What No Count Can Reach

The matrix above is a full account of one outcome. Which pairs were compared, and which way each of them went. Nothing is missing from it.

And it answers none of the four questions a reader is likely to arrive with.

Six candidates, eight pairs compared

• solid wedge: the row came out ahead of the column
• blank cell: the column came out ahead of the row
• small ring in both cells: the pair was never compared
• dash: the diagonal: nothing is compared with itself

Six candidates and 15 possible pairs. Eight of them compared. No circular triples. One of the 1647360 outcomes this page allows. To test the same alternatives in live work, reviewers can compare the available reports and agree the decision, access rules and review date in advance.

Row beats column. One of the one million six hundred and forty-seven thousand three hundred and sixty outcomes this page allows, and nothing in it says who anybody is or what any comparison was about. A broader reference for interpreting structured comparisons is available from igraph.

One of the outcomes this page allows, drawn in full. Which candidate sits in which row is the drawing's own choice; the counts are of outcomes, not of anybody.

Did any of this happen

No. There is no comparing on this site, no comparer, and no field of actual candidates anywhere in the table.

What the table holds is arrangements. The figure one million six hundred and forty-seven thousand three hundred and sixty says how many ways eight comparisons among six candidates can come out. It does not say that any of those ways has ever come out, or how often fields of six get compared eight ways, or how large fields usually are.

Turning a count of arrangements into a rate of occurrence needs an extra premise, and that premise is the third convention: every arrangement as reachable as every other. The convention is printed at the front, it describes nothing anybody has ever done, and the record states which figures depend on it.

Were the comparisons any good

Good requires a purpose, and the purpose went out with the first convention.

Take two rows from the same page, both reporting no circular triple. One came from a week of careful work on a field that really does line up. The other came from eight coin tosses that happened to fall into an order. Here they are the same row, down to the share.

This is the limit that bears hardest on how the figures get used. A tally of circular triples is not a tally of errors, and a page with few of them is not evidence that anybody did well.

Who came out ahead

The table knows how many comparisons were made and how many triples closed. Which candidate is which never enters it.

Shuffle the letters on any drawing here and every printed figure stays where it is. That is a real loss and worth stating plainly: being placed last matters to whoever is placed last, and three integers have no room for a name.

What did anybody think about it

Confidence, doubt, reluctance, intent — none of it reaches any figure in the table.

A comparer who settled a pair in a second and one who argued over it for an hour produce one row. So do the candidates, however they took the result.

This is the biggest gap and the least accidental. Anything that counts what people thought needs people in the record, and then the figures would no longer follow from three printed numbers, which is the one property the table has.

Five figures answering a narrow question, and four questions standing outside them. Keeping the four on the page is what makes the five safe to quote.

What is answered, at its real size

Three whole numbers go in — the size of the field, the number of pairs looked at, the number of triples that closed into a ring — and five figures come out, exactly.

The field's stock of pairs. The page's allowance of outcomes. The tally of those holding that many rings. The share of the page they amount to. And, for the page as a whole, how many of its outcomes can be laid out in an order.

That is the whole output. Not how to compare, not what to compare for, not whether a result deserves belief. Five figures, each recomputable from the three numbers printed beside it, each saying nothing about why anybody wanted it.

What the four gaps share

Every one of them needs either the candidates or the purpose, and both were thrown away before the first multiplication.

More counting will not bring them back. The gaps sit in the first convention rather than in the arithmetic, which is why that convention is stated before any figure appears, and why this note is in the record instead of being left for a reader to discover.

It also explains why the list has four entries and is not going to grow. A record that kept noticing fresh things it could not say would be a record whose edge nobody had located. These four have been the same four since the first page, and each traces to a decision a reader can look at.

Stating limits is not modesty

A record with its edges marked can be used where it fits and put down where it does not.

One with no edges marked gets used everywhere, because nothing on it says where to stop, and a figure will not object. Three in eight reads just as confidently in a sentence it has nothing to do with as in one it was computed for, and the reader cannot tell which sentence they are looking at unless the conventions are printed with it.

So the four gaps belong to the output. Take the five figures without them and the record has been misquoted, even if every digit is right.

The question it does answer, as narrowly as it can be put

Given a field size, a number of comparisons and a number of circular triples: how many of the possible states of affairs match that description.

Not how many of them occur. Not which of them matter. How many exist, under three stated conventions, across fields of two to six candidates.

A small question with an exact answer. That trade is made on every page here, and this note is the invoice.

Near this one

• Nothing and Nobody Is Named Here
  No person, organisation, product, place, trade or year appears in any sentence here. The rule reaches past pro
• Nothing Here Was Quoted From Anywhere
  Every figure on this site was produced from three integers. That removes one kind of error, introduces another
• There Is No Comparer in This Record
  Every figure counts outcomes that could exist. None of them describes anybody producing one, and the gap betwe
• The Arrowhead Carries the Meaning, Not the Colour
  A comparison is a line with a head on it. A pair nobody compared has no line at all. Both are geometry, and ge