Comparing less

The Biggest Page Is Not the Busiest

Six candidates: the widest page is ten comparisons, not fifteen. Two factors pull against each other, and the product peaks two-thirds of the way along.

at most 3075072 outcomes

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Six candidates with ten of their fifteen pairs compared: three million seventy-five thousand and seventy-two outcomes.

With all fifteen compared: thirty-two thousand seven hundred and sixty-eight. Ninety-four times fewer.

Six candidates, either side of the peak

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. Teams comparing routes, outcomes and corrections can see which records can support the test, then check whether the resulting evidence improves the decision in practice.

Six, ten and fourteen pairs compared. The page sizes are three hundred and twenty thousand three hundred and twenty, three million seventy-five thousand and seventy-two, and two hundred and forty-five thousand seven hundred and sixty. A separate technical or research perspective relevant to the comparison can be found at NIST.

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.

Why the fuller page is smaller

Because a page counts two things multiplied together, and one of them runs out.

The count of possible selections of pairs is a binomial coefficient: one at no comparisons, six thousand four hundred and thirty-five in the middle, and back to one once every pair is in.

The number of ways the comparisons can come out is two to the number of comparisons: it doubles all the way along and never turns.

Their product peaks where the fall of the first overtakes the climb of the second, which for fifteen pairs is at ten.

The whole row for six candidates

By number of comparisons, the outcomes each page allows: one, thirty, four hundred and twenty, three thousand six hundred and forty, twenty-one thousand eight hundred and forty, ninety-six thousand and ninety-six, three hundred and twenty thousand three hundred and twenty, eight hundred and twenty-three thousand six hundred and eighty, one million six hundred and forty-seven thousand three hundred and sixty, two million five hundred and sixty-two thousand five hundred and sixty, three million seventy-five thousand and seventy-two, two million seven hundred and ninety-five thousand five hundred and twenty, one million eight hundred and sixty-three thousand six hundred and eighty, eight hundred and sixty thousand one hundred and sixty, two hundred and forty-five thousand seven hundred and sixty, thirty-two thousand seven hundred and sixty-eight.

Up to ten, then down. The two ends are tiny for opposite reasons.

The widest page in this record describes comparing two pairs in three. That is not a corner of the subject; it is the middle of it.

What the peak is not

A statement about where the comparing should stop.

The peak is where the most states of affairs are possible, which is a fact about counting rather than about work. Nothing follows from it about where to stop, and the figures that do bear on stopping — how many outcomes contain no ring, how many rank the field — peak in different places.

The order-settling count peaks at ten as well, for a reason that has nothing to do with this one: it is the number of orders times a binomial coefficient over the pairs left free, and that coefficient peaks in the middle of its own range.

The three rows

Six comparisons: three hundred and twenty thousand three hundred and twenty outcomes, two hundred and fifty-three thousand nine hundred and twenty with no circular triple.

Ten: three million seventy-five thousand and seventy-two outcomes, seven hundred and sixty-seven thousand two hundred and sixty-eight.

Fourteen: two hundred and forty-five thousand seven hundred and sixty, nine thousand.

Those first and last pages are within a third of each other in size and have almost nothing else in common: on the first, four outcomes in five contain no circle; on the last, one in twenty-seven.

The same peak on the smaller fields

Five candidates: the widest page is seven of the ten pairs, with fifteen thousand three hundred and sixty outcomes.

Four candidates: four of six pairs, with two hundred and forty.

Three candidates: two of three pairs, with twelve.

In each case the peak is about two-thirds of the way along, and in each case the fully compared page is far smaller — one thousand and twenty-four, sixty-four and eight.

So the shape is the same at every size the record holds, which is weak evidence that it is a property of the arithmetic rather than of the particular numbers. The strong evidence is the argument: a binomial coefficient times a rising power peaks to the right of the coefficient's own peak, always.

Why the record is organised by page at all

Because every share printed here is a share of a page total.

A count means nothing without knowing how much it is counted out of, and the page total is the only honest denominator. The record could have printed everything against the grand total of fourteen million three hundred and forty-eight thousand nine hundred and seven, and every figure would have become unreadable.

So the forty pages are the units, their totals are printed on every row, and nothing on this site is a share of anything else.

What the page sizes are not

A measure of how much comparing is going on.

A page with three million outcomes is not busier than a page with thirty-two thousand; it is less determined. The large pages are large because many different plans produce the same number of comparisons, and the small ones are small because almost no plan does.

So the peak at ten comparisons is a peak in ambiguity about which pairs were looked at, not a peak in effort. Reading it as the latter would be the natural mistake and would get the whole table backwards.