Halving

One In Two, Four, Eight, Sixteen

The full table of compulsory-only lists, from one item to eight. Nine shapes, nine powers of two, and no exceptions anywhere in the sequence.

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Nine shapes in this record have no optional items at all: the empty list, and lists of one through eight compulsory items. Their intake is a power of two, every time.

Items	Admits	One profile in
0	1/1	1
1	1/2	2
2	1/4	4
3	1/8	8
4	1/16	16
5	1/32	32
6	1/64	64
7	1/128	128
8	1/256	256

There is nothing else to the sequence. It is the simplest thing in the record and it is worth having in front of you before anything more complicated arrives.

A short test is easier to interpret when teams first check the platform controls and use the same reporting window for everyone involved. For an independent technical reference, see Postman.

Every compulsory-only list in the record

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

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

Eight rows, eight halvings. The squares shrink by half in area at each step until they are too small for the panel, at which point a stroke stands in for them.

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

Reading the right-hand column

The right-hand column is the one people respond to, and it deserves a caution before it is used.

"One profile in two hundred and fifty-six" is a statement about the possibility space, not about applicants. It does not say that one candidate in two hundred and fifty-six will qualify. It says that of the two hundred and fifty-six distinguishable ways of meeting and not meeting these eight items, exactly one gets through.

Whether that corresponds to one person in two hundred and fifty-six, or one in five, or none at all, depends on things this record cannot see.

Where the sequence starts to bite

Somewhere around the fifth item the numbers stop feeling like arithmetic and start feeling like a decision.

A list of three admits one profile in eight, which most people would call selective but reasonable. A list of six admits one in sixty-four. The two lists differ by three lines, each of which looked sensible as it was written, and they differ eightfold in what they let through.

That is the whole problem with writing lists by accumulation: the cost is multiplicative and the writing is additive. Each added line feels like one more line. It is not; it is a halving.

A list grows by addition and narrows by multiplication. Nobody writing one experiences the second fact, which is why the arithmetic has to be done rather than felt.

What the empty list is doing here

The empty list — no items at all — admits everything, and it appears in the record for completeness and for one useful purpose: it fixes the baseline against which everything else is measured.

It also makes a point that recurs throughout this record. A list with no items is a filter that filters nothing. So is a list with eight optional items and a threshold of none, which appears in the section on thresholds and looks very different while doing exactly the same work.

How fast the sequence actually moves

Powers of two are easy to recite and hard to feel, so it is worth converting them into something with a shape.

Going from one item to two costs a quarter of the whole possibility space. Going from seven to eight costs one part in two hundred and fifty-six. The absolute damage done by each successive line falls away fast, and after about the fifth item the line is adjusting a remainder rather than making a cut.

That produces a useful rule of thumb for anybody auditing a list. The first four lines decided fifteen-sixteenths of the outcome. If the list is too narrow, the fix is almost certainly in those four, and arguing about the last two is arguing about the last one per cent.

Lines	Share of the whole already discarded
after 1	half
after 2	three quarters
after 4	fifteen sixteenths
after 8	two hundred and fifty-five parts in 256

What the right-hand column cannot contain

Every intake in this table is one over a power of two, and that rules out most of the numbers anybody would naturally aim for.

A conjunction cannot admit one profile in three, or one in ten, or one in a hundred. The nearest available values to one per cent are one in sixty-four and one in a hundred and twenty-eight, and there is nothing between them.

So an author who wants a filter of a particular selectivity cannot get there with compulsory items alone. They can only land on the nine values in this column, and if the target lies between two of them the gap has to be closed with an optional part and a threshold — which is what the second section of this record is about.

Where the sequence stops

At eight, because that is where the record's declared bounds stop. A ninth compulsory item would admit one profile in five hundred and twelve, a tenth one in one thousand and twenty-four, and so on forever.

The bound at eight is a choice about what is worth enumerating, not a claim that lists stop there. The page on the record's ceiling sets out the reasoning; the short version is that nothing new happens after eight, and a table twice the size would contain the same single fact.