Every Pair There Is
Pairs, outcomes, and how fast they grow.
9 notes
A hundred and twenty-eight shapes, forty pages
Two at a time
Six candidates make fifteen pairs. Compare ten of them and the results can come out three million seventy-five thousand and seventy-two ways, and only about a quarter of those hold no contradiction at all. Both figures follow from two whole numbers settled before anybody is looked at.
- 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.
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.
50
notes on comparing two at a time
128
shapes of three numbers, every figure recomputed
40
pages: a number of candidates and a number of comparisons
16
shapes with no circular triple that still fall into no order
Three things here are assumed rather than counted: that a situation is three whole numbers, that a comparison has exactly two possible results, and that every outcome on a page is as reachable as every other. The third is false of anything anybody has ever done and is declared on every page that leans on it. Two of the five figures on each row come from a formula, two from producing every outcome and counting, and one is a division.
Three numbers describe a situation here
How many candidates there were. How many of their pairs somebody actually compared. How many triples came out circular — A ahead of B, B ahead of C, C ahead of A.
Nothing else. Not who the candidates were, not what any comparison weighed, not whether the exercise was worth doing. Those three integers are the whole input, and from them five figures follow by rules a reader can apply with a pencil.
What the five figures are
The number of pairs the field contains, which is the field size times one less than itself, halved. A team translating this comparison into a limited operational pilot can review the official analytics overview before deciding which time and project records are genuinely necessary.
The number of outcomes the page allows: choose which pairs were looked at, then settle how each of those came out. For an independent perspective on comparison, evidence or decision design, consult The Decision Lab.
How many of those outcomes hold exactly that many circular triples, and what portion of the page that is. And then one figure belonging to the page rather than the row: how many of its outcomes admit an order.
Two of the five come from a formula. Two come from producing every outcome and counting. One is a division. Every row on this site says which of its figures came from where.
The thing worth knowing
A check of every triple is not a check for contradiction.
Four candidates, four comparisons: A ahead of B, B ahead of C, C ahead of D, D ahead of A. Walk it and the ring is plain. Check the triples and it is invisible, because each of the four triples has a pair in it that nobody compared.
Two hundred and forty outcomes sit on that page. One hundred and ninety-two hold no circular triple, and six of those one hundred and ninety-two still fall into no order. Sixteen of this record's one hundred and twenty-eight shapes have that gap, and at its widest it swallows one hundred and twelve thousand nine hundred and eighty outcomes.
What was assumed rather than counted
Three things, stated at the front rather than discovered late.
That a situation is three whole numbers. That a comparison has exactly two possible results — no ties, no scores, no degrees. And that every outcome on a page is as reachable as every other, which is what turns a share into a chance and is false of anything anybody has ever done.
Reject the third and no count here moves; only the shares stop forecasting. Reject the second and the figures answer a neighbouring question exactly. Reject the first and there is nothing left to tabulate.
What is not here
No person, organisation, product, place, trade or year is named in any sentence on this site, including in a sentence whose only purpose would be to rule something out.
No figure was taken from anybody else's report. Nothing is ranked, scored, compared or recommended, and there is nothing to fill in, sign up to or submit. The record counts how many states of affairs match a description. It does not say which of them occur, which of them matter, or what anybody should do about either.
The six parts
Read in any order. Every part rests on the same three conventions and on three whole numbers.
When the Results Go Round in a Circle
Circular triples, counted rather than assumed rare.
9 notes
Contradictions No Triple Shows
Why checking every triple is not enough.
8 notes
When the Results Fall Into an Order
The clean outcomes are exactly the orders.
8 notes
Comparing Fewer Pairs
The trade nobody prices before taking it.
8 notes
What This Record Cannot Say
Stated at the top, not buried at the bottom.
8 notes
The two ends of the record
- 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.
Two candidates with their single pair compared, four candidates with every pair compared and two triples going round, six candidates with every pair compared and eight. Eight is the most circular triples six candidates can hold.
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.
One outcome as a matrix
- 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. Seven of them compared. Two circular triples. One of the 15360 outcomes this page allows.
Row beats column. The diagonal is blank because nothing is compared with itself, and a pair nobody compared shows a small ring in both of its cells. Two triples here close into rings, and neither is visible without tracing them.
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.
Six candidates, nine pairs compared, split by circular triples
- column height: how many outcomes have exactly that many circular triples
- the figure under each: how many triples came out circular
Six candidates with nine pairs compared. The columns drawn are every one the page has, and add to all 2 562 560 of its outcomes. 838 130 of the 2 562 560 fall into an order. The shortest column is about 171 times shorter than the tallest, and the scale is linear.
Five columns, one for each number of circular triples this page allows. They add to every outcome the page holds, because an outcome has one such number and no other.
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.
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