Four hundred and five shapes

What a list of requirements admits

Every item that must be met halves the intake. Every item that need not be sits under a rule which is usually unwritten. The median shape in this record admits one profile in thirty-nine.

Start at the halving What the rule does

filled: a requirement that must be met
open: one of the optional ones
area of the inner square: the share admitted
a stroke instead: too small for this panel to draw

Six shapes drawn; two of them admit too little for the square to be drawn, and carry a stroke instead.

Computed from the shape of the list. The shape is a stated convention; the count of admitted profiles is arithmetic on subsets.

notes on what a list of requirements admits

405

shapes of list, every one of them computed

the median shape admits one profile in this many

152

shapes admit less than one in a hundred

One thing here is invented: that each item on a list is counted as a separate coin, which fixes every denominator at two to the power of the number of items. It is a unit of measurement and not a claim about anybody. Every other figure is counted, and comes apart under a pencil.

The question

Somebody writes a list of requirements. Eight items, say — four that really matter and four that would be nice. Then a rule, usually unstated: all of the first four, and two or three of the rest.

What fraction of possible candidates does that list admit?

Almost nobody computes it. The list gets written by adding items that each seem reasonable on their own, and the shape it ends up with is whatever those additions happened to produce. The result is a filter whose selectivity nobody chose and nobody knows.

Why the arithmetic is worth doing

Because the answer is not intuitive in either direction, and the errors run both ways.

Before turning the list into a live operating rule, readers can see the current platform overview and identify which records would genuinely help a decision. For an independent technical reference, see Zapier.

A list of eight compulsory items admits one profile in two hundred and fifty-six. That is far narrower than most people picture when they write the eighth line, and each line looked harmless as it went in.

A list of eight optional items needing four of them admits about one in six. Longer than the first list, far wider. The length of a list says almost nothing about what it admits; the rule attached to it says almost everything — and the rule is the part that usually goes unwritten.

The finding that matters most

Twenty-one different shapes in this record admit exactly the same share: one profile in two hundred and fifty-six. They run from eight items to sixteen.

A list twice the length of another can be precisely as selective. So length cannot be read as rigour, and rigour cannot be read from length — not approximately, not as a rule of thumb. The only thing that tells you what a list admits is the arithmetic.

The limits, stated at the top rather than the bottom

The calculations remain the core of this record. Practical platform guides are maintained separately, use stated workflow scenarios and link to current official product information for verification.

All of that moves. A page carrying it keeps its confident tone long after the facts beneath it have shifted, and nothing in the text ever signals the moment it stopped being right — so the reader acts on it anyway. Subsets do not move. That is the whole reason this site is about subsets.

And it makes no claim about people. Every figure here counts subsets — each requirement treated as a separate coin — which is a stated way of measuring, not a description of any population. The median shape in this record admits one in thirty-nine, and that is a fact about filters, not about anybody's chances.

What is left is small and durable: given a list and a rule, exactly what gets through. Four hundred and five shapes, all computed, none estimated.

The six parts

Read in any order. Every part stands on the same one convention and on binomial coefficients.

Every item halves what is left

A list whose items must all be met halves its own intake with each one added.

Part one · 9 notes

How many of them

The rule attached to a list decides far more than its length.

Part two · 9 notes

Equal length, unequal filter

Two lists of the same length can differ by a factor of two hundred and fifty-six.

Part three · 8 notes

What one change is worth

Eight single edits, measured against the same starting list.

Part four · 8 notes

Different lists, one filter

Twenty-one shapes admit exactly one profile in two hundred and fifty-six.

Part five · 8 notes

What the arithmetic cannot say

Every figure here counts subsets. None of them counts people.

Part six · 8 notes

Six optional items, every rule that can sit over them

Seven thresholds drawn. The tread is flat because the threshold is a whole number: there is nothing between one item and the next. A tread at the top admits everybody, which is a list that filters nothing at all.

A filled dot is an item that must be met, an open one an item that need not be. The staircase is flat at both ends and steep through the middle, which is where every real rule sits.

Computed from the threshold. Each item is counted as a separate coin: this is the unit of measurement, not an estimate of any population.

Four of the twenty-one shapes that admit one profile in two hundred and fifty-six

Four shapes drawn, running from 8 items to 16. They are not variants of one list; they are different lists that happen to be the same filter.

Eight lines, eight lines, twelve lines and sixteen lines. One filter. The last is a sixteen-item list whose second half changes nothing at all.

Computed separately for each shape shown. They agree because the arithmetic makes them agree, not because they were chosen to.

Elsewhere on this site

Practical software guides

Three ways to compare connected-work platforms

The guides use different structures and list sizes: an eight-platform field guide, a twelve-tool comparison matrix and a fifteen-platform decision catalog.

8 platforms · field guide