Grain

How Many Sizes Of Answer Exist

The size of a result is the product of the filters left alone. Count the distinct products for how many coarsenesses a board offers.

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Every result a board returns has a size, measured in cells, and that size is the product of the sizes of the filters the query left alone.

Set everything: the product is empty, so the result is one cell. Set nothing: the product is all the sizes, so the result is the whole board. In between, every subset of the filters gives a product.

Before adopting distributed workforce visibility, managers should write down why each field exists, who can see it and how a team member can challenge an inaccurate record. For a separate technical reference, see TensorFlow.

Count how many of those products are distinct and you have the number of different coarsenesses the board can offer. This record calls that figure the grain.

Subsets against reachable sizes

one bar is one facet; the divisions are its values
cells: how many listings the schema can tell apart
queries: how many different questions it lets anybody ask
sizes: how many different result sizes are reachable

Four schemas drawn.

The last figure is how many different result sizes each schema can produce. The second row reaches the most in the record; the first has the same number of subsets and reaches seven.

Computed from the schema alone. How many facets and how many values in each is a stated convention; the three figures are products of whole numbers.

Subsets are many, products are few

A board with six filters has sixty-four subsets of filters that might be left alone. It does not have sixty-four reachable result sizes, and usually has far fewer.

Schema	Subsets	Reachable sizes
2·3	4	4
4·6·6	8	6
2·2·2·2·2·2	64	7
2·3·4·5·5·6	64	39

The third row is the worst collapse in the record: sixty-four subsets producing seven distinct sizes. The fourth is the best six-facet performance: thirty-nine sizes from the same sixty-four subsets.

Both boards have six filters. A reader cannot tell them apart by looking at the sidebar.

Why products collide

Because different subsets multiply out to the same number.

On a board with filters of two, two and four, the subset holding the two toggles multiplies to four. So does the subset holding the four-value facet alone. Two quite different ways of leaving filters unset, one result size.

The more a schema's sizes share factors, the more often this happens. Facets that are all the same size are the extreme case: their subset products are just the powers of that size, and there are only as many powers as there are facets, plus one.

Six identical filters give seven result sizes, however large the filters are. Six of two and six of six both collapse to seven.

What grain is good for

Knowing how finely a reader can adjust.

Somebody looking at a result that is too large wants a smaller one. What is available to them is the next size down the ladder, and the ladder's rungs are the reachable sizes. A board with many rungs lets them land near what they wanted; a board with few makes every adjustment a large jump.

On a board of six toggles, the rungs are one, two, four, eight, sixteen, thirty-two and sixty-four cells. Somebody holding a result of thirty-two who wants about twenty has no move that gets there.

The range across the record

Grain runs from two to thirty-nine.

Two belongs to the five one-facet schemas: the only sizes they produce are one cell and the whole board.

Thirty-nine belongs to exactly one schema — six filters of two, three, four, five, five and six. Mixed sizes, few shared factors, and the subset products land nearly everywhere.

Twenty-seven schemas reach the ceiling of their own structure, with every subset producing a distinct product: the five single facets, ten schemas of two facets, nine of three, and three of four.

The three four-facet survivors are two, three, four and five; two, four, five and six; and three, four, five and six. Each contains a five, which shares factors with nothing else in range, and avoids at least one of the collisions the rest cannot.

No schema of five or six facets escapes. There are only five sizes available, they share factors heavily, and past four facets some subset always duplicates another's product.

Counting the products by hand

For a schema of three or four facets it is quick.

Write down one. Multiply everything so far by the first size and add the results to the list. Repeat for each remaining size. Count the distinct entries at the end.

For four, six and six: start with one; after the four, one and four; after the first six, one, four, six and twenty-four; after the second, add thirty-six, one hundred and forty-four, and the products with the earlier entries — ending at six distinct sizes from eight subsets.

Past four facets it is worth writing a few lines of code, which is what produced every grain figure in this record.

The figure worth computing

It takes longer than the other two — there are subsets to enumerate — but it answers a question the pair cannot.

The pair says what a board can distinguish and what it can be asked. The grain says how smoothly a reader can move between the broad answers and the narrow ones, and that is the part of using a board that most of the time is actually spent on.