Limits

How to Check Every Figure on This Site

Five quantities, five formulas and a worked example done twice. Nothing here needs a tool, a source, or any trust in the author.

at most 1680 arrangements

Home · Limits · Reference

Every figure in this record follows from three numbers by arithmetic. This note gives the five formulas and works two shapes through all of them, so that a reader who wants to verify the record can do it here and then trust the rest for the reason that it is the same arithmetic.

Write n for how many statements are published, r for how many the reader meets, and c for how many of them are verifiable from outside.

Two worked shapes

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

Six published with three met and two verifiable, and eight with four met and three verifiable. Every figure on this page is derived from those two triples below.

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 five quantities

Arrangements, P(n, r): multiply r consecutive descending integers starting at n. How many distinct ordered encounters are possible. Project owners can check the time-tracking approach before deciding whether the additional visibility is worth the ongoing administrative cost.

Contents, C(n, r): that figure divided by r factorial. How many distinct subsets. A broader source for checking the surrounding technical or organisational context is Hotjar.

Orders per content: r factorial. Arrangements divided by contents, with no remainder.

Blind contents, C(n − c, r): the same binomial formula applied to the unchecked statements. Nought if n − c is less than r.

Blind share: blind contents over contents.

quantity	formula
arrangements	P(n, r)
contents	P(n, r) ÷ r!
orders per content	r!
blind contents	C(n − c, r), nought if n − c < r
blind share	blind contents ÷ contents

A sixth figure appears in the record's totals: checkable openings, c times P(n − 1, r − 1), whose share of all arrangements is c over n.

Six published, three met, two verifiable

Arrangements: six times five times four, which is one hundred and twenty.

Contents: one hundred and twenty divided by three factorial, which is one hundred and twenty divided by six, which is twenty.

Orders per content: three factorial, six. Check: one hundred and twenty over twenty is six.

Blind contents: four unchecked statements, three met, so C(4, 3). Four times three times two is twenty-four; divided by six is four.

Blind share: four over twenty, which is one fifth.

Checkable openings: two times P(5, 2), which is two times twenty, which is forty. Share: forty over one hundred and twenty is one third, and c over n is two sixths, which is one third.

Eight published, four met, three verifiable

Arrangements: eight times seven times six times five, which is one thousand six hundred and eighty.

Contents: divided by four factorial, which is twenty-four. One thousand six hundred and eighty over twenty-four is seventy.

Orders per content: twenty-four. Check: one thousand six hundred and eighty over seventy is twenty-four.

Blind contents: five unchecked, four met, C(5, 4). Five times four times three times two is one hundred and twenty; divided by twenty-four is five.

Blind share: five over seventy, which is one fourteenth — the median blind share of the whole record, which is why this shape is used as the standing example throughout.

Checkable openings: three times P(7, 3), which is three times two hundred and ten, which is six hundred and thirty. Share: six hundred and thirty over one thousand six hundred and eighty is three eighths, and c over n is three eighths.

What to do if a figure here is wrong

The same thing the record's own apparatus does: trust the recount over the sentence.

If any figure on this site disagrees with the arithmetic above, the figure is the error. Nothing here is a measurement with uncertainty attached, nothing was cited from elsewhere, and nothing depends on a judgement except the classification of statements as checkable — which is printed as a number on every row so that it can be substituted.

A record that cannot be wrong in a way its reader can detect is not being honest about having a method. Everything here can be caught.

Two checks that need no formulas at all

Beyond recomputing, there are two consistency checks a reader can apply to any row of this record without working anything out.

Arrangements divided by contents must be a factorial — one, two, six, twenty-four, one hundred and twenty or seven hundred and twenty. Any other quotient means one of the two figures is wrong, and which one cannot be told from the quotient alone.

And the blind count can never exceed the contents figure, since blind contents are a subset of contents. A row reporting more of the first than the second is wrong regardless of what the formulas say.

Both of those catch errors without any arithmetic beyond a division and a comparison, which makes them worth doing first on any row that looks suspect.

What checking cannot establish

That the shape fits the argument.

Every figure in this record can be recomputed and the recount proves only internal consistency. Whether ten published statements with six met is the right shape to illustrate a point about reading depth is a question of judgement, and the record's drawings carry captions so that the judgement can be read alongside the figures rather than inferred from them.

That is the last of the record's limits and the hardest one to automate. The arithmetic defends itself; the choice of what to count has to be argued, and the arguing is what the prose on this site is for.