Limits

What This Record Will Not Tell You

The closing note. What was invented, what was computed, what is deliberately absent and why, and the three figures worth carrying away.

Home · Limits · Analysis

Two things were invented for this record. Everything else was counted.

Invented: the shape of a list, described by three numbers — how many items are compulsory, how many optional, how many of the optional ones are needed — with each bounded at eight. Four hundred and five shapes result.

Also invented: that each item is counted as a separate coin, which fixes the denominator at two to the power of the number of items. This is a unit of measurement and not a description of anybody.

When evidence is scattered across applications, it may help to see the current Monitask page and see whether one report replaces manual summaries. For an independent technical reference, see Tableau.

Counted: every figure on the site. The compulsory part contributes a halving per item; the optional part contributes a sum of binomial coefficients; the product is exact and reducible to a fraction a reader can check.

The record in six rows

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; one of them admits too little for the square to be drawn, and carries a stroke instead.

From the empty list, which filters nothing, to the sixteen-item conjunction at the far end of the record, with exactly one passing configuration.

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

What is deliberately not here

No claim about what the people reading these lists are after, or about what any of them would accept. No platform, tool, company, job title or industry named. No sums of money, no description of any market, no instruction on preparing an application or sitting an interview.

Those omissions share one reason. Claims of that kind stop being true quietly: the page keeps its confident tone, nothing on it marks the moment it expired, and somebody acts on it anyway. Losing an opportunity to a stale page is a real loss. The arithmetic of subsets does not go stale, so that is what this site holds.

What the record cannot do

It cannot say how many people a list will admit, because it counts profiles and not people.

It cannot say whether a filter is too strict, because that depends on the work.

It cannot handle rules of a different shape — substitutions, conditional items, weighted scores — which are different filters with different arithmetic.

It cannot see that items on a real list travel together, which means its absolute figures understate the intake wherever they do.

The record measures documents under a stated convention. Everything it says is about the document in front of you, and nothing it says is about the people who will read it.

The three figures worth carrying

Every compulsory item halves the intake. Eight of them admit one profile in two hundred and fifty-six. That single sentence reproduces most of the first section.

The rule decides more than the length. Six optional items requiring three admit twenty-one in thirty-two — wider than one single compulsory item at one in two. Length is not strictness.

Twenty-one different shapes admit exactly one profile in two hundred and fifty-six, running from eight items to sixteen. A list cannot be judged by reading it.

How to check any figure here

Two operations, and nothing else is needed.

For the compulsory part, halve once per item. Three compulsory items contribute one eighth; eight contribute one in two hundred and fifty-six.

For the optional part, take the row of binomial coefficients for the number of optional items and add it from the threshold upward. Six optional items give one, six, fifteen, twenty, fifteen, six, one; a threshold of three means twenty plus fifteen plus six plus one, which is forty-two, over sixty-four.

Multiply the two and reduce. Three compulsory with six optional at a threshold of three gives forty-two over five hundred and twelve, which is twenty-one in two hundred and fifty-six. Every number on this site was produced that way and every one of them comes apart the same way.

The one habit worth forming

Write the shape down next to the list.

Three numbers — compulsory, optional, threshold — and the fraction they produce. It takes a line. What it buys is that the next person to edit the document starts from a known position instead of an unknown one, which is the single thing that prevents a filter drifting by a factor of a hundred and sixty-eight in three defensible steps.

Everything else on this site is an elaboration of that line. The classes, the ladder, the eight edits and the staircase all exist to show what happens when the line is missing, which is the ordinary case.

Where to start, holding a list

Three lines of work, in this order.

Count the compulsory items and halve once per item. Count the optional items, find the threshold, and add the binomial coefficients from the threshold upward. Multiply, reduce, and write the fraction next to the list.

If the threshold is not written anywhere, that is the finding. Compute the two extremes instead — a threshold of none and a threshold of all — and the gap between them is what the document has left unsaid. On three compulsory items with six optional, that gap is a factor of sixty-four, and closing it costs one sentence.

And then the decision

Whether a particular filter is right for a particular piece of work is not a question this record can answer. It has no access to the work, the field, the candidates, or what happens if nobody qualifies.

What it provides is the figure — exact, checkable, and available in under a minute to anybody holding a list. That figure is the smaller half of what a person deciding needs to know, and it is the half that is usually missing.