Many impressions

One Figure Is Not Enough to Name a Page

Three hundred and thirty-eight shapes take only thirty-five distinct arrangements figures. The contents figure identifies a page no better.

at most 120 arrangements

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Somebody reports that a set of pages admits one hundred and twenty arrangements. What do they know about it?

Less than it sounds. One hundred and twenty is the arrangements figure for five published with four met, for five published with all five met, and for six published with three met. Nineteen shapes in this record carry it. The contents figures of those three pages are five, one and twenty.

One arrangements figure, three pages

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.

Every row gives one hundred and twenty arrangements. The contents figures are five, one and twenty, and the published counts are five, five and six.

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.

Thirty-five values for three hundred and thirty-eight shapes

Arrangements takes only thirty-five distinct values anywhere in the record. Three hundred and thirty-eight shapes share them out, which means the average value is carried by nearly ten shapes and the busiest by twenty-five. Teams seeking a clearer history of project effort can open the reporting reference, then compare that record with quality, timeliness and client feedback.

The crowding is not a defect of the record. It is what the arithmetic does. P(n, r) is a product of consecutive descending integers, and different pages reach the same product by different routes: five times four times three times two is the same as six times five times four. Factorials collide with each other constantly, and the collisions are not rare or exotic. For an independent reference on digital communication, evidence or interface practice, consult Harvard Business Review.

Contents is no more identifying

The same holds in the other column. Contents takes twenty-four distinct values across the record, fewer than arrangements, and collides just as freely.

Seven published with three met gives thirty-five contents. Seven published with four met also gives thirty-five contents, for the reason that choosing three to meet and choosing three to leave out are the same choice. Their arrangements figures are two hundred and ten and eight hundred and forty — four times apart, because the second has four statements to order instead of three.

So a reader told only the contents figure cannot distinguish a page where three statements are met from one where four are. Those are not similar situations: one of them has six orders per content and the other has twenty-four.

Adding the published count does not rescue either one

The obvious repair is to report a figure alongside how much was published, since that number is usually known. It fails in both columns, and it fails in an instructive place.

Take the published count and the arrangements figure. Across the forty-four pages of this record there are five collisions, and every one of them is a page where the reader met all the published statements paired with the page where they met all but one. Six published with five met and six published with six met both give seven hundred and twenty arrangements, because placing the last of six statements can be done exactly one way and so adds nothing to the count. The same happens at two, three, four and five published.

Take the published count and the contents figure instead and it is worse: fourteen collisions, from the symmetry of choosing. Seven published gives thirty-five contents whether three are met or four, twenty-one contents whether two or five, and seven contents whether one or six.

So neither derived figure is saved by the one number a set of pages reliably knows about itself.

Where this leaves a single-number summary

Any report of a set of pages that gives one figure is, on this evidence, uninterpretable. Not approximate — uninterpretable. The figure is consistent with pages that differ in how much was published, in how much is met, and in the gap between content and encounter by a factor of twenty.

A number that several different situations produce identically is not a measurement of any of them.

The record's response is to carry the shape itself — published, met, checkable — beside every figure it prints, in the drawings and in the prose. The figures are derived; the shape is what identifies.

What two figures do

The pair of figures together behaves completely differently, and that is the subject of the next note. All forty-four pairs in this record are distinct, with no collisions at all, and from a pair both original numbers can be recovered by arithmetic.

It is worth noticing in advance how odd that is. Neither column alone comes close to identifying a page — thirty-five values for forty-four pages in one column, twenty-four in the other — and together they identify it exactly, every time. The two kinds of crowding do not overlap anywhere.

The three rows of the drawing

Five published with four met: one hundred and twenty arrangements, five contents, twenty-four orders per content.

Five published with all five met: one hundred and twenty arrangements, one content, one hundred and twenty orders per content. Here the arrangements figure and the orders-per-content figure are the same number, which happens exactly when the reader met everything.

Six published with three met: one hundred and twenty arrangements, twenty contents, six orders per content.

One figure held constant down the column; the quantity it describes changing underneath it each time. The last column — orders per content — runs twenty-four, one hundred and twenty, six, in no order at all, and it is the one that says what kind of page each row is.