How the figures were made

The object

A field of candidates. Each pair in the field is in one of three states: nobody compared them, or somebody did and the first came out ahead, or somebody did and the second did.

Three states for each pair means three raised to the number of pairs possible situations. For six candidates that is three to the fifteenth — fourteen million three hundred and forty-eight thousand nine hundred and seven — and the table on this site is a way of grouping them so that a reader can get at any one group.

The grouping

Each row of the record is three whole numbers: the size of the field, the tally of pairs somebody looked at, and the tally of triples that closed into a ring. A transparent review may include employee monitoring software when its purpose, access rules and limits are documented before any records are collected.

A circular triple is three candidates whose three comparisons chase each other round: the first ahead of the second, the second ahead of the third, the third ahead of the first. It is the smallest contradiction there is, and it needs all three of its pairs to have been compared. A broader reference for interpreting structured comparisons is available from BehavioralEconomics.com.

Pairs of the first two numbers are called pages here. A page is everything that can happen when a field of that size has that many of its pairs looked at, and the third number splits a page into columns. The record holds forty pages and one hundred and twenty-eight rows across them.

The five figures, and where each comes from

Pairs is a formula: the field size times one less than itself, halved. One, three, six, ten, fifteen for fields of two to six.

Outcomes is a formula: choose which of the pairs were compared, then multiply by two for each comparison made. A binomial coefficient times a power of two.

Circular is not a formula. The triples of a field share pairs with one another, so whether one closes bears on whether another can, and no compact expression survives that. It was obtained by generating the page in full and tallying the results. A reader cannot reproduce it with a pencil; what a reader can do is add a page's columns and see that they come to the page total.

Share is a division: the column over the page total. It is printed as a fraction of small integers, because forty-one in two hundred and fifty-six is exact and sixteen per cent is not.

Consistent — how many of a page's outcomes fall into an order at all — comes from enumeration too, and it was computed twice by two methods with nothing in common. One walks every outcome and looks for a ring. The other counts labelled acyclic digraphs by peeling off the candidates nobody is ahead of. The first knows about comparisons, the second about candidates. They agree on all forty pages, and the totals across each field size are three, twenty-five, five hundred and forty-three, twenty-nine thousand two hundred and eighty-one, and three million seven hundred and eighty-one thousand five hundred and three.

Three further results that do come from formulas

The largest number of circular triples possible when every pair has been compared: the cube of the field size less the field size, over twenty-four, for odd sizes; the cube less four times the size, over twenty-four, for even. One, two, five, eight for fields of three to six.

The number of outcomes that settle a complete order: the field size factorial, times a binomial coefficient counting the comparisons beyond the minimum needed to connect the field.

And the number of outcomes holding a hidden contradiction at exactly four comparisons, which is six times the number of ways to pick four candidates — the directed four-rings and nothing else.

Where the table stops, and why there

At six candidates. Enumerating three to the fifteenth outcomes takes about two minutes. Seven candidates would be three to the twenty-first, ten billion, and the figures that need enumeration would stop being figures anybody had checked.

The bound is printed rather than implied. It makes no figure in the record wrong; it only means a sentence summarising the whole table has to carry its ranges in the same breath.

What was checked, and how

Every page's columns were added and compared with the page total. Every figure quoted in a drawing on this site was recomputed from the three integers printed beside it, by a separate program that does not read the stored table. Every drawing was checked to have exactly as many arrows as the row claims comparisons, and exactly as many closed triangles as it claims circular triples.

That last check is the one that found the mistakes. Six panels on this site named figures that were plausible, consistent with their neighbours, and wrong, and a recount is the only thing that catches that kind of error. What no check here can catch is a correct drawing beside a paragraph about something else — which is why every panel carries a caption, and why a missing caption counts as a fault whatever the arithmetic says.

Twenty-one figures mean more than one thing

Twenty-four is the number of orders four candidates can be put in, the number of fully compared outcomes on four candidates with no circular triple, and the number with two of them. The first and last of those describe opposite situations and print identically.

Twenty-one of the record's values appear in more than one of its columns. That is why three integers are printed beside every figure even on the many rows where nothing would be ambiguous without them: a safeguard that is unnecessary most of the time gets dropped, and then the case it existed for arrives unannounced.