One Outcome, Taken Through Both Checks

The matrix at the head of this note is one outcome from a page of thirteen thousand four hundred and forty.

Five candidates, six of their ten pairs compared, and not one circular triple in it. This note runs both checks on it by hand.

Five candidates, six pairs compared

• solid wedge: the row came out ahead of the column
• blank cell: the column came out ahead of the row
• small ring in both cells: the pair was never compared
• dash: the diagonal: nothing is compared with itself

Five candidates and ten possible pairs. Six of them compared. No circular triples. One of the 13440 outcomes this page allows. To test the same alternatives in live work, reviewers can compare the setup with the written criteria and agree the decision, access rules and review date in advance.

Row beats column. Of the thirteen thousand four hundred and forty outcomes this page allows, eight thousand five hundred have no circular triple — and seven thousand nine hundred and sixty of those fall into an order. A broader reference for interpreting structured comparisons is available from Yale University.

One of the outcomes this page allows, drawn in full. Which candidate sits in which row is the drawing's own choice; the counts are of outcomes, not of anybody.

The triple check

Five candidates have ten triples. Take each in turn.

A triple can only be circular if all three of its pairs were compared, and six comparisons out of ten leave four pairs untouched. Most triples therefore have a gap and are settled at once: nothing to check.

For the triples with all three pairs compared, read the three results. If two of them point the same way round the triangle and the third points back, that triple is circular. None of them does.

The check passes. The count of circular triples is nought, which is what the drawing reports.

The walk

Pick any candidate and follow the comparisons.

Step from a candidate to anybody they came out behind. Keep stepping. Note each candidate as it is passed.

If the walk reaches a candidate it has already passed, the comparisons contain a ring, and a ring of any length means no order can hold them. On this outcome the walk closes after four steps.

Two checks, two answers. The first says nothing is wrong; the second finds a ring of four.

Why the first was not careless

Because the ring of four has no triple to be found in.

Take the four candidates on the ring. Any three of them leave out one of the ring's links and include one pair the ring does not use. If that pair was compared it might close a triple — but on this outcome it was not, which is exactly why the ring was available to form in the first place.

Long rings need gaps, and a gap is precisely what stops a triple from being checkable. The two facts are the same fact.

The check and the contradiction need the same thing: an uncompared pair. Where one is possible the other is invisible.

What the page says about how often

Eight thousand five hundred of the thirteen thousand four hundred and forty outcomes have no circular triple. Seven thousand nine hundred and sixty of those fall into an order.

Five hundred and forty do not. That is about one in sixteen of the outcomes that pass the triple check, on a page where six of ten pairs have been compared.

Doing this without a drawing

The walk needs no matrix and no record.

Write the comparisons down as they are made: an arrow from the candidate who came out ahead to the one who came out behind. When the comparing stops, start anywhere and follow arrows. If you come back to a name you have already written, stop — the comparisons cannot be ordered, and the names you passed are the ring.

That takes about as long as there are comparisons. Examining all ten triples of a field of five takes longer and finds less.

Why the walk is the right shape of check

It follows the thing that makes a contradiction.

A contradiction is a path that comes back. Looking at triples is looking at a fixed-size piece of a path, and a path can be longer than any fixed size. So a check built on triples is a check with a length limit, and the limit is three.

The walk has no limit. It follows paths as far as they go, which is exactly as far as a contradiction can hide.

That is the general lesson and it is worth more than the counts: when something is defined by a chain of any length, a check that inspects fixed-size pieces will miss some of it, and the number of pieces it inspects does not help.

What the record adds

Only the counting.

The walk is older than this record and needs none of it. What three whole numbers buy is the scale of the thing: how many of the possible outcomes have a ring, how many have one that a triple check would miss, and how both figures move as the comparing goes on.

Those are not available from any single set of comparisons, however carefully it is examined. They are available before the first comparison is made.

What a reader should take from this page

The walk, and nothing else.

The figures are here because the record is a record of figures, but the usable part of this note is the procedure: write each comparison as an arrow, then follow arrows from any starting point. Come back to a name and the comparisons cannot be ordered.

It needs no table, no field size and no count of anything. It works on fields larger than this record covers, and it is the only part of this site that is any use while the comparing is still going on.

Near this one

• What the Gap Costs
  On the middle pages of this record, one outcome in eight passes the obvious check and settles nothing. Closing
• A Circle That Takes Four Steps
  A ahead of B, B ahead of C, C ahead of D, D ahead of A. Four comparisons that cannot be put in order, and not
• Six Outcomes, and the Smallest Case There Is
  Four candidates, four comparisons: one hundred and ninety-two outcomes hold no circular triple, and six of tho
• Sixteen Shapes Where the Third Number Omits
  On sixteen of this record's hundred and twenty-eight shapes the count of circular triples is nought and some o