Many impressions

Two Figures Name the Page Exactly

Neither figure identifies a page alone. All forty-four pairs are distinct, and both original numbers come back out of a pair with two divisions.

at most 840 arrangements

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Forty-four distinct pages appear in this record — forty-four combinations of how much was published and how much was met. Each produces a pair of figures: arrangements and contents.

All forty-four pairs are different. Not nearly all: all of them, with no collisions anywhere in the table. And the correspondence runs both ways, so a pair of figures can be turned back into the page that produced it.

Same contents, different 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

Two shapes drawn; all of them give the same 35 different contents; the left-to-right order is the drawing's own and carries no meaning.

Both rows give thirty-five contents. The arrangements figures are two hundred and ten and eight hundred and forty, and dividing one by the other recovers which row is which.

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.

The recovery, in two divisions

Divide arrangements by contents. The answer is r factorial, and the six factorials this record can produce are all different — one, two, six, twenty-four, one hundred and twenty, seven hundred and twenty — so the quotient names how many statements the reader met. If screenshots are part of a documented workflow, employee screen monitoring should be tested with employee-visible controls and the smallest necessary capture rate.

Now take the contents figure, which is C(n, r), and with r already known there is only one n that produces it. Read it off. For additional background relevant to the interpretation of online work and information, see Khan Academy.

Worked on the first row of the drawing: two hundred and ten divided by thirty-five is six, which is three factorial, so three statements were met. Thirty-five contents with three met means seven published, because C(7, 3) is thirty-five. The pair has been inverted.

Worked on the second row: eight hundred and forty divided by thirty-five is twenty-four, which is four factorial, so four statements were met. Thirty-five contents with four met means seven published again, because C(7, 4) is also thirty-five.

The two rows share a contents figure and are told apart entirely by the quotient.

Why this is not a lucky property of this table

The identification is a consequence of the arithmetic rather than a feature of how the record was assembled.

Arrangements divided by contents is r factorial by construction — it is the count of orders of one content, and it is the same for every content. The factorial is strictly increasing, so distinct values of r give distinct quotients, and the quotient determines r with no ambiguity. With r fixed, C(n, r) is strictly increasing in n, so the contents figure determines n.

Both steps are one-to-one, which makes the composition one-to-one. No two pages can collide, however large the table grows.

The claim is not that no collisions were found. It is that a collision is impossible, and the difference between those two statements is the difference between a survey and a proof.

Nowhere else here is the right thing to say stronger than a count. Everywhere else the record reports what it counted; this once, it reports what cannot happen.

It is checked anyway

The argument above is correct and was still verified by recomputation when the table was built: every pair collected, every page collected, the two sets counted, and the counts compared. Both come to forty-four.

The reason for checking a proof is not doubt about the mathematics. It is that the table is produced by a program, and a program can fail to implement a correct argument. A proof covers the arithmetic; the recount covers the implementation. They are different claims and both can fail separately.

What the pair still cannot tell anybody

It identifies the page completely and says nothing about the third number.

A page is a pair: published and met. A shape is a triple: published, met, and how many of the published statements a reader can check from outside. Forty-four pages carry three hundred and thirty-eight shapes between them, so a pair of figures leaves the third number entirely open — and it is the number that decides whether a reader can confirm anything at all.

The practical consequence appears later in the record. In one hundred and ninety-nine of the three hundred and thirty-eight shapes, a reader can come away having met nothing checkable. Both figures discussed here are identical across shapes that differ enormously in that respect, which is why the record carries a third column and a third kind of drawing.

What a third figure would have to do

The pair fails to say anything about verifiability, so the obvious question is whether some third derived figure would complete the identification of a shape rather than a page.

The blind count almost does and not quite. Adding it separates two hundred and forty-three of the three hundred and thirty-eight shapes and leaves ninety-five indistinguishable — all of them shapes where nothing can be blind, since once the unchecked pool is too small the count is nought and stays nought however much higher the checkable count goes.

So no figure derived from the three numbers recovers them completely. The record's answer is to print the three numbers themselves on every row of every drawing, and to treat the derived quantities as assertions about them rather than as a substitute for them.

Forty-four, twice

Forty-four is the number of pages, and therefore the number of pairs. Forty-four is also, separately, the tally of wholly blind shapes — those publishing nothing a reader could check.

These are unconnected. The first follows from the ranges the record covers — two to ten published, one to six met, with met never exceeding published. The second is the count of shapes with nothing checkable published at all, which is one shape per page, and so it comes to forty-four for a reason that only looks like the first one.

One page, one shape with no checkable statement. Written out that way the agreement stops being a coincidence and becomes a third fact, which is the better way to hold it.