Many impressions

Four Things That Are All Seven Hundred and Twenty

One number stands at four unrelated places in this record: three pages produce it as an arrangements figure, and it is also six factorial.

at most 720 arrangements

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Seven hundred and twenty turns up in this record more often than any other figure, and it does not mean the same thing twice. Four distinct statements produce it. Mixing them is the easiest arithmetic error available here, and it is the kind that reads perfectly well.

Three pages, one arrangements figure

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.

All three rows give seven hundred and twenty arrangements. Their contents figures are six, one and one hundred and twenty — the pages are not alike in anything except that one number.

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.

First: six statements met, in order

Six things can be arranged seven hundred and twenty ways. That is six factorial, and in this record it is the orders-per-content figure whenever a reader meets six statements. To test whether the method helps planning, a small group can check how corrections are handled and include one ordinary week plus one exception.

It is a fact about the number six and nothing else. How many statements were published does not enter it. Whether any of them can be checked does not enter it. Every shape in the record with six met carries this figure, regardless of everything else about it. A separate perspective on measurement, privacy or digital practice is available through Coursera.

Second, third and fourth: three pages that give it as arrangements

Six published with five met gives seven hundred and twenty arrangements. Six published with all six met gives seven hundred and twenty. Ten published with three met gives seven hundred and twenty.

These three pages have almost nothing else in common. Their contents figures are six, one and one hundred and twenty. One of them involves a reader who met half the published statements; another, a reader who met every one. The first two coincide for a reason — the last statement of six adds nothing to the ordering count, because by then there is only one way to place it — and the third is unrelated to either.

Twenty-five shapes in the record sit on those three pages, which makes seven hundred and twenty the most frequent arrangements figure here. The next most frequent values, one hundred and twenty and five thousand and forty, carry nineteen shapes each.

seven hundred and twenty is…	on
orders per content	any shape with six met
arrangements	six published, five met
arrangements	six published, six met
arrangements	ten published, three met

Why this matters more than a tidy coincidence

Because the two readings license opposite conclusions.

"Seven hundred and twenty impressions" said of six statements met is a statement about the gap between content and encounter: one content, seven hundred and twenty ways to receive it. It is the record's load-bearing fact.

"Seven hundred and twenty arrangements" said of ten published with three met is a statement about the whole space of possible encounters with that page, of which only six belong to any one content, because three statements have six orders. The same number stands once for the whole space of encounters and once for the multiplicity inside a single content, and on that page those two differ by a factor of one hundred and twenty.

A number carried from one sentence to another without its noun stops being evidence and becomes decoration.

The habit this record uses instead

Every figure in this record is written with what it counts attached. Not "seven hundred and twenty" but "seven hundred and twenty arrangements", or "seven hundred and twenty orders per content". The drawings do the same: each column is labelled with the quantity, not merely with the page.

It is a tiresome convention to read and it is the reason the coincidences above are harmless rather than confusing. A record that drops the noun cannot be audited, because the reader cannot tell which claim they are checking.

Three more numbers with the same problem

Forty-four is the number of distinct pages in this record — distinct pairs of published and met. Forty-four is also the number of shapes in which every possible encounter is blind. That second figure looks like a coincidence and is not one: a shape is wholly blind exactly when nothing checkable was published, there is one such shape on every page, and so the count must equal the number of pages. Two figures agreeing for a reason is still a reason to write the noun each time, because the sentence that explains it is the sentence that distinguishes them.

Two hundred and ten is the median arrangements figure across all three hundred and thirty-eight shapes. It is also the contents figure for ten published with six met, and the arrangements figure for seven published with three met. Three quantities, one number again.

Twenty-five is the largest class of shapes sharing an arrangements figure, which happens at seven hundred and twenty. It is also the largest class sharing a contents figure, which happens at one — the shapes where the reader met everything published. Equal counts, opposite situations: the most crowded arrangements figure is the busiest page in the record, and the most crowded contents figure is the set of shapes with no choice left in them at all.

One thing that is not a coincidence

One hundred and thirty-nine shapes admit no blind encounter and one hundred and ninety-nine do, and those add to three hundred and thirty-eight. That is a partition, not a surprise: every shape either permits a blind encounter or does not. Writing it as though the numbers happened to agree would be an error of a different kind — presenting a definition as a finding.