Telling apart

Two Cells, And Forty-Six Thousand

The narrowest schema resolves two listings, the widest forty-six thousand six hundred and fifty-six. Both are ordinary boards.

Telling apartReference2 to 46 656

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The smallest schema this record holds is one filter offering two choices. It tells two listings apart.

The largest is six filters of six choices each. It tells forty-six thousand six hundred and fifty-six listings apart.

For a workflow with several handoffs, it is useful to open the implementation guide and test one ordinary project rather than a polished demonstration. For a separate technical reference, see Python.

Both are structures a board might plausibly have, and between them lies a factor of more than twenty thousand.

The two ends and the middle

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

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

Three figures on the record's full range of resolution. The scale is logarithmic, and even so the median sits well left of centre.

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

What the small end is actually like

A single filter of two choices is not a hypothetical. Plenty of collections of listings offer exactly one toggle and nothing else.

Such a board partitions everything it holds into two compartments. Every listing on one side of the toggle is, to the board, interchangeable with every other listing on that side. A reader can ask three different questions of it: one side, the other side, or everything.

Three questions is the floor of this record. It is reached by exactly one schema, and it is worth stating plainly rather than treating as an absurd corner: a board with one toggle has given its readers three possible queries, total, for as long as the board exists.

What the large end is actually like

Six filters of six is a heavy sidebar — thirty-six controls' worth of choices spread across six dropdowns — and it resolves forty-six thousand six hundred and fifty-six kinds of listing.

Whether that resolution is reached depends on the listings. A board with forty-six thousand six hundred and fifty-six cells and four hundred listings has, at best, four hundred cells occupied and the rest empty. The ceiling is real but it is a ceiling, not an achievement.

Resolution is a capacity. Nothing requires a board to use it, and a board with more cells than listings is mostly empty compartments.

Where the middle sits

The median cell count across the record is eight hundred and sixty-four. Seven schemas reach it, from four facets up to six.

On a logarithmic scale running from two to forty-six thousand six hundred and fifty-six, eight hundred and sixty-four sits well left of the midpoint. The distribution is not symmetric: there are far more ways to build a modest schema than an extreme one, and the extremes are each reached exactly once.

Figure	Cells	Schemas reaching it
smallest	2	1
median	864	7
largest	46656	1

The two ends being unique is not a coincidence. Only one schema can be the smallest, since there is one way to build it; the same at the top.

The span, read carefully

A factor of twenty thousand sounds like it ought to separate categories of board rather than variations within one. It does not.

Both ends of the range are ordinary structures. One toggle is a design choice people make. Six dropdowns of six is a design choice people make. Neither is an abuse of the format, and a reader moving between two such boards will not necessarily register that one of them resolves twenty thousand times as much as the other.

What a reader registers is the sidebar: short or long, simple or busy. That impression tracks the number of controls, which — as the class of seven shows — is close to uninformative about resolution.

Moving from one end towards the other

Getting from two cells to forty-six thousand six hundred and fifty-six takes a specific sequence, and it is worth seeing how short it is.

Step	Schema	Cells
start	2	2
widen to six	6	6
add five more facets of six	6·6·6·6·6·6	46656

Six edits in total: four to widen the first facet, then five new facets. Each one is an ordinary change that a board might make in a quarter, and together they span the entire range of the record.

The resolution multiplies by more than twenty thousand across those six steps. Nothing about any individual step would read as dramatic in a change log.

Why the record stops where it does

Six at both ends, and the page on method says that it is a decision.

The honest reason is checkability. Six filters of six gives a largest denominator of one hundred and seventeen thousand six hundred and forty-nine for the question count, and that is about as far as figures can go while a reader can still verify them with a pencil in a reasonable time.

Push the bounds out and the structure of the record does not change — cell counts still collide, pairs still identify, the grain still collapses — but the tables stop being things anybody checks. The record would then be asking to be believed rather than verified, which is the one thing it is built to avoid.