Going round

The Page That Comes Out Symmetric

Four candidates fully compared split twenty-four, sixteen, twenty-four. No other page in the record does it, and nothing produced the symmetry.

64 outcomes

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Four candidates, every pair compared. Twenty-four outcomes with no circular triple, sixteen with one, twenty-four with two.

Twenty-four, sixteen, twenty-four. The first and last match, and the three columns read the same in either direction. To test the same alternatives in live work, reviewers can review the product with the team and agree the decision, access rules and review date in advance.

Four candidates, all six pairs compared

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

Four candidates with six pairs compared. The columns drawn are every one the page has, and add to all 64 of its outcomes. 24 of the 64 fall into an order. A broader reference for interpreting structured comparisons is available from Julia.

Three columns adding to all sixty-four outcomes the page allows. The outer two are equal, and nothing in the arithmetic required them to be.

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.

Why it is not a law

Because no other page does anything of the kind.

Five candidates fully compared: one hundred and twenty, one hundred and twenty, two hundred and forty, two hundred and forty, two hundred and eighty, twenty-four. Six candidates fully compared: seven hundred and twenty, nine hundred and sixty, two thousand two hundred and forty, two thousand eight hundred and eighty, six thousand two hundred and forty, three thousand six hundred and forty-eight, eight thousand six hundred and forty, four thousand eight hundred, two thousand six hundred and forty.

Neither reads the same both ways. The second is not even one-humped: it dips at five circles and rises again at six.

So the four-candidate page has something its two nearest relatives lack, and coincidence is the word for that.

Where the two twenty-fours come from

Two arguments sharing no step produce them, which is as clear a sign of accident as this subject offers.

The left-hand twenty-four counts outcomes with no circle. With every pair compared those are exactly the orders of four candidates, and there are four factorial of them. Twenty-four.

The right-hand twenty-four counts outcomes with two circles — the most four candidates can hold. Reaching that ceiling requires the wins spread as evenly as four candidates allow, which means two candidates ahead twice and two ahead once; counting the arrangements that do that gives twenty-four by a route with no factorial anywhere in it.

A factorial on one side, a count of near-regular arrangements on the other. Two routes to the same number with nothing in common.

When two arguments land on one number, that is not evidence they are describing one thing. More often it is evidence that the number is small.

The record has one other case like it

Five candidates fully compared report one hundred and twenty twice in a row, and then two hundred and forty twice in a row.

Those are adjacent columns rather than opposite ones, so the page does not look symmetric — it looks like a staircase. The equality is just as accidental and rather more likely to pass unnoticed, because a reader scanning a column of figures sees a repeated value as a pattern in the data rather than as two different things that happen to be the same size.

The trap this page sets

A reader who meets four-into-six first will carry the symmetry forward, and the record has a set of figures ready to reward that expectation on exactly one page and punish it on the rest.

The defence is the one used throughout this site: a pattern seen on a single page is not yet a pattern. Forty pages are available, and testing a claim against a second costs nothing.

What symmetry would have to mean

If a page's columns were symmetric for a reason, there would be a way to pair the outcomes off: take any outcome with no circle and produce exactly one with two, reversibly.

No such pairing is available here. The obvious candidate — reversing every comparison — does not work: reversing an order gives another order, so it maps the no-circle column onto itself and leaves the two-circle column exactly where it was.

Twenty-four outcomes rank cleanly and twenty-four are as tangled as four candidates get, and nothing sends one set to the other. That is what an unexplained agreement looks like, and the honest thing to do with it is to print it and say so.

Twenty-four, and what it means here

Twenty-four is the number this record is most careful with.

It is the count of outcomes with no circle on this page, the count with two circles on the same page, the number of orders four candidates can take, and the count of outcomes with five circles when five candidates are fully compared. Four different things, one figure.

A sentence containing twenty-four and no noun is not reporting anything.

How this one came to light

Not by catching the eye.

Every page's columns were listed and set against each other while the record was being assembled. One page turned out to have two columns reporting the same figure, and it was the same page whose columns happen to read alike in both directions.

A reader might have spotted it unaided, and the five-candidate page where two neighbouring columns agree as well, and would have had no way of judging whether either mattered. The count supplies that: one page in forty.

The habit it argues for

Try a claim on a second page before writing it down.

This record holds forty of them. A pattern read off one page costs nothing to test against another, and the test would have caught every coincidence in this record before it reached a sentence.