Three ends

Why This Record Stops At Three

A fourth end adds no new behaviour. Once only the outermost two matter, every further end is inside the span or becomes the new extreme.

This record computes two ends and three ends and stops. A fourth is not left out for want of effort; it is left out because the answer is already known and writing it down would add nothing.

The rule does not change

For two ends, the shared time is the length of the working day minus the distance between them, floored at zero.

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For three ends, it is the length of the working day minus the distance between the outermost two, floored at zero. The middle end does not appear.

For four ends, five ends or fifty, it is the same sentence. Every end is either inside the span of the extremes, in which case it contributes nothing, or outside it, in which case it becomes one of the extremes and the span widens accordingly.

There is only ever one measurement, however many people are involved: the distance from the earliest to the latest. Everybody in between is along for the ride.

What this means for a growing group

The implication is sharper than the arithmetic looks.

Adding somebody inside the span is free. Not nearly free — exactly free. The shared hours do not move at all.

Adding somebody outside it costs an hour per hour, immediately and at full rate, and the cost is paid by everybody rather than by the newcomer.

So the question when a group grows is never "can this person make the overlap work". It is "is this person inside the span we already have", and that is a yes-or-no question with no middle ground.

Why this is the useful form of the result

Groups do not usually choose all their positions at once. They acquire them one at a time, and each acquisition is considered on its own terms — a good candidate, a necessary office, a person who was already there.

Each of those decisions looks small. The one that moves an extreme is not small, and it looks exactly like the ones that are. A group that has drifted to a span of eight hours did not decide to; it made six reasonable decisions, one of which happened to be the expensive one, and nobody could tell which at the time.

The one thing that does change with more ends

Not the arithmetic — the likelihood. With more ends, the chance that at least one of them lands outside the current span rises, and the span only ever widens. It cannot narrow unless somebody leaves or moves.

That is a ratchet. The shared hours of a growing group are non-increasing by construction, and the only question is how fast.

What a four-end table would actually contain

Suppose it were computed anyway. Four ends, each at one of twenty-five whole-hour gaps, gives a table with many thousands of entries.

Every one of them would be the same number: eight minus the distance between the earliest and the latest, floored at zero. Not approximately — exactly, in every entry.

The table would therefore contain no information that is not in the sentence above it. It would be large, it would look authoritative, and anybody consulting it would have been better served by the sentence.

That is the test a table has to pass to be worth printing: does looking something up in it tell you anything you could not have derived? For two ends and three, the answer is yes, because the mechanism is not obvious until it has been seen worked out. For four, the mechanism has already been established and the table would be a restatement.

A record that computes everything it can compute is not thorough; it is padded. Thoroughness is computing everything that answers a question and then stopping.

Where the record does stop, and why there

Three reasons, and they happen to coincide.

Three ends is where the mechanism becomes visible. Two ends do not reveal that only the extremes matter, because with two ends every participant is an extreme. Three is the smallest number that has a middle, and so the smallest number that demonstrates the rule.

Three ends is where the intuition fails. Pairwise reasoning works for two and breaks for three, which is the practical result of this section. Nobody reasons pairwise about four ends and gets it more wrong than they do about three.

Past three, the results are identical in form. Every further end is either inside the span or redefines it, and the arithmetic is a repetition.

So three is where the record ends and where the question is finished, which is a more satisfying place to stop than running out of patience would have been.

What would have to be true for a fourth-end table to be worth having

Working days of different lengths at different ends, or broken days in the mix. Both of those break the simple rule, and both are already treated in the section on shapes for the two-end case.

Extending them to four ends is arithmetic nobody needs: the shapes section establishes what a long or broken day does, and this section establishes that only the extremes matter. Putting the two together is left to the reader, which is what a record should do when the combination holds no surprise.