Grain

Two Times Six, Three Times Four

Result sizes collide because the five facet sizes share factors. One of them shares nothing, and it is why any schema escapes at all.

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The five facet sizes in this record are two, three, four, five and six. Break them into prime factors and the problem is immediate.

Size	Factors
2	2
3	3
4	2·2
5	5
6	2·3

Four of the five are built from twos and threes. Only the five stands outside.

A clear baseline makes operational time tracking easier to evaluate because planned effort, actual time and delivered work can be compared over the same interval. For a separate technical reference, see fast.ai.

The collision that starts everything

Two times six is twelve. Three times four is twelve.

So on a board with filters of two, three, four and six, a query that leaves the two and the six alone returns a result of the same size as one that leaves the three and the four alone. Two genuinely different queries, two genuinely different sets of cells, one size.

That single identity — two sixes equal three fours — is responsible for a large share of the grain collapse in the record, because schemas containing those four sizes are common.

The board can tell the reader how many results there are. It cannot tell them which combination of unset filters produced that number, and a number alone does not say.

What repetition does on top

Repetition is worse than sharing, because it collides immediately rather than in combination.

Two facets of the same size give two subsets of size one with the same product. Three give three. Six identical facets give fifteen subsets of size two, all producing the same result size, and so on up — which is why six identical filters reach seven sizes out of sixty-four subsets.

Source of collision	Example	Severity
repeated size	2·2	immediate and total
shared factors	2·6 against 3·4	partial
no shared factors	5 against anything	none

Why the five is special

It is the only size in range that is prime and does not divide any other.

Three is prime but divides six. Two is prime and divides four and six. Five divides nothing here and nothing here builds it.

So any subset containing a five produces a product divisible by five, and no subset without one can match it. The five cleanly splits the subsets into two halves that cannot collide across the divide, and the collisions that remain happen only within each half.

That is why all three four-facet schemas with no collapse contain a five, and why the schema with the most reachable sizes in the record contains two of them.

What this would look like past the bounds

The scarcity of coprime sizes is an artefact of stopping at six.

With sizes up to eleven, there would be seven and eleven to work with as well, and schemas avoiding collapse would be easier to build. The structure of the argument would not change — products collide when factors are shared — but the specific statements in this record about how many schemas escape would not carry over.

The page on going beyond the bounds says this in general. It is worth repeating here because the grain figures are the ones most sensitive to where the bounds were drawn: the resolution and question counts scale smoothly past six, and the collision structure does not.

The collision table

Every collision among products of the five sizes traces back to a small set of identities.

Identity	Consequence
2·2 = 4	a four collides with two toggles
2·3 = 6	a six collides with a two and a three
2·6 = 3·4	two pairs give twelve
4·6 = 2·2·6	and so on upward

Everything else follows from combining these. There is no collision anywhere in the record that does not come from twos and threes being shared between four and six.

That is why removing a four or a six from a schema improves its grain more than removing anything else, and why the five never hurts.

The reading for somebody choosing filter sizes

Count the shared factors before counting the facets.

A board deciding between a six-value facet and a five-value one is, on resolution, choosing between six and five — a small difference. On grain it is choosing between a size that collides with half the board and one that collides with nothing.

That is the larger of the two effects, it is free, and it is invisible unless somebody factorises.