Figure

Every count is a choice of pins times a choice of place constraints

Drawn here at the parameters it defaults to, with every essay that calls it.

For each size and each number of pins, how many independent sets there are, beside what a direct sum predicts: the number of ways to choose that many pins from two, times the number of independent sets of PLACE constraints of the remaining size. All 12 agree exactly. That is a much stronger statement than a budget of 2n — it says the pins and the place constraints do not interact at all, so a picture's eight numbers are two separate accounts rather than one, and nothing can be moved between them.

It is one of 73 figures drawn by the same construction, and the drawing above is this one with nothing changed.

1 essay calls it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

A picture of four places, and the one constraint that is about the sheet. The best flat picture of four world cities, with the six distances it could be asked to hold drawn between the dots and the twelve bearings implied by them. Every one of those is a statement about how two places stand to one another, so every one is blind to where the whole picture sits on the paper. The cross is a constraint of a third kind: a stated position for one place — this dot goes here — which is two numbers and is blind to nothing. Adding both of them takes the largest independent set from 6 to 8.

The impossibility

Two numbers about the paper buy nothing about the places

A picture of n places has four freedoms, and a constraint holds only what its own blindness leaves: distances reach 2n − 3, bearings 2n − 3, the two together 2n − 2. Position is reached by neither, because nothing about how places stand to one another says where on the sheet to put them. A stated position does reach it, and two of them take the budget to 2n exactly. What the count conceals is that the extra two buy nothing: a largest picture holds six place constraints with the pins and six without, the same 7,596 of them, holding the same pairs at the same rates. The constraint system is a direct sum, and a pin is a different currency.

The whole library