Nothing in threes

A Circle That Takes Four Steps

A ahead of B, B ahead of C, C ahead of D, D ahead of A. Four comparisons that cannot be put in order, and not one of the four triples is circular.

240 outcomes

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Four candidates. The first came out ahead of the second, the second ahead of the third, the third ahead of the fourth, and the fourth ahead of the first.

Four comparisons, going round. No order can hold them.

Four candidates, four 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

Four candidates and six possible pairs. Four of them compared. No circular triples. One of the 240 outcomes this page allows.

Row beats column. Four comparisons close into a ring of four, and every triple in it is missing one of its three pairs, so the record counts no circular triple here at all. If several people repeat this comparison, managers may explore the available controls while interpreting every signal beside completed work and participant feedback.

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. Readers examining the figures against practical research methods can also see Our World in Data.

Why no triple is circular

Take any three of the four candidates. Among those three, one of the pairs was never compared.

The ring uses four pairs: first with second, second with third, third with fourth, fourth with first. The two pairs it does not use are first with third and second with fourth. So every triple contains one of those gaps, and a triple with a gap in it is not a circular triple — it is two comparisons that agree about nothing in particular.

The record therefore counts this outcome as having no circular triple. Nought. And the outcome still falls into no order.

This is the thing the record exists to point at

A check of every triple passes here.

That is not a failure of the check. It is a fact about rings: checking triples finds rings of three, and this ring has four links. To find it a reader has to follow the comparisons round, which is a different operation from looking at three candidates at a time.

Every triple is fine. The whole is not. The two statements are compatible and the second is the one that matters.

How many such outcomes there are

On this page — four candidates with four pairs compared — there are two hundred and forty outcomes.

One hundred and ninety-two have no circular triple. One hundred and eighty-six fall into an order. The six that do not are exactly the four-step rings: three ways to choose which pairs form the ring, and two directions to go round it.

Three times two is six. The smallest invisible contradiction in this record, counted by hand in one line.

The count of four-step rings has a formula

Unlike most figures in this part, this one can be written down.

Choose the four candidates on the ring: C(n, 4) ways. Arrange four candidates in a ring: three distinct rings, since fixing one candidate leaves three ways to pair the others and each ring is counted once. Choose a direction: two.

Six times C(n, 4). For four candidates that is six; for five, thirty; for six, ninety.

Those are exactly the hidden counts at four comparisons on each of those pages — six, thirty and ninety outcomes that pass a triple check and fall into no order. With exactly four comparisons there is nothing else a hidden contradiction could be.

Longer rings as well

Five comparisons can close a ring of five, six a ring of six. The record's largest field holds rings up to six links long.

Each of them is invisible to a triple check for the same reason: a ring of length k uses k pairs, and any three candidates on it have at least one pair the ring does not use. The longer the ring, the more gaps there are in each triple, and the less a triple check has to work with.

So the gap between "no circular triple" and "falls into an order" is not one special case. It is a whole family, it starts at four candidates and four comparisons, and it grows from there.

What the matrix shows

Row beats column. A filled wedge in a cell means the row's candidate came out ahead of the column's.

The diagonal is blank, because nothing is compared with itself. A pair that was never compared shows a small ring in both of its cells, and in this outcome there are two such pairs.

Follow the wedges and the four-step circle appears: each row with a wedge leads to the column it points at, which is a row with another wedge, and four steps later the path is back where it began. The matrix makes that easy to trace, which is one reason this record draws the same outcome two ways.

Why it is worth a part of the record

Because the triple is the natural unit of checking and it is not sufficient.

Three candidates are the fewest that can contradict each other, so a triple is where anybody looks first. The record's own third number counts triples. And the arithmetic says plainly that a column of noughts in that third number does not mean what a reader would reasonably take it to mean.

How often it does not mean that, exactly, is the next few notes.

What this costs the third number

Its claim to be a complete account of contradiction.

The third number of a shape counts circular triples, and the record prints it on every row because it is what the drawings show and what the obvious check would find. It is not a count of contradictions, and the record never calls it one.

The count of contradictions is the page total minus the count that falls into an order, and that is a different figure in a different column, derived a different way.