Sixteen Shapes Where the Third Number Omits

The record holds one hundred and twenty-eight shapes. On sixteen of them the third number is nought while the page's orderable count comes out below that shape's own.

Those sixteen are where a triple check passes and the comparisons still cannot be ordered. A transparent review may include project time data for comparing processes when its purpose, access rules and limits are documented before any records are collected.

Three of the sixteen, smallest to largest

• 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. Rows differ in how many candidates. Which candidate is which is the drawing's own choice and carries no meaning. A broader reference for interpreting structured comparisons is available from Visual Capitalist.

Each row counts outcomes with no circular triple at all. Six of the first row's outcomes fall into no order, five hundred and forty of the second's, and one hundred and twelve thousand nine hundred and eighty of the third's.

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.

Every one of them

Four candidates with four pairs compared: one hundred and ninety-two clean of triples, one hundred and eighty-six orderable, six short.

Five candidates, by number of comparisons. Four: thirty short. Five: two hundred and sixty-four. Six: five hundred and forty. Seven: three hundred and sixty. Eight: sixty.

Six candidates. Four: ninety short. Five: one thousand seven hundred and sixty-four. Six: thirteen thousand four hundred and forty. Seven: forty-eight thousand nine hundred and sixty. Eight: ninety-seven thousand seven hundred and seventy. Nine: one hundred and twelve thousand nine hundred and eighty. Ten: seventy-six thousand two hundred and forty-eight. Eleven: twenty-nine thousand five hundred and twenty. Twelve: six thousand one hundred and twenty. Thirteen: five hundred and forty.

Where the sixteen are not

At the ends.

With fewer than four comparisons there is no room for a ring of four, so the third number tells the whole truth. With fourteen or fifteen comparisons out of fifteen it tells the whole truth again, and that second fact is the interesting one.

When nearly every pair has been compared, a long ring cannot avoid producing a short one. Take a ring of four in a field where every pair is compared: the pair joining the first and third candidates has a result, and whichever way it goes, it closes a triple into a circle. So on a fully compared page the triple check is sufficient after all.

The triple check is safe when almost nothing has been compared and safe again when almost everything has. It fails in the middle, which is where all the comparing happens.

The worst of them

Six candidates with nine pairs compared: nine hundred and fifty-one thousand one hundred and ten outcomes with no circular triple, eight hundred and thirty-eight thousand one hundred and thirty that fall into an order.

One hundred and twelve thousand nine hundred and eighty outcomes pass the triple check and are not orderable. That is about one in eight of the outcomes that pass.

Nine comparisons out of fifteen is three-fifths of the way through the work. It is where a comparer would plausibly stop.

How the shape of it moves

The count of hidden cases rises and then falls with the amount of comparing, and the peak is near the middle.

For six candidates it runs ninety, one thousand seven hundred and sixty-four, thirteen thousand four hundred and forty, forty-eight thousand nine hundred and sixty, ninety-seven thousand seven hundred and seventy, one hundred and twelve thousand nine hundred and eighty, seventy-six thousand two hundred and forty-eight, twenty-nine thousand five hundred and twenty, six thousand one hundred and twenty, five hundred and forty.

Up by a factor of twenty, then of seven, then of three and a half, then of two, then by a tenth — and then down, steeply, as the comparing fills in the gaps that let long rings hide.

The three families inside the list

The sixteen are not one phenomenon repeated.

At exactly four comparisons the hidden outcomes are four-step rings and nothing else, and there are six times C(n, 4) of them: six, thirty, ninety.

In the middle of each pool's range the hidden outcomes are rings of four, five and six mixed together with any number of untouched pairs, and no formula covers them.

At the top of each pool's range — fourteen and fifteen comparisons for a field of six — there are none, because a long ring in a nearly complete field always drags a short one along with it.

So the list has a floor with a formula, a middle that had to be enumerated, and a ceiling at zero. Reading it as a single curve would miss that its two ends are there for different reasons.

What to do with the list

Not very much, and that is worth saying.

The list does not say which outcomes are the hidden ones, because the record counts rather than identifies. What it establishes is the size of a blind spot in the obvious check, on every page of a small table, exactly.

A reader with a field of six and nine comparisons made now knows that passing a triple check leaves about an eighth of the remaining possibilities unaccounted for. What to do about that is the subject of the next note, and the answer is short.

Why the list is printed rather than summarised

Because the summary would be a single share and the share moves by a factor of twenty across the list.

One in thirty-two at the smallest shape, one in eight at the largest, and everything between. A reader given a mean would have a figure that describes no page in the record and would have no way to know which page theirs resembles.

Sixteen lines of figures is more to read and it is what the record has to offer. Each line is a page somebody might be on.

Near this one

• 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
• One Outcome, Taken Through Both Checks
  Five candidates, six comparisons, no circular triple anywhere. The triple check passes and the walk finds a ri