The End In The Middle

The arithmetic of three ends depends only on the two extremes. The middle end — whichever of the three sits between the other two — makes no difference to how many hours all three share. It can move freely within the span and the answer does not change.

That is a mathematical fact and a misleading summary of the situation, because the middle end is doing most of the work.

If a team already has strong project data, it can still explore the available workforce tools and compare the setup without committing to deployment. A useful external reference for the surrounding topic is American Psychological Association.

What the arithmetic says

Three ends at gaps of zero, two and five from a reference. The extremes are zero and five, five hours apart, so the three share three hours. Move the middle end from two to three and the answer is still three hours. Move it to one, or to four: still three.

The middle end only begins to matter when it stops being the middle end — when it moves outside the span and becomes an extreme itself.

In the first three rows the middle end has moved across four hours and changed nothing. In the fourth it has moved past the far extreme and cost two hours.

What the arithmetic cannot see

The person in the middle overlaps with both of the others for longer than either of them overlaps with the other. In the first row of that table, the middle end shares seven hours with one colleague and four with the other, while the two extremes share only three with each other.

Which means the middle end is the only one who can reasonably talk to everybody, and so, in practice, becomes the route by which everything travels. Questions arrive there. Context accumulates there. The two extremes stop addressing each other directly because it is inconvenient, and address the middle instead.

The person in the middle has the best position on the clock and the worst position in the work. Being reachable by everybody is not a privilege.

Why this record cannot help

It counts hours of simultaneous availability. It has no measure for how much is being relayed, who is holding the context, or what it costs to be the junction.

Those are real costs and they fall predictably on one person, and the fact that a record of this kind cannot see them is worth stating plainly rather than leaving as an implication.

The middle end's actual position

Take three ends at gaps of zero, three and six from a reference. The middle one is at three.

With the first end, the middle shares five hours. With the third end, the middle shares five hours. The two extremes share with each other two hours. All three together share two hours.

So the person in the middle has five hours with each colleague individually, and the group as a whole has two. The middle end's pairwise position is more than twice as good as the group's.

That gap between the pairwise figure and the group figure is what turns into a relay. It is simply easier for either extreme to talk to the middle than to the other extreme, and over time the easier path becomes the only path.

What accumulates

Context, mostly. The person who talks to everybody ends up being the only person who knows everything, not because they sought it but because every exchange passed through them.

That has two consequences the record cannot measure and that follow from its arithmetic nonetheless. The middle end becomes hard to replace, because the knowledge is not written anywhere. And the middle end becomes a bottleneck, because anything that requires their attention waits for it, and everything requires their attention.

Neither is anybody's fault. The geometry put one person at the junction and the work flowed the way water does.

The best clock position in a group is a liability, and the person who has it did not choose it and often cannot explain why their days are fuller than everybody else's.

The one thing the numbers do suggest

If the two extremes are seven hours apart or less, they share something directly, and the relay is optional. If they are eight apart or more, they share nothing, the three-end arrangement is one of the two hundred and thirty-three, and every exchange between them must pass through the middle or wait a day.

So the question "how far apart are the outer two" is not only about whether the three can meet. It is also about whether one person is about to become a permanent intermediary, and that is probably the more consequential answer.

Next along

• 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 o
• The Single Hour Everybody Can Attend
  Twenty-two three-end arrangements leave exactly one shared hour. One hour is a worse position than it sounds.
• Why A Third End Is Not A Third As Bad
  Two hundred and thirty-three of the three hundred and twenty-five three-end arrangements leave no hour at all.
• The Two Hundred And Thirty-Three
  Nearly three quarters of the three-end arrangements leave nothing. They share one property, and it predicts ev