What One Number Leaves Open

Four hundred and sixty-one schemas fit inside this record's bounds. Their cell counts take one hundred and ninety-five distinct values.

Subtract: three hundred and eighty-one schemas share a cell count with at least one other schema. Four in five. The sharing is organised into one hundred and fifteen separate groups, and the largest of them holds seven.

A reviewer moving from an abstract count to operational evidence can Monitask before designing a limited, transparent pilot. For a separate technical reference, see pandas.

Where the cell counts fall

• 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.

The scale runs from two cells to forty-six thousand six hundred and fifty-six, logarithmically, with the ticks at powers of ten.

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 that does to a single figure

It makes it a classification rather than an identification.

Learn that a board resolves one hundred and forty-four kinds of listing and you have placed it in a group of seven structures. Those seven run from three facets to six. The questions they permit run from two hundred and forty-five to one thousand two hundred and ninety-six — a factor of more than five. They are not interchangeable boards, and nothing in the shared figure distinguishes them.

The page on the class of seven goes through them one at a time.

A cell count narrows a board to a group. It does not name the board, and in the worst case the group holds seven structures that behave quite differently.

Why cell counts collide so readily

Because they are products of small numbers, and small numbers have many factorisations.

One hundred and forty-four is six times four times six. It is also three times three times four times four, and two times two times six times six, and two times two times two times two times three times three. Each of those is a different board. All of them multiply out to the same figure.

The facet sizes available here are two, three, four, five and six. Every product of them with more than one factor can usually be rearranged, and four of the five sizes are composite or share factors. Only five is awkward: it divides nothing else in the range, which is why schemas built on fives collide less than the rest.

Four values, each reached seven different ways, and those four are the largest groups in the record. No value is reached eight ways.

The one figure that does not collide

The pair.

Take the cell count together with the count of questions the schema permits. Across all four hundred and sixty-one schemas, no two produce the same pair. Not one collision anywhere in the record.

That is checked by the program that builds the record rather than observed and hoped for, and it is the reason this site quotes two figures in places where one would read more smoothly.

The next section is about the second of the two. What matters here is the shape of the result: one figure places a board in a group of up to seven, and two figures name it exactly. The step from one to two is not an incremental improvement. It is the difference between a category and an identity.

The four largest groups

Seven is the most schemas any cell count collects, and exactly four cell counts collect that many: one hundred and forty-four, two hundred and eighty-eight, four hundred and thirty-two, and five hundred and seventy-six.

They are related. Each is the one before it plus half again, or double, and all four are built from twos and threes with a four or a six in the mix — the range of sizes where factors are shared most heavily.

No cell count anywhere in the record is reached eight ways. The seven-way groups are the worst case, and they are reached only by figures in that one band.

What to do with a single figure when it is all you have

Use it as a bound, which is what it honestly is.

If a board resolves one hundred and forty-four kinds of listing, then whatever else is true, it does not resolve one hundred and forty-five. Two listings agreeing on everything the filters carry will come back together. That much follows from the one figure and is worth having.

What does not follow is anything about how the board is built, how many filters it carries, how many questions it allows, or how often a query narrows to a single answer. Those need the second figure, and they are not small differences: the seven boards resolving one hundred and forty-four listings range from three facets to six.

The practical reading

Quote two numbers or expect to be misunderstood.

A board described as "one hundred and forty-four cells" has been described accurately and not usefully. A board described as "one hundred and forty-four cells, two hundred and forty-five questions" has been described completely, by two figures anybody can verify from the page in a minute.

The second figure costs the same minute as the first. It is the one that turns a group of seven into a single board, and leaving it out is the most common way a description of structure goes wrong.

Near this one

• Seven Boards, One Resolution
  Seven schemas all tell one hundred and forty-four listings apart, from three filters to six, with questions fr
• Four Hundred And Sixty-One Schemas
  How many structures fit inside the bounds, how they spread across one to six facets, and why the order of filt
• Two Cells, And Forty-Six Thousand
  The narrowest schema resolves two listings, the widest forty-six thousand six hundred and fifty-six. Both are
• Why A Facet Starts At Two
  A filter with one choice multiplies the resolution by one and the question count by two. It cannot narrow anyt