Every pair

How Many Comparisons There Are

Four candidates make six pairs. Six make fifteen. Both the number of comparisons available and the ways they can come out follow from one whole number.

at most 32768 outcomes

Home · Every pair · Definition

Put four candidates side by side and compare them two at a time. How many comparisons is that?

Six.

Five candidates make ten. Six make fifteen. Seven would make twenty-one, and the record stops before them.

Three sizes of field, every pair compared

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. Rows differ in how many candidates. Which candidate is which is the drawing's own choice and carries no meaning. The comparison described here can be extended with activity reports for process analysis, provided the team states what the resulting evidence can and cannot prove.

Each dot is one candidate; a line with a head is a pair that was compared, the head pointing at whoever came out behind. Every pair is compared in all three rows, and in each the figure on the right is how many of the outcomes leave no circular triple at all. A separate technical or research perspective relevant to the comparison can be found at The Open University.

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.

Where the figure comes from

Every pair, counted once. Choosing two out of n can be done in n(n − 1)/2 ways, which is what C(n, 2) means, and that is the whole derivation.

Four candidates: four times three, halved. Six. Six candidates: six times five, halved. Fifteen.

Nothing about the candidates enters. Who they are, what they were being compared for, whether the comparisons were careful or careless — none of it is in the figure, and none of it could be.

The second number, which is much larger

Six comparisons can come out sixty-four different ways, because each of the six has two possible results and they are independent.

Ten comparisons: one thousand and twenty-four. Fifteen: thirty-two thousand seven hundred and sixty-eight.

So four candidates, compared exhaustively, give a space of sixty-four possible verdicts. Six candidates give a space of over thirty-two thousand. The pool barely changed; the space multiplied by five hundred.

Why that gap matters

Because the comparisons are usually the thing that feels like work, and the space of outcomes is the thing that is never counted.

Going from four candidates to six adds nine comparisons — a morning, perhaps. It multiplies the number of ways those comparisons could come out by five hundred and twelve. One of those two figures is the one anybody would quote, and it is the smaller one.

The number of comparisons grows like the square of the pool. The number of ways they can come out grows like two to the power of that square.

Nothing is compared with itself, and no pair twice

Two restrictions do quiet work in every figure here.

A candidate is not compared with themselves, which is why the figure is n(n − 1)/2 rather than n squared. In a matrix of results this shows as a blank diagonal, and the blank is the whole explanation of the matrix's shape.

And a pair is compared at most once. Comparing the same two candidates twice and getting different answers is a real thing that happens and it is a different counting problem; this record has one result per pair or none.

Those two together are what make the count exact. Every pair appears once in the arithmetic, in one of three states, and nothing else about any pair is recorded.

The third number

Every drawing on this site carries a third figure that the two above cannot see: how many of the comparisons came out circular.

A circular triple is three candidates where the first came out ahead of the second, the second ahead of the third, and the third ahead of the first. Nothing is wrong with any single comparison. Together they cannot be arranged in any order at all.

In the first row of the drawing — four candidates, all six pairs compared — twenty-four of the sixty-four outcomes contain no circular triple. In the second, one hundred and twenty of the one thousand and twenty-four. In the third, seven hundred and twenty of the thirty-two thousand seven hundred and sixty-eight.

Three in eight, then about one in nine, then about one in forty-five. The proportion collapses, and the only thing that changed was how many candidates were in the room.

Not every pair has to be compared

The rows above all compare every pair, and that is the exception rather than the rule.

A pair can also simply not be compared. That is the third state of every pair in this record, and it is what makes the count larger than two to the power of the comparisons: each pair is not compared, or compared with one result, or compared with the other.

Three states, fifteen pairs, and a pool of six admits fourteen million three hundred and forty-eight thousand nine hundred and seven possible states of affairs. The record holds all of them, sorted into pages by how many comparisons were actually made.

Reading the drawing

Each dot is one candidate, placed on a circle. A line between two dots means that pair was compared, and the head of the line points at whoever came out behind.

No line means nobody compared that pair. That absence is the whole of the third state and it is drawn by drawing nothing, which is the only way an absence can honestly be drawn.

Where on the circle any candidate sits was settled by the drawing and means nothing. What the record holds is a count of candidates and a count of compared pairs; identities never enter it, and relabelling the dots leaves every figure exactly where it was.

What this part covers

How the two figures behave when a candidate is added and when fewer pairs are compared. The widest page in the record. Why the number of ways to choose which pairs to compare is counted at all. And how little a single figure identifies, which is the reason every row of every drawing here prints three whole numbers beside it.