What is met

The Widest Point Is the Middle

Contents peaks when a reader has met about half a set and falls away on both sides, which makes the halfway reader the least predictable one.

at most 151200 arrangements

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Here is a sequence that does something unexpected. Ten statements published. A reader meets two: forty-five possible contents. Meets five: two hundred and fifty-two. Meets six: two hundred and ten.

It went down. The reader got further through the set and the number of possible states of their information shrank. A campaign team may look at the current analytics options, assign one owner for the pilot and review both useful findings and false signals at the end.

Ten published, met two, five and six

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.

Ten published in every row. The contents figure climbs from forty-five to two hundred and fifty-two and then drops back to two hundred and ten as the reader meets more.

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.

Why it turns over

Choosing which statements a reader met is the same act as choosing which ones they missed. Six met out of ten is four missed out of ten, and there are as many ways to pick four as to pick six. For additional background relevant to the interpretation of online work and information, see OpenStreetMap.

So the contents figure for six met equals the figure for four met, the figure for seven equals three, and so on. The sequence is a mirror, and the mirror's centre is at five — half of ten. That is where the figure is largest, and it falls away symmetrically in both directions.

At the extremes it collapses entirely. A reader who met one statement out of ten is in one of ten states. A reader who met all ten is in one state. Neither of those is hard to think about. The one in the middle, with two hundred and fifty-two states, is.

The halfway reader is the hard case

Three readers of the same set of ten statements:

The first met two. Easy to reason about: forty-five pairs, and most of the set is simply absent from their account of it.

The third met nine. Also easy: they are missing one statement, and there are only ten ways for that to happen.

The second met five. Two hundred and fifty-two possibilities, every one of them a half-complete account, and no two of them wrong in the same way. This reader has enough to form a confident impression and not enough for the impression to be determined by the set.

The reader who has seen half of something is the one whose conclusions are least predictable. The reader who has seen almost none and the reader who has seen almost all are both easier cases.

Where the peak sits for other sizes

For an even number of statements the peak is at exactly half. Ten published peaks at five met, with two hundred and fifty-two contents.

For an odd number the peak is shared by the two values either side of the middle. Nine published peaks at four met and five met equally, both giving one hundred and twenty-six contents — which is why those two shapes turn up repeatedly in this record with identical contents figures and very different arrangements figures.

The record stops at six met, so for the larger pages the peak is inside the range and for the smaller ones it is not. Ten published reaches its peak at five and then descends; four published has its peak at two and the record's upper limit on r never comes into it.

What this does to a plan

A set of pages that wants its readers to arrive at a consistent impression has two directions it can push in, and the arithmetic says they are not equivalent.

Push toward readers meeting less, and the contents figure comes down toward the lower end, where few combinations are possible. The cost is that each of those combinations is thin.

Push toward readers meeting more, and the figure comes down toward one. The cost is everything that makes a reader stop before the end.

What does not work is landing in the middle and hoping. The middle is exactly where the number of possible outcomes is maximised, and a half-read set of ten statements is the least determinate situation available on that page.

The two readers either side of the peak

Four met out of ten and six met out of ten both give two hundred and ten contents. They are a mirrored pair, equal by the symmetry of choosing, and they sit one step down from the peak on each side.

Their arrangements figures are five thousand and forty and one hundred and fifty-one thousand two hundred — thirty times apart. The reader who met six is in a space thirty times larger than the reader who met four while facing exactly the same number of possible contents.

Which is the clearest available demonstration that the two columns are measuring different things. Hold contents fixed and arrangements can still move by a factor of thirty on a single page, because arrangements counts order and contents does not.

Reading the drawing

The first row shows ten published with two met: eight short pale strokes, two full ones, ninety arrangements and forty-five contents.

The second shows five met: two hundred and fifty-two contents, the largest figure in the record, from thirty thousand two hundred and forty arrangements.

The third shows six met: two hundred and ten contents from one hundred and fifty-one thousand two hundred arrangements. The arrangements figure is five times the second row's and the contents figure is smaller.

Four checkable statements are published in every row, and that number is doing nothing to either figure. It has its own part of the record, and it is the only one of the three numbers that touches whether the reader can confirm a word of it.