Six Outcomes, and the Smallest Case There Is

Four candidates with four of their six pairs compared.

Two hundred and forty possible outcomes. One hundred and ninety-two of them contain no circular triple. One hundred and eighty-six fall into an order.

Six do neither.

Four candidates, four pairs compared

• 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

Two outcomes drawn. Every row is the same number of candidates. Which candidate is which is the drawing's own choice and carries no meaning. Before adding another reporting layer, a project lead can see the official service description and test only the smallest useful setup on one representative process.

Both shapes of one page. The first row is the one hundred and ninety-two outcomes with no circular triple; six of those still fall into no order, and nothing in the figures says which six. For additional background on data, visualisation and evaluation, review material from The Data Visualisation Catalogue.

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.

Where the six come from

They are the four-step rings, and there are exactly six of them.

Four candidates can be arranged in a ring three different ways: the ring can pair the first candidate with the second and third, or with the second and fourth, or with the third and fourth. Each of those three choices leaves two pairs uncompared, which is exactly the four comparisons this page allows.

Then the ring can be traversed in either direction. Three times two is six.

Why this is the smallest case

Because a ring of four needs four candidates and four comparisons, and nothing smaller can do it.

Three candidates can only contradict themselves with a circular triple, which the third number counts. Four candidates with three comparisons cannot close a ring at all — three comparisons can close a ring only if all three fall on one triple, and then it is a circular triple and the record sees it.

So four and four is the floor, and below it the third number tells the whole truth. Above it, never again.

The figures on this page

Of the two hundred and forty outcomes: one hundred and ninety-two with no circular triple, forty-eight with one.

Of the one hundred and ninety-two: one hundred and eighty-six fall into an order, six do not.

Of the forty-eight with a circular triple: none falls into an order, because a circular triple is itself a ring and a ring of any length prevents an order.

So the page's count of outcomes that fall into an order is one hundred and eighty-six, which is neither of the two figures the third number offers.

One hundred and ninety-two have nothing wrong with any triple. One hundred and eighty-six have nothing wrong at all. The gap is six, and it is invisible to the only check most readers would think to run.

How rare six is, and why rarity is the wrong frame

Six out of two hundred and forty is one outcome in forty. Six out of the one hundred and ninety-two that pass the triple check is one in thirty-two.

A reader could decide that one in thirty-two is a risk worth ignoring. The record's objection is not that the rate is high; it is that the rate is not the point. The six outcomes are not unlikely versions of the other one hundred and eighty-six. They are a different kind of thing, and no amount of checking triples will ever find them.

On larger pages the share is bigger as well. Six candidates with nine comparisons: one hundred and twelve thousand nine hundred and eighty outcomes pass the triple check and fall into no order, which is about one in eight of the outcomes that pass it.

The same count on larger fields

Four comparisons on a larger field give the same kind of outcome and more of them.

Five candidates with four comparisons: thirty hidden outcomes. Six candidates with four: ninety. In both cases the comparisons form a four-step ring somewhere in the field and the remaining candidates are untouched.

The figures are six times the number of ways to choose four candidates from the field — six, thirty, ninety for fields of four, five and six. That is the one place in this part of the record where a hidden count has a formula, and it works only because four comparisons leave no room for anything else.

What the two rows show

Both rows are four candidates with four pairs compared, so both report two hundred and forty outcomes.

The first row's third number is nought circular triples: one hundred and ninety-two outcomes. The second's is one: forty-eight.

One hundred and ninety-two plus forty-eight is two hundred and forty, which is the page, and the drawing checks the addition rather than asserting it. What neither row can report is the six, because the six are not a different number of circular triples — they have the same number as the one hundred and eighty-six, which is none.

Why the smallest case is the one to remember

Because it is small enough to hold whole.

Four candidates, four comparisons, six bad outcomes out of two hundred and forty. A reader can picture the ring, count the three ways to draw it, double for the two directions, and have the entire result in mind without any table at all.

Every larger case is the same phenomenon with more room, and none of them can be held that way. If one figure from this part is going to survive being carried off the page, it should be the six.

Near this one

• 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
• Why Checking Every Triple Is Not Enough
  The triple is the natural unit of checking and it misses every ring longer than three. On the middle pages of
• Counting the Outcomes That Fall Into an Order
  The fifth figure on every row counts outcomes with no ring of any length. It is the only one in this record th
• Two Routes to the Same Number
  One method walks the comparisons, the other counts who could be first. They share no step and agree on all for