Every pair

What the Outcome Count Leaves Out

Three million possible states of affairs, and not one of them is a statement about anybody. Closing the first part of the record.

at most 3075072 outcomes

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Two candidates and one comparison: two outcomes.

Six candidates with ten comparisons: three million seventy-five thousand and seventy-two.

Between those two rows lies everything this part of the record has counted, and it is worth being exact about what the counting was of.

The record's whole span

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. If several people repeat this comparison, managers may read the current platform outline while interpreting every signal beside completed work and participant feedback.

Two candidates with their single pair compared, five with seven pairs, six with ten. The smallest page in the record and the widest, with one in between. Readers examining the figures against practical research methods can also see dscout.

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.

What the figure says

How many states of affairs are possible, given a pool size and a number of comparisons.

That is a fact about two whole numbers. It is available before anybody is compared with anybody, it does not change while the comparing is going on, and it would be the same figure if the candidates were books, routes or machine parts.

What it does not say

Not which outcome happened. The record has no comparing in it, no comparer, and no account of how any real set of comparisons gets made. Every one of the three million is counted once and none is preferred.

Not whether the comparisons were any good. Good requires knowing what the comparing was for, which was left out of the three numbers at the start and cannot be smuggled back in.

Not who came out ahead. The record knows how many pairs were compared and how many triples closed into circles, and never which candidate is which. Shuffle the letters on any drawing here and not one figure moves.

Not how often any of this happens. The figures count possibilities under the printed convention that each outcome is as reachable as the next. Comparing done by anybody is nothing like that, and in what way it differs is outside everything here.

The one thing it does establish

That the space is larger than it looks, and that its size is settled by two decisions taken before anybody looks at a name.

How many candidates go in, and how many pairs get compared. Those two fix the number of possible states of affairs exactly — three million seventy-five thousand and seventy-two for six candidates and ten comparisons — and nothing done during the comparing alters it.

The size of the space is not a result of the comparing. Two decisions settle it, and both are commonly made on grounds that never touch the arithmetic.

The two figures a reader should be able to rebuild

Anybody who can produce these two without looking anything up has the whole of this part.

The number of pairs in a field of six: six times five, halved. Fifteen.

The number of outcomes when ten of those fifteen are compared: fifteen choose ten, which is three thousand and three, times two to the tenth, which is one thousand and twenty-four. Three million seventy-five thousand and seventy-two.

Two derivations, no table, and between them they account for two of the five figures on every row of this record.

What this part established

A pool of n candidates has C(n, 2) pairs: six for four candidates, ten for five, fifteen for six.

Every pair is in one of three states — not compared, compared one way, compared the other — so a pool of six admits fourteen million three hundred and forty-eight thousand nine hundred and seven states of affairs in all.

The record splits that total into forty pages by how many comparisons were actually made. Each page's total multiplies the available selections of pairs by two raised to the comparisons made, and the widest is six candidates with ten comparisons rather than fifteen, because the choosing and the landing pull in opposite directions.

Adding one candidate adds n − 1 comparisons and multiplies the space by two to that power. The comparisons grow like a square, the outcomes like two to the power of a square, and the outcomes that rank cleanly like a factorial — which against the other two is hardly growing at all.

And a single figure identifies almost nothing: twenty-one values in this record mean more than one thing, and even two figures together leave eight shapes indistinguishable.

What comes next

The third number, which neither of the first two can see.

Three candidates, three comparisons, and the possibility that the first came out ahead of the second, the second ahead of the third, and the third ahead of the first. How often that happens, how it behaves as the field grows, and why three comparisons are the fewest that can do it.

One figure to carry out of this part

Fifteen pairs, from six candidates.

Everything else in this record is built on that: the outcome counts are powers of two over subsets of those fifteen, the triples are the twenty ways to choose three of the six candidates, and the record's ceiling is where three to the fifteenth stops being enumerable.

A reader who remembers only that a field of six makes fifteen pairs has the hook for all of it, and the rest can be rebuilt from the row in front of them.