One more facet
When A Board Runs Out Of Moves
A board runs out of moves at six facets of six. Past the bounds the method still works and the statements about collisions do not.
One more facetAnalysis×k against ×(k+1)/k
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Inside this record, a board can add facets until it has six and widen them until each offers six choices. At that point it has forty-six thousand six hundred and fifty-six cells and no moves left.
That ceiling is a property of the bounds, not of boards. Real collections of listings carry filters with dozens of options and sometimes more than six filters. This page is about which parts of the record survive past the edge and which do not.
A short operational test is easier to interpret when teams first check the available controls and use one reporting window for everyone involved. For a separate technical reference, see PostgreSQL.
Three boards against the ceiling
- the upright: where this figure falls on the record's range
- the ticks are powers of ten; the scale is logarithmic
Three figures placed; the range is the record's own, over cells.
A board of two filters, the median of the record, and the largest schema the bounds allow, on the record's range of resolution.
Placed on the full range the record holds. The scale is the record's own, and the drawing refuses a scale that is not.
What carries over
The multipliers. A new facet of k multiplies the resolution by k at any size. A new value in a facet of k multiplies it by k plus one over k. Neither statement mentions a bound.
The two products. Cells are the sizes multiplied; questions are the sizes each plus one, multiplied. A board with nine filters of eleven choices has eleven to the ninth cells and twelve to the ninth questions, and both are computed the same way.
Commuting. Products do not care about order at any size.
The share. Cells over questions is still the product of size over size plus one, one fraction per filter.
What does not carry over
Every statement about collisions.
The claim that three hundred and eighty-one schemas share a cell count is about these four hundred and sixty-one schemas. The claim that no two share a pair is about the same set. Both were computed by enumerating the bounded range, and neither is guaranteed anywhere else.
Every statement about grain structure. The finding that only a five escapes the shared factors depends entirely on the available sizes stopping at six. With sizes up to eleven there would be sevens and elevens to work with, and the schemas that avoid collapse would be a different set with a different description.
| Claim | Holds past the bounds? |
|---|---|
| a new facet of k multiplies cells by k | yes |
| the share is the product of k over k+1 | yes |
| 381 schemas share a cell count | no — this set only |
| the pair is unique | no — this set only |
| only the five is coprime in range | no — depends on the range |
The method is general. The census is not, and every census figure in this record is a statement about four hundred and sixty-one schemas rather than about boards.
Why the bound was set at six
Checkability, as the page on the extremes says.
Six facets of six gives one hundred and seventeen thousand six hundred and forty-nine questions, which is the largest figure anywhere here. It can be verified by hand in a minute: seven to the sixth power. Push the bounds to nine and eleven and the figures become things to be taken on trust.
A record asking to be trusted rather than checked would be a different kind of document, and the whole of the method section is an argument for not writing one.
Why the census cannot simply be extended
It could be, and the result would be a different record.
Enumerating schemas with facets up to eleven and as many as nine facets produces hundreds of thousands of structures. The collision statements would have to be recomputed, and there is no reason to expect the same answers: more sizes means more primes, which means fewer collisions, which could well mean cell counts become unique and the central finding of this record dissolves at larger scales.
That would be worth knowing and it is not knowable from here. What can be said is that inside the bounds, where boards of modest size live, one figure is not enough — and modest boards are most boards.
What a board past the bounds should do
Compute its own figures and ignore the tables.
Multiply the sizes. Add one to each and multiply again. Write both numbers down. Every rate in this section applies, every comparison between two such boards works, and the only thing lost is the ability to look the board up in a census that does not reach it.
That loss is smaller than it sounds. The census exists to show that single figures collide and pairs do not — a point already made and already carried. A board outside the bounds has the same reason to quote both numbers as a board inside them, and the same minute's work to get them.
Near this one
- What No Edit Can Buy
Three things no edit reaches: a distinction with no filter, a result size off the ladder, and a reader's knowl - The Two Things A Board Can Change
Add a filter, or add a choice to one that exists. The only two moves, and one multiplies the resolution while - Adding A Filter
A new facet of size k multiplies the cells by k and the questions by k plus one. The largest edit, and the onl - Adding A Choice
One more option inside a filter multiplies the cells by k plus one over k. The cheap edit, and nearly always t