, fewer than one in ten do.

Reaching oneReference64/729 to 6/7

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The share of questions that reach a single cell runs from six sevenths down to sixty-four over seven hundred and twenty-nine.

Nearly eighty-six per cent at one end. Under nine at the other. Both ends are reached by exactly one schema, and the two schemas are as different as the record allows.

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The two extremes, by resolution

the upright: where this figure falls on the record's range
the ticks are powers of ten; the scale is logarithmic

Two figures placed; the range is the record's own, over cells.

The schema with the highest share of pinning questions and the schema with the lowest, placed on the record's range of resolution. The high-share board resolves six kinds of listing; the low-share board resolves sixty-four.

Placed on the full range the record holds. The scale is the record's own, and the drawing refuses a scale that is not.

The high end: one filter of six choices

Six cells, seven questions. Six of those seven set the filter and reach a single cell; the seventh leaves it alone and returns everything.

So on the simplest useful board imaginable, almost every available question pins something down. A reader has six precise questions and one broad one.

That is the highest share any schema here reaches, and it is reached by the schema with the least resolution of any multi-valued facet in the record. The share is highest exactly where there is least to tell apart.

The low end: six filters of two choices

Sixty-four cells, seven hundred and twenty-nine questions. Only sixty-four of those questions set all six toggles.

Six hundred and sixty-five questions — more than nine in ten — leave at least one toggle alone and return a group.

Schema	Cells	Questions	Share
6	6	7	6/7
2·2·4·4	64	225	64/225
2·2·2·2·2·2	64	729	64/729

The second and third rows have the same resolution — sixty-four cells — and their shares differ by more than a factor of three. Same number of distinguishable listings, wildly different proportion of questions that get down to one.

Two boards resolving sixty-four kinds of listing: on one, nearly three questions in ten reach a single kind; on the other, fewer than one in ten.

Why the toggles do so badly

Because a facet of two multiplies the share by two thirds, and a facet of six multiplies it by six sevenths.

Two thirds is a heavy cut. Six of them compound to sixty-four over seven hundred and twenty-nine. Six sevenths is a light one: six of those would compound to about four tenths, and that schema — six facets of six — is the largest board in the record.

So the smallest six-facet schema has the lowest share in the whole record and the largest six-facet schema has a share more than four times higher. The direction is the opposite of what the sizes suggest and it follows directly from the arithmetic.

What this means for a sidebar of toggles

A long row of yes-or-no switches is a common and reasonable design. It is also the structure that pushes this share down fastest.

Each toggle adds a factor of two to the resolution and a factor of three to the questions. The resolution climbs steadily; the share of questions that pin anything down falls by a third each time.

The practical consequence is that readers of such a board will almost always be looking at a group. To reach one kind of listing they must make a decision about every single toggle, including the ones they have no view on — and a reader with no view on a toggle will leave it alone, which is exactly what keeps them in the group.

The middle of the range

Between six sevenths and sixty-four over seven hundred and twenty-nine sits everything else, and the median is sixty-four over two hundred and twenty-five.

That median is nearer the bottom than the top, for the same reason every median here is: the enumeration is dominated by long schemas, and long schemas have low shares.

A board with two or three filters will sit well above the median. A board with six will sit below it almost regardless of the sizes — the only six-facet schemas with shares above the median are the ones built from large facets, and those are rare in practice because large facets are harder to design.

The shape of the whole finding

The share is high where the board is simple and low where it is elaborate, and resolution runs the other way.

That is not a defect to be fixed. It is the trade the structure makes: more facets buy more resolution and more expressiveness, and they charge for it in how much work a reader must do to get down to one thing.

Knowing the rate is the useful part, and the rate is one fraction per facet: size over size plus one, multiplied together. The next page is that derivation.