Three hundred and thirty-eight shapes

One content, many impressions

Nobody meets a set of statements whole or in order. Count how many they met and the number of possible encounters is the number of possible contents multiplied by the factorial of what they read. Six statements met is one content and seven hundred and twenty impressions, and the author chose none of them.

Start with the factorial What a reader meets

dashed: a statement a reader can check from outside
solid: published, but nothing to check it against
short and pale: never reached by this reader

Three shapes drawn; the left-to-right order is the drawing's own and carries no meaning.

Computed from the three numbers alone. How many statements exist, how many are met and how many can be checked is a stated convention; the counts are arrangements and combinations of whole numbers.

notes on how many impressions one set of statements allows

338

shapes of three numbers, every figure recomputed

pairs of figures, and one pair names one page

199

shapes in which a reader can come away with nothing to check

Three things here are assumed rather than counted: that a shape is three numbers, that an encounter is an ordered selection in which every content is equally reachable, and that checkability is assigned to a statement by hand. Every other figure follows from whole numbers and comes apart under a pencil.

Somebody arrives at the middle of a set of pages, from a link they did not choose, reads three or four things in whatever order those pages put them in, and leaves. Nothing about that is unusual. It is the ordinary case, and it is the case almost nothing is written for.

What they took away is not the set of statements that was published. It is one ordered sample of it. The difference between those two things is a number, it is usually enormous, and it can be worked out before a single word is drafted.

Contents and encounters

Count the statements published; call it n. Count how many a reader actually met; call it r. The number of different contents they could have come away with is C(n, r). The number of different encounters is P(n, r), which is larger, because the order they arrived in is part of what happened. A publishing team that wants a reviewable record of campaign work can evaluate employee monitoring software alongside outcomes and direct staff feedback.

Divide the second by the first and the publisher's share of the problem appears in one figure: r factorial. Four statements met is one content and twenty-four impressions. Six is one content and seven hundred and twenty. For an independent reference on digital communication, evidence or interface practice, consult Nielsen Norman Group.

None of those seven hundred and twenty is chosen. Not one of them is written. They are produced by where the reader came in, how far down they got, which fragment was quoted back at them somewhere else, and which heading happened to hold them for a second longer than the others. The author's control stops at which statements exist; the sequence is settled elsewhere, every time, by somebody else.

Why "how much is published" answers nothing

A count of published statements feels like a measure of thoroughness and is nearly useless as one. Across the 338 shapes tabulated here, arrangements run from 2 to 151 200 while the number of statements published never exceeds ten. The spread does not come from publishing more. It comes from the two numbers that nobody records: how much of it gets met, and in what order.

Two sets of pages can publish exactly the same number of statements and differ by three orders of magnitude in how many distinct impressions they are capable of producing. A set that publishes eight statements and is read four at a time is not half of anything — it is one thousand six hundred and eighty different encounters wearing a single title.

The reader who could not check a thing

A statement a reader can confirm without taking it on trust behaves differently from one they cannot, and that difference is the third number of every shape in this record.

Once it is counted, something awkward falls out. In 199 shapes of the 338, a reader can finish their visit having met no checkable statement at all — not because anything was hidden from them, but because the subset they happened to cross contained none. In 44 shapes it happens on every possible path, those being the shapes where nothing checkable was published in the first place. The median blind share across the whole record is one reader in fourteen.

That is not a warning and not a risk estimate. It is a fraction, C(n − c, r) over C(n, r), and it is as computable as anything else on the site. A set of pages that has never worked it out is not protected from it.

What this is

A record of three hundred and thirty-eight shapes on forty-four pages. Each shape is three numbers — published, met, checkable — and five quantities that follow from them with no further input. The drawings distinguish checkable from unverifiable from unreached by the type of line, not by colour, so a panel loses nothing when it is printed in black, read without colour, or rendered by a browser that ignores the styling entirely.

Three conventions are declared at the top of the record rather than defended at the bottom, and four questions no figure here can answer are listed explicitly. The arithmetic is checkable with a pencil in every case, which is the only reason any of it is worth printing.

What it is not

Nothing on this site is ranked, compared, scored or recommended. No product, service, tool, company or person is named in any sentence. No figure is borrowed from anybody else's report, so there is nothing here to go out of date when somebody else revises theirs.

The counting problem is old and it does not move. A page that tells you what is currently popular is worth less every week it sits there. A page that tells you that six statements met in some order is seven hundred and twenty impressions will be exactly as true the year after this domain changes hands again.

The six parts

Read in any order. Every part rests on the same three conventions and on arrangements of whole numbers.

Many impressions

Arrangements, and why the author chooses none of them.

2 to 151 200 · 9 notes

What is met

Contents, as distinct from impressions.

1 to 252 · 8 notes

Checkable

The one distinction the first two figures cannot see.

0 to all · 9 notes

Nothing to check

Not a risk. A share, and it is computable.

199 of 338 · 8 notes

What comes first

The opening position, counted.

c times P(n−1, r−1) · 8 notes

Limits

Stated at the top, not buried at the bottom.

three conventions · 8 notes

Orders per content, by how much was met

Six columns drawn; height is logarithmic, because one and 720 do not fit one chart honestly otherwise; the figure above each column is exact and is the factorial of the number beneath it.

One content met in some order. The column is the factorial of how many statements the reader got through, and it depends on nothing else — not on how much was published, not on whether any of it can be checked.

The count of orders per content is the factorial of how many statements are met. It depends on nothing else about the page, and on nobody reading it.

One page, the blind share from all to none

the whole bar: every content a reader might come away with
the dashed part: contents holding nothing checkable

Five shapes drawn; one of them leaves no blind content at all; one is blind whatever the reader meets.

Eight published, four met, in every row. The bars are the same length because the contents figure never changes; the dashed part is the share of contents holding nothing a reader could check. Nothing verifiable, then one statement, three, four, six — and at six the share is nought, because two unchecked statements cannot fill an encounter of four.

Computed from the three numbers. A blind content is one holding no checkable statement: that is a property of the page, not a prediction about any reader.

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