What is met

Met and Missed Are the Same Count

Choosing which statements were met is choosing which were missed, so the contents figure mirrors. Two shapes agree on it for opposite reasons.

at most 15120 arrangements

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Thirty-five ways to choose three statements out of seven. Thirty-five ways to choose four out of seven. The same number, and not by coincidence.

Picking the three a reader met is the same act as picking the four they missed. One choice, described from either end. The count has to be identical because there is only one thing being counted. Readers who want to examine the reporting side can read the setup notes and note which controls are available to employees and administrators.

Shapes that agree on contents

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

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

Seven published with three met and with four met give thirty-five contents each; nine published with four and five met give one hundred and twenty-six each. Arrangements differ by four and by five.

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 mirror in full

For any page, the contents figure for r met equals the figure for n − r met. Seven published: one met and six met both give seven; two and five both give twenty-one; three and four both give thirty-five. Readers looking beyond this site's model can compare it with material from United Nations.

Nine published: one and eight give nine, two and seven give thirty-six, three and six give eighty-four, four and five give one hundred and twenty-six.

The record only reaches six met, so the upper halves of these mirrors are mostly outside it. What is inside is the consequence: on pages where both halves of a mirrored pair fall within range, two shapes carry the same contents figure while describing readers in opposite situations.

Two readers, one figure, opposite cases

A reader who met three of seven statements is missing four. A reader who met four is missing three. Both face thirty-five possible contents.

What differs is everything else about them. The first has six orders per content and the second has twenty-four, because three things can be sequenced six ways and four things twenty-four. The first is more likely to have missed a checkable statement. The first can be disjoint from another three-statement reader, since twice three is less than seven; two four-statement readers on that page must overlap.

So the shared contents figure conceals a different situation on each side, and the record's response is to never print the figure without the shape beside it.

The symmetry is in the counting, not in the reading. A mirrored pair of shapes agrees on one number and on nothing a reader would notice.

Where the arrangements figures part company

Contents mirrors; arrangements does not.

Seven published with three met gives two hundred and ten arrangements. With four met it gives eight hundred and forty — four times as many, because there is one more statement in the sequence and it can go in any of four places.

Nine published with four met gives three thousand and twenty-four, with five met fifteen thousand one hundred and twenty. Five times as many, by the same reasoning.

The ratio between a mirrored pair's arrangements figures is always the larger value of r. That is a small fact and it is the mechanism behind something established earlier in this record: the pair of figures identifies a page because the quotient of the two recovers how much was met, and the mirror is exactly where the contents figure fails on its own.

A use for the mirror

There is one practical consequence, and it is a way of checking intuition cheaply.

Asked how many ways a reader might have met six of ten statements, most people begin multiplying. The mirror says: it is the same as the number of ways they might have missed four, which is the number of ways to choose four from ten, which is two hundred and ten. Four choices rather than six, and the same answer.

The record uses this throughout. The largest contents figure on the ten-published page is at five met, because five is where the mirror folds and the two halves meet. Every figure above and below it has a partner, and the partners are equal.

Where the record truncates the mirror

The mirror is complete in the arithmetic and incomplete in this table, because the record stops at six met.

Seven published pairs one with six, two with five, three with four — all inside the range. Ten published pairs four with six, five with five, and would pair three with seven, two with eight and one with nine, none of which the table reaches.

So on the larger pages only the lower half of each mirror is present, and a reader looking for a shape's partner will sometimes not find it here. The table's edge is at fault and not the symmetry: the partner exists, its contents figure is identical, and this record simply does not reach it.

What the mirror does not do

It does not make a half-read set equivalent to a half-missed one. The same count of possibilities covers two different kinds of reader, and only one number about them agrees.

Nor does it extend to the third column. The blind count — contents holding no checkable statement — is not symmetric in r at all, because it depends on how many of the published statements are checkable and that number does not mirror. Nine published with four met and nine with five met both give one hundred and twenty-six contents; with three checkable, the first has fifteen blind contents and the second has six.

Same contents figure, and a reader of the second shape is two and a half times less likely to come away unable to verify anything. That is the third number doing work neither of the first two can see, and it has its own part of this record.