Grain

Two Rungs, Or Thirty-Nine

The ladder of reachable result sizes runs from two rungs to thirty-nine. The schema at the top does not look special from the sidebar.

GrainReference2 to 39

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The number of different result sizes a schema can produce runs from two to thirty-nine.

Two belongs to the five single-facet schemas. A result is either one cell or the whole board.

A manager can review the project tracking approach, run one complete cycle and invite the team to challenge any misleading interpretation. For a separate technical reference, see Hugging Face.

Thirty-nine belongs to exactly one schema in the record: six filters of two, three, four, five, five and six choices.

Grain across the record

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 result sizes.

Placed on the record's own range of reachable result sizes. The six-identical-filter case sits near the bottom despite having the most subsets available to it.

Placed on the full range the record holds. The scale is the record's own, and the drawing refuses a scale that is not.

The schema at the top

Three thousand six hundred cells, fifteen thousand one hundred and twenty questions, and thirty-nine reachable sizes from sixty-four subsets.

Nothing about it stands out. Its resolution is well above the median of eight hundred and sixty-four but nowhere near the ceiling of forty-six thousand six hundred and fifty-six. Its question count is middling. In a sidebar it is six controls, like a great many other schemas.

What distinguishes it is that its facet sizes are nearly all different — four distinct sizes across six facets, with only the five repeated.

Schema	Facets	Cells	Sizes
2·3·4·5·5·6	six	3600	39
2·2·2·2·2·2	six	64	7
6·6·6·6·6·6	six	46656	7

Three six-facet boards. One offers thirty-nine positions on the ladder of coarseness; the other two offer seven each.

Why nearly all different, rather than all different

All six sizes different is impossible: there are only five sizes in the record. So six facets must repeat at least one, and the best available is to repeat exactly one.

Which one to repeat matters. Repeating the five is best, because five shares no factors with the rest — two fives give a twenty-five that nothing else can produce, and every subset containing exactly one five is distinguishable from the subsets containing none or both.

Repeating the two would be worse: twos are already present inside the four and the six, so an extra two creates products that collide with existing ones.

With six filters and five sizes available, one size must repeat. The record's best grain comes from repeating the one that shares nothing with the others.

The shape of the distribution

Grain is not spread evenly. The commonest value is twelve, reached by sixty-four schemas; twenty-four is reached by fifty-three, and sixteen by forty-three. Most of the record sits between eight and twenty-four.

Beyond thirty the record thins sharply. Eleven schemas reach thirty-two, seven reach thirty, and above that the counts fall to single figures: one schema reaches thirty-four, one reaches thirty-eight, and one reaches thirty-nine.

The reason is the same shortage of coprime sizes. Getting many distinct subset products requires facet sizes that do not share factors, and the available sizes share factors heavily: two divides four and six, three divides six, and only five stands apart.

What a reader gains from more rungs

Narrowing that lands closer to what they wanted.

On seven rungs the jumps are large and fixed. On thirty-nine the reader has many more intermediate positions, so a query that is too broad can be tightened by a modest amount rather than by a whole factor of six.

It is the difference between a dial and a switch, and it costs nothing: both boards have six controls, and the one with more rungs also resolves more than fifty times what the toggle board resolves.

The schema one rung down

Thirty-eight is reached by exactly one schema as well, and the pair of them shows how fine the margin is.

A single change to the facet sizes moves a board between thirty-nine rungs and thirty-eight, or down to the low twenties, without affecting the sidebar's appearance at all. The figures that readers can see — the number of controls, the number of options in each — move hardly at all across that range.

So grain is the figure most sensitive to choices that look arbitrary, and the one least visible from the page. It is also, unlike the other two, not recoverable from the pair: two boards with identical resolution and identical question counts have identical schemas, so identical grain — but a board described only by its resolution could have any grain its class allows.

The one practical conclusion of this section

When choosing facet sizes, prefer sizes that do not share factors.

It is free. It does not reduce resolution, it does not reduce the question count, and it does not add controls. It changes only how many distinct coarsenesses the board can produce, and on a six-facet board that figure can be seven or thirty-nine depending on a choice nobody is currently making deliberately.