Shapes

All Six Hundred And Twenty-Five Pairs

Five shapes at each end and twenty-five gaps: six hundred and twenty-five arrangements, all computed. The extremes, and how many leave nothing.

Five shapes can sit at either end, which gives twenty-five pairings. Each pairing is evaluated at twenty-five whole-hour gaps. Six hundred and twenty-five arrangements in total, every one of them computed rather than sampled.

Computing all of them was cheap. Choosing a subset would have required a claim about which arrangements are common, and this record has no basis for such a claim.

For recurring work across clocks, reviewers can learn more from the official source and test whether the record improves planning rather than adding administration. A useful external reference for the surrounding topic is Notion.

The extremes

The largest shared time anywhere in the record is nine hours, and there is exactly one arrangement that reaches it: two long days at no gap. Nine hours is the length of the long shape, and two identical days at no gap share all of it.

The smallest non-zero shared time is one hour, reached by many arrangements at the far edge of their range.

The widest gap that leaves anything at all is nine hours, reached only by two split days against each other.

How many leave nothing

Every pairing has a point past which it leaves nothing, and the pairings differ considerably in where that point is.

Pairing	Gaps leaving nothing, out of 25
split with split	6
long with split	7
long with long	8
split with an unbroken eight-hour day	8
long with an eight-hour day	9
two unbroken eight-hour days	10

The ordering is the opposite of what the middle of the range suggests. The pairings that survive furthest — the split ones — are the ones that share least in the ordinary cases. Reach and depth trade against each other throughout.

Counting how many arrangements survive and counting how much they leave are two different questions with two different winners. Any summary that uses one to answer the other is wrong.

Three ways to read the full table

The six hundred and twenty-five arrangements can be sliced three ways, and each slice answers a different question.

By pairing. Fix the two shapes and sweep the gap. This gives a column of twenty-five numbers showing how that pairing degrades, and it is the right view for a pair who know their shapes and want to see what would happen if the distance changed.

By gap. Fix the gap and sweep the shapes. This gives twenty-five numbers showing what rearranging working hours would buy at a fixed distance, and it is the right view for a pair who cannot change the distance and want to know what else is available.

By outcome. Collect every arrangement that leaves a given number of hours. This is the view that produces the aggregate claims on this page, and it is the least useful of the three for anybody with an actual arrangement.

Most of this record is written in the first view, because that is the question people arrive with. The second view is the one that contains the actionable answers, and the section on choosing a shape is built on it.

What the extremes are worth knowing

The single nine-hour arrangement — two long days at no gap — is in the record as a ceiling rather than as a recommendation. It establishes that no arrangement anywhere here exceeds the length of the shorter working day, which is the sanity check every other figure has to pass.

The single widest reach — two split days at a gap of nine — is in the record for the same reason from the other direction. It establishes the outer boundary of what any rearrangement can achieve, and the section on stretching uses it as the stopping point.

Between those two the table is dense and unremarkable, which is the correct state for a reference. The interesting cases are at the edges and they have been named.

A complete table's job is to be looked up, not read. Its aggregate figures are a description of its own shape, and nobody should mistake them for findings about the world.

What the table is not

It is not a ranking of shapes. The column above counts breadth of survival, which is a reasonable thing to care about if the gap is wide and an irrelevant thing to care about if the gap is two hours.

It is also not a distribution. Nothing here says the arrangements are equally likely, because nothing here knows what is likely. Each row is a complete enumeration of a pairing across every gap the record considers, and that is all.

Using it

Find the pairing, find the gap, read the answer. The record exists so that a specific arrangement can be looked up rather than estimated, and the aggregate figures on this page are a description of the table rather than a substitute for it.

The two pages that follow pull out the pairing that fails soonest and the one that survives longest, since those are the two rows people reach for first.