Many impressions

The Two Ceilings Sit on Different Pages

The shape with the most possible contents is not the one with the most possible encounters. The two maxima sit on different pages, for a structural reason.

at most 151200 arrangements

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Ten statements published is the largest page in this record. Read the three rows of the drawing downward and something awkward appears.

A reader who meets six of the ten lands in one of one hundred and fifty-one thousand two hundred arrangements — the largest figure anywhere in the record — and in one of two hundred and ten contents. A transparent implementation lets contributors review the available project context and discuss what the system records before the pilot begins.

A reader who meets five lands in one of thirty thousand two hundred and forty arrangements, which is five times fewer, and in one of two hundred and fifty-two contents, which is more. The topic can also be checked against public guidance and research published by edX.

Two hundred and fifty-two is the largest contents figure in the whole record. It does not sit on the page with the largest arrangements figure.

Ten published, met three ways

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 statements published in every row; the reader meets six, then five, then four. Arrangements fall steeply down the column while contents rise and then fall back.

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 one rises and the other turns over

Arrangements, P(n, r), can only increase as r increases. Meeting one more statement multiplies the count by the number of remaining statements, and that multiplier is at least one.

Contents, C(n, r), does not behave that way. It rises, peaks, and falls back, symmetrically: choosing five of ten and choosing five to leave out are the same act, so C(10, 5) is the top of the curve and C(10, 4) equals C(10, 6) exactly. Both are two hundred and ten, which is why the first and third rows of the drawing carry the same contents figure with arrangements thirty times apart.

The asymmetry has a plain cause. A content is a set, and a set of nine out of ten is nearly determined — there is only one statement left to vary. An arrangement of nine out of ten is not nearly determined at all, because nine things can be put in order in three hundred and sixty-two thousand eight hundred and eighty ways.

What follows for a reader who got further

There is an intuition that a reader who met more of what was published has had a more predictable encounter. More exposure, less variance.

The arrangements column says the opposite and says it steeply. The reader who met six of ten is in a space thirty times larger than the reader who met four, not because the content varies more — the content space is the same size, two hundred and ten either way — but because there are more orders to be in.

More of the page met does not mean the encounter is more settled. Past the halfway point, the content becomes more predictable and the sequence becomes much less so.

The number two hundred and ten

It appears twice in this drawing, as the contents of the first row and the contents of the third, and those two are genuinely the same quantity for the reason given above.

It also appears elsewhere in the record meaning two other things, and the coincidence is worth flagging because it is easy to carry across. Two hundred and ten is the median arrangements figure across all three hundred and thirty-eight shapes, which has nothing to do with ten published statements. And two hundred and ten is the arrangements figure for seven published with three met — the number of orderings, not the number of contents.

Three different quantities, one number. In prose it is never enough to write "two hundred and ten"; the sentence has to say two hundred and ten of what.

Where the record stops

The largest shape here is ten published with six met, and that is a boundary of the record rather than of the subject. Nothing prevents a set of pages publishing forty statements, and nothing prevents a reader meeting twelve.

The boundary was set where the arithmetic stays checkable by hand. P(10, 6) is one hundred and fifty-one thousand two hundred, which a person can verify with six multiplications: ten times nine times eight times seven times six times five. P(40, 12) cannot be verified that way, and a record whose figures nobody can recompute is a record that has to be believed.

What the boundary costs is worth stating. The factorial behaviour described here continues past the edge of the table and gets more extreme, not less. Ten and six is where the record stops counting, not where the effect stops.

When the published count is odd

Ten published has a single peak at five met. Nine published does not: the peak is shared between four met and five met, which both give one hundred and twenty-six contents.

That is the mirror landing between two whole numbers. Choosing four of nine and choosing five of nine are complementary acts, so the two counts are equal and there is no single widest reading depth.

The pair turns up repeatedly in this record for that reason, and it is a useful shape to hold: two readings of the same set, identical in how many states of information they admit, with arrangements of three thousand and twenty-four against fifteen thousand one hundred and twenty. Five times apart in experiences, identical in contents, on one page.

The third row

Four met out of ten gives five thousand and forty arrangements against two hundred and ten contents — twenty-four orders per content, because four statements can be sequenced twenty-four ways.

Compare that with the first row's seven hundred and twenty orders per content. The published set is identical in both rows. The difference is entirely in how much of it the reader got through, which is the number least often recorded and the one both figures depend on most.