How to Check Every Figure on This Site

Five candidates, seven pairs compared, two circular triples.

One shape of the hundred and twenty-eight here, taken through every figure on its row.

One shape, worked through every rule

• line with a head: a pair compared; the head points at whoever came out behind
• no line: a pair nobody compared
• closed ring of three: a circular triple: no order can hold all three

One outcome drawn. Every row is the same number of candidates. Which candidate is which is the drawing's own choice and carries no meaning. A team translating this comparison into a limited operational pilot can review the tool beside the baseline before deciding which time and project records are genuinely necessary.

Five candidates, seven of their ten pairs compared, two triples circular. Every figure in this note is derived from those three integers below. For an independent perspective on comparison, evidence or decision design, consult FlowingData.

Computed from the three numbers alone. Who the candidates are and what any comparison was about is outside the record; the counts are of outcomes, not of anybody.

Pairs

Five times four, halved. Ten.

The only figure on the row that needs just one of the three numbers, and the only one nobody has ever got wrong.

Outcomes

Pick which seven of the ten pairs were looked at: ten choose seven, which is one hundred and twenty. Then settle how each of the seven came out: two to the seventh, one hundred and twenty-eight.

One hundred and twenty times one hundred and twenty-eight is fifteen thousand three hundred and sixty.

A binomial coefficient and a power of two, and two of the five figures are now checked.

The column for two circular triples

Two thousand four hundred and sixty, and no line of arithmetic yields it.

The ten triples of a five-candidate field share pairs with one another, so whether one closes bears on whether another can. No compact expression survives that, and the figure comes from producing every outcome of the page and counting.

What a reader can do without a machine is the addition. This page runs six thousand nine hundred, five thousand eight hundred and sixty, two thousand four hundred and sixty, one hundred and forty. They total fifteen thousand three hundred and sixty.

Share

Two thousand four hundred and sixty over fifteen thousand three hundred and sixty.

Both divide by twenty: one hundred and twenty-three over seven hundred and sixty-eight. Both divide again by three: forty-one over two hundred and fifty-six. Roughly one outcome in six.

Shares appear here as fractions of small integers because forty-one in two hundred and fifty-six is exact and sixteen per cent is not.

The column for outcomes admitting an order

Six thousand five hundred and forty, and this is the figure to handle carefully.

It falls below the six thousand nine hundred with no circular triple, and the difference of three hundred and sixty counts outcomes whose shortest ring runs to four or five links.

No line produces it either. What a reader can confirm is where it sits: never above the no-circle column, since every orderable outcome is free of circles, and never below the count that ranks the field, which here is two thousand four hundred.

Two thousand four hundred, six thousand five hundred and forty, six thousand nine hundred — three figures in the order the argument demands.

Two tests needing no arithmetic

A column count can never exceed its page total, because it counts part of that page.

And the columns of one page must total it, because each outcome has one number of circular triples and no other. Either test catches an error on sight, and neither needs anything beyond the row.

Matching the total settles nothing on its own. Failing to match it proves something is broken, and finding that out before publishing is why the test is run.

Which order to work in

Four of the five figures take a minute or two apiece. Two tests take seconds and belong first.

Set the column count against the page total; if it is larger, stop, the row is wrong. Add the page's columns and compare with the page total; a mismatch puts the fault on that page without saying which figure holds it.

Recompute afterwards, beginning with the page total, since a wrong page total makes every share on the page wrong too. Then the pair count, which is one multiplication. Then, given a machine, the enumerated columns.

When something fails

Five figures out of three integers localises a fault quickly.

A wrong page total means either the binomial coefficient or the power of two, and each can be redone in a line. A right page total with a wrong column puts the fault in the enumeration or the transcription, and the addition says which page.

If every figure survives and the row still looks wrong, arithmetic is not the problem. The shape is, or the sentence beside it.

The same five checks on a different row

Two candidates, their single pair compared, no circular triple.

One pair, from two times one halved. Two outcomes: one choice of which pair to look at, times two for how it came out. Both outcomes have no circular triple, so the column equals the page and the share is one. Both fall into an order, since two candidates with one comparison between them always do.

Every test passes trivially, and that is worth doing once. A row where the answers are obvious is how a reader confirms they have understood what the five figures are before trying a row where they cannot tell by looking.

The limit of all of it

None of these tests asks whether the shape suits the argument it was printed next to.

Recomputation shows a row agrees with itself. A panel may hold three faultless integers, five faultless figures drawn from them, and no relation at all to the surrounding sentences, and every test on this site will pass it.

Reading the caption against the prose is the only guard. That is why no panel here is allowed to go without one, and why a missing caption counts as a fault however perfect the arithmetic.

Near this one

• What This Record Is Not For
  Closing with the uses it does not support, the one argument it does make, and the single reason a counting pro
• Three Decisions Behind Every Figure
  Each of the three conventions enters the arithmetic at a particular line. Here is which line, and what the cou
• What No Count Can Reach
  Five exact figures from three whole numbers, and four questions they do not touch. Listing them is what makes
• Nothing and Nobody Is Named Here
  No person, organisation, product, place, trade or year appears in any sentence here. The rule reaches past pro