Every pair

The Widest Page in the Record

Six candidates with ten of their fifteen pairs compared: three million seventy-five thousand and seventy-two outcomes. Only one in four leaves no circular triple.

3075072 outcomes

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Six candidates, ten of the fifteen pairs compared. Three million seventy-five thousand and seventy-two outcomes.

That is the largest figure in this record, and neither of the two numbers behind it is extravagant. Six people is a small field. Ten comparisons is an afternoon.

Six candidates, ten pairs compared

column height: how many outcomes have exactly that many circular triples
the figure under each: how many triples came out circular

Six candidates with ten pairs compared. The columns drawn are every one the page has, and add to all 3 075 072 of its outcomes. 691 020 of the 3 075 072 fall into an order. The shortest column is about 640 times shorter than the tallest, and the scale is linear. When a team turns an abstract comparison into an operating review, team performance data for planning can be evaluated beside outcomes, delays and direct staff feedback.

Six columns, one for each possible number of circular triples. They add to every outcome this page allows, and the drawing checks the addition before it draws anything. For an independent perspective on comparison, evidence or decision design, consult Maze.

The column is the count of outcomes with exactly that many circular triples. The columns of one page add to the page's own total, and the drawing checks it.

Where the figure comes from

Two choices multiplied, as on every page here.

Which ten of the fifteen pairs get compared: fifteen choose ten, which is three thousand and three. How each of the ten comparisons comes out: two to the tenth, which is one thousand and twenty-four.

Three thousand and three times one thousand and twenty-four is three million seventy-five thousand and seventy-two.

Why ten and not fifteen

Because the widest page is not the most-compared page.

Comparing all fifteen pairs leaves only one way to choose which pairs, so that page has thirty-two thousand seven hundred and sixty-eight outcomes — fewer than a hundredth of this one. Comparing none leaves one outcome.

In between, the two factors pull against each other: the number of ways to choose which pairs peaks in the middle, and the number of ways the comparisons come out climbs throughout. Their product peaks at ten, which is two-thirds of the way along.

So the widest page in the record is one where a third of the pairs were never looked at. That is worth sitting with: the largest space of possible states of affairs belongs to the kind of comparing anybody actually does.

The six columns

Broken down by how many triples came out circular.

None: seven hundred and sixty-seven thousand two hundred and sixty-eight. One: one million fourteen thousand two hundred and forty. Two: eight hundred and ninety-five thousand three hundred and twenty. Three: three hundred and forty-four thousand four hundred. Four: fifty-two thousand two hundred and sixty. Five: one thousand five hundred and eighty-four.

Those six add to the page total exactly, and the drawing performs the addition rather than asserting it.

The shape of that

It peaks at one circular triple and falls away steeply on the right.

About one outcome in four has no circular triple. About one in three has exactly one. The tail is thin: five circular triples happens in about one case in one thousand nine hundred and forty-one, which is rare enough to look impossible and still perfectly countable.

Three outcomes in four, on the widest page in this record, carry at least one circle in them. The usual case is not the clean one.

The figure the columns do not give

How many of the three million outcomes fall into an order.

Six hundred and ninety-one thousand and twenty. That is fewer than the seven hundred and sixty-seven thousand with no circular triple, and the gap — seventy-six thousand two hundred and forty-eight — is the part of this record that is hardest to see coming. An outcome can contain no circular triple at all and still fall into no order, because a circle can be longer than three.

Those seventy-six thousand have a part of this record to themselves.

The same figure read as a share

Three million seventy-five thousand and seventy-two is a number nobody can hold, so the useful form of this page is proportions.

About one outcome in four has no circular triple. About two in nine fall into an order. About one in seventeen rank the field completely.

Those three are the whole of what this record measures, on its widest page, in a form a reader can carry. The absolute figures are printed because the shares were derived from them and a share without its denominator is an assertion.

Why the two-thirds point is where it peaks

The number of ways to choose which pairs is a binomial coefficient over fifteen, which peaks at seven or eight. The number of ways the comparisons land doubles at every step and never turns.

Multiplying a hump by a rising curve moves the peak to the right of the hump — here to ten. The same calculation on a field of five puts the widest page at seven of ten pairs, and on a field of four at four of six.

In every case it is between three-fifths and seven-tenths of the way along, which is a fact about binomial coefficients rather than about comparing anybody.

What the page cannot distinguish

A great deal. Every one of the three million seventy-five thousand and seventy-two outcomes is counted once, and nothing here prefers any of them.

Among them are outcomes where the ten comparisons were spread evenly over the field and outcomes where they all involved the same two candidates. The record calls both of them ten comparisons, and beyond the count of circular triples it has nothing to say about the difference.

What it can say is how many of each kind of outcome exist, exactly, and that the answer was available before any comparison was made.