Three ends

The Two Hundred And Thirty-Three

Nearly three quarters of the three-end arrangements leave nothing. They share one property, and it predicts every one of them.

Two hundred and thirty-three of the three hundred and twenty-five three-end arrangements leave no hour at which all three are at work. That is a little over seventy per cent, and it is the single figure this record would most like to be wrong about.

They have one thing in common, and it is enough to identify every one of them without computing anything.

Operational context is most useful when managers see the full product details and read the output beside the work product rather than on its own. A useful external reference for the surrounding topic is Nature.

The last live band before the floor

every cell prints the hours all three share
reversed: no hour at all

28 arrangements drawn; no of them leave no hour the three can all attend; the grid is a half: the two gaps are interchangeable, so the other half would repeat it.

Each row holds one value throughout: the figure depends on the widest separation and on nothing else. One step further out in either direction and every cell here reverses.

Computed from the two gaps. The working day is a stated convention; the intersection is arithmetic.

The rule

Take the three ends. Find the two that are furthest apart — not the two that are being compared, the two at the extremes. If those two are eight hours apart or more, the arrangement leaves nothing.

That is the whole test. The middle end is irrelevant to it. A third end sitting between two others can neither save an arrangement nor break one; it only matters if it is itself one of the extremes.

Widest separation among the three	Result
7 hours or less	something survives
8 hours or more	nothing survives

The threshold is eight because that is the length of the working day in this part of the record. For three nine-hour days it would be nine, and for three seven-hour days, seven.

Three ends are not three relationships. They are one measurement — the distance between the outermost two — with a third person standing somewhere in the middle of it.

Why the count is so high

The grid of three-end arrangements runs from twelve hours behind to twelve ahead in both directions. Within that grid, the region where the outermost two ends are within seven hours of each other is small.

Picture it as a band. The surviving arrangements lie inside a band around the diagonal, and everything outside it is dead. The band is fourteen hours wide in a space twenty-four hours across, and the geometry does the rest.

What the dead arrangements are like in practice

They are not arrangements with a little shared time. They are arrangements with none, which means every exchange between at least one pair of the three is a handover rather than a conversation.

Such arrangements run, and some of them run well, but they run on written artefacts rather than on simultaneity. Decisions wait a day. Questions get asked in a form that anticipates the answer not arriving for sixteen hours. Documents matter more than meetings because documents are the only thing that works.

That is a coherent way to organise work, and it is a different way, and the two hundred and thirty-three are all in it.

Why the middle end is irrelevant and feels like it should not be

The rule says the middle end contributes nothing. That is correct and it reliably surprises people, so it is worth seeing why.

The shared time of a group begins when the last person starts and ends when the first person finishes. Those are both properties of the extremes: the last to start is one extreme, the first to finish is the other. Anybody whose day begins earlier than the last start and ends later than the first finish is at work through the whole of the shared stretch, and so adds no constraint.

A middle end can only matter by ceasing to be a middle end. If it moves outside the span, it becomes one of the extremes, and the span widens by however far it went.

So the influence of a third person on a group's shared time is binary: zero if they are inside, costly if they are outside, with nothing in between.

The consequence for where people sit

This produces a rule of thumb sharper than most things in this record.

Adding somebody inside the existing span is free. Not approximately free. The shared hours do not move at all, and the group can absorb any number of such people without losing a minute.

Adding somebody outside it costs an hour of everybody's shared time for every hour they are outside. The cost is immediate, it is paid by every existing member, and it does not diminish.

A group can therefore be large and have excellent shared time, or small and have none, and the size has nothing to do with it. What matters is whether the group fits inside a window narrower than a working day.

The question is never how many people are in the group. It is how wide the group is, and those two quantities are unrelated.

What this means for the two hundred and thirty-three

They are not two hundred and thirty-three different problems. They are one problem — a span of eight hours or more — appearing in two hundred and thirty-three configurations.

Which is why the record does not rank them, and why anybody in one of them should stop examining the configuration and look at the span. There is exactly one number to change, and changing anything else is motion without effect.

The one consolation

Among arrangements that leave nothing there is nothing to rank, which means there is nothing to optimise either. An arrangement with a widest separation of eight hours and one with a widest separation of twelve are in exactly the same position.

For anybody in that region, the only move that changes anything is a move that brings the outermost two ends to within seven hours of each other. Closing part of the distance is worth precisely nothing, and knowing that saves a great deal of effort spent closing part of the distance.