Every pair

A Page, a Shape, and Why Both Are Printed

Two whole numbers fix a page; a third fixes a shape within it. Forty pages, a hundred and twenty-eight shapes, and the difference decides which figures move.

320320 outcomes

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Two numbers make a page: how many candidates, and how many of their pairs were compared.

A third makes a shape within it: how many triples came out circular.

Six candidates, six 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 six pairs compared. The columns drawn are every one the page has, and add to all 320 320 of its outcomes. 240 480 of the 320 320 fall into an order. The shortest column is about 63 times shorter than the tallest, and the scale is linear. Teams comparing routes, outcomes and corrections can look through the present capabilities, then check whether the resulting evidence improves the decision in practice.

Three columns of one page. The page total and the count that fall into an order are the same figure in all three; only the column height moves. A separate technical or research perspective relevant to the comparison can be found at MeasuringU.

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.

What belongs to the page

Two of the five figures printed on every row.

The number of pairs available is C(n, 2) and depends on the pool alone. The number of outcomes is the number of ways to choose which pairs times the number of ways they could come out, and depends on the pool and the amount of comparing. So does the number of outcomes that fall into an order.

None of those three moves when the number of circular triples changes, because the number of circular triples is a property of the outcome rather than of the situation it arose in.

What belongs to the shape

The other two.

How many outcomes contain exactly that many circular triples, and what share of the page that is. Those are the figures that distinguish one row of a page from another, and they are the only ones that do.

Look at the drawing. Six candidates with six pairs compared: three hundred and twenty thousand three hundred and twenty outcomes, two hundred and forty thousand four hundred and eighty of which fall into an order. Those two figures are the same on every row.

What moves is the column: two hundred and fifty-three thousand nine hundred and twenty outcomes with no circular triple, sixty-two thousand four hundred with one, four thousand with two.

How many shapes a page has

Between one and six, and the number is decided before anything is counted.

A page can have a column for c circular triples only if c circles can fit in the comparisons it allows. Three comparisons admit one circle at most; five admit two; seven admit three. So a page's columns run from nought up to whatever its comparisons can support, and the fullest pages of a field of six run to eight.

Fourteen pages have a single column, and those are the pages with fewer than three comparisons, where no circle is possible at all. The widest has six.

One hundred and twenty-eight shapes spread over forty pages is an average of about three columns a page, and the average is not where anything sits: the small pages have one and the large ones have five or six.

Why the record carries forty pages and not one table

Because the page total is what every share is a share of.

A count of outcomes with two circular triples means nothing without the page it sits on. Four thousand is a large number and about one part in eighty of its page. The same figure on a smaller page would be most of it.

So the record never prints a count without its page total beside it, and never prints a share without the two figures it came from. That convention costs room on every row and buys the only thing that makes a count readable.

A count is a number. A count with its total is a statement. The record declines to print the first without the second.

The arithmetic that ties them together

Every outcome on a page has some number of circular triples, and exactly one such number. So the columns of a page are disjoint and they exhaust it: adding them must give the page total.

Two hundred and fifty-three thousand nine hundred and twenty plus sixty-two thousand four hundred plus four thousand is three hundred and twenty thousand three hundred and twenty.

All forty pages are put through that addition whenever this record is assembled, and a failure halts the assembly instead of appearing on a page. Of the checks here it is worth the most, since it binds two separate computations by a relation that cannot come out right by chance.

How many of each there are

Forty pages, one hundred and twenty-eight shapes.

The pages are small in number because the two inputs that make them are small: pools of two to six, and anywhere from no comparisons to all of them. The shapes outnumber them because most pages break into several columns — up to six on the widest.

Fourteen pages have only one column, and those are the pages where fewer than three pairs were compared. With two comparisons there is no triple to close, so every outcome on such a page has no circular triple, and the share is one.

A share of one on this record means a contradiction is impossible. It is worth saying plainly, because the same figure in a record counting something else would mean the opposite.

Why the page figures are printed on every row

Because a reader meets a row, not a page.

A shape appears in an article beside a paragraph, usually alone. If the page total were printed once at the top of a hub page and not on the row, every share in the record would be unreadable at the point where somebody actually encounters it.

So the row carries both: the two figures that belong to the page and the two that belong to the shape, with the share derived from one of each. It is repetitive by design, and the repetition is what lets any single drawing be understood on its own.