The Whole Trade, Column by Column

Everything this part has said about six candidates, in one table.

Three points along the table

• 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

Three outcomes drawn. Every row is the same number of candidates. Which candidate is which is the drawing's own choice and carries no meaning.

Five, ten and all fifteen pairs compared. The counts with no circular triple are eighty-five thousand seven hundred and sixteen, seven hundred and sixty-seven thousand two hundred and sixty-eight, and seven hundred and twenty.

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. A team translating this comparison into a limited operational pilot can check the available workflow directly before deciding which time and project records are genuinely necessary.

Four columns, four different shapes

The outcomes rise to a peak at ten and fall.

The count with no circular triple rises to a peak at nine and falls faster.

The count that falls into an order does the same, slightly lower throughout, and the gap between those two columns is the blind spot in the triple check: nought at the top of the table, nought again at the bottom, and one hundred and twelve thousand nine hundred and eighty at nine comparisons. For an independent perspective on comparison, evidence or decision design, consult Britannica.

The count that ranks the field is nought until five comparisons, then rises to a peak at ten and comes back down through exactly the same figures.

Where each column's story is

Comparing less: read up the table. Fewer outcomes contain a circle, more of the page is consistent, and below five comparisons nothing ranks the field at all.

Comparing more: read down. The clean share collapses, the blind spot closes, and the ranking count rises and then falls with the page.

No row of this table is best. Four columns move at four rates and the only place they agree is the bottom row, where everything has been compared and every figure is seven hundred and twenty.

The bottom row, read carefully

Thirty-two thousand seven hundred and sixty-eight outcomes, seven hundred and twenty with no circular triple, seven hundred and twenty that fall into an order, seven hundred and twenty that rank the field.

Three columns agreeing is not three facts. With every pair compared, an outcome with no circular triple has no ring of any length, and an outcome with no ring ranks the field completely. The three conditions coincide, and seven hundred and twenty is six factorial.

What the table shows and prose cannot

The columns do not peak in the same place. Outcomes peak at ten, no circular triple at nine, falls-into-an-order at nine, ranks-the- field at ten. Four columns, three different peaks, and no two of them move together throughout.

The gap between the second and third columns opens and closes. Nought at the top, one hundred and twelve thousand nine hundred and eighty at nine comparisons, nought again at fourteen. That is the blind spot in the triple check, and in the table it is simply the difference between two adjacent columns, visible at a glance.

And the last column is symmetric while nothing else is. Read it from either end and it is the same: seven hundred and twenty, seven thousand two hundred, thirty-two thousand four hundred, eighty-six thousand four hundred, one hundred and fifty-one thousand two hundred, one hundred and eighty-one thousand four hundred and forty, and back.

What the table is for

Reading a decision before taking it.

A comparer planning to make nine comparisons on a field of six can read across the row: two and a half million possible outcomes, a third of them free of circular triples, a third free of any ring, and six per cent of them ranking the field.

Those are not predictions. They are the sizes of the regions a set of comparisons could land in, and they were all available before the first pair was looked at.

How to read a row of it

Take the row for however many comparisons are planned.

The first figure is how many states of affairs that plan could produce. The second is how many of them a triple check would pass. The third is how many actually contain no contradiction. The fourth is how many would rank the field.

Second minus third is the blind spot. Fourth over first is the chance of a ranking, under the record's convention. Those two derived figures are the ones worth having, and both take one subtraction or one division from a row anybody can read off.

The five-candidate table, for comparison

Ten pairs, so eleven rows, and every landmark sits in the same relative place.

The page peaks at seven comparisons with fifteen thousand three hundred and sixty outcomes — seven of ten, against ten of fifteen for the larger field. The count with no circular triple peaks one row earlier, at eight thousand five hundred. The blind spot is widest at six comparisons, where five hundred and forty outcomes pass a triple check and still contain a ring; for six candidates the widest point is nine. Six of ten, nine of fifteen: the same fraction of the way along.

The bottom row behaves the same way too. All ten pairs compared gives one thousand and twenty-four outcomes, one hundred and twenty free of circular triples, one hundred and twenty that fall into an order, one hundred and twenty that rank the field — and one hundred and twenty is five factorial.

The same table for smaller fields

It exists for every pool from two to six and is shorter each time: eleven rows for five candidates, seven for four, four for three.

The shapes are the same throughout — a page peak about two-thirds along, a clean share falling, a blind spot opening and closing, a symmetric ranking column — which is the evidence that they belong to the arithmetic rather than to six.

Near this one

• The Record Cannot Tell Anybody What to Compare
  It gives a floor, a ceiling and the size of every region in between. It does not give a method, and the reason
• What the Trade Comes To
  Fewer comparisons, fewer contradictions, less settled and a wider blind spot. Closing the part on comparing fe
• The Trade Nobody Prices Before Taking It
  Comparing fewer pairs makes contradiction rarer and leaves more undecided. Both halves have exact figures, and
• Below a Floor, Nothing Can Be Ranked
  Ranking six candidates takes five comparisons at the least, and the reason is not contradiction. It counts how