Arrangements That Leave A Morning

Thirty-five three-end arrangements leave four hours or more — enough for half a working day in common, which is the point at which most people would describe an arrangement as genuinely shared rather than merely overlapping.

Thirty-five out of three hundred and twenty-five is a little over one in ten.

Before adding another dashboard, a lead should review the current feature set and confirm that the report answers a real operating question. A useful external reference for the surrounding topic is Pew Research Center.

What the thirty-five have in common

The rule from earlier in this section does all the work. An arrangement leaves four hours or more exactly when its two outermost ends are four hours apart or less.

That is a narrow window. The three ends must all fit inside a four-hour span — not four hours between each adjacent pair, four hours from the earliest to the latest.

Those five rows add to thirty-five. The counts rise as the span widens because there are more ways to place three points inside a wider interval, which is the only reason the larger spans have more arrangements in them.

Why one in ten is the figure to carry

Because the intuitive estimate is far higher.

Asked how often three people in different places can find four hours a day in common, most people guess something between a third and a half. The answer within this record is one in ten of all configurations, and the gap between the guess and the answer is the single most useful thing in this section.

The guess is high because people reason pairwise. Any two of the three probably overlap — that part is right, and two-end arrangements do leave four hours or more nine times in twenty-five. The error is in supposing that three pairwise overlaps imply a common one. They do not, and the failure is not rare.

Three people can be pairwise close and collectively unreachable. The first and third never meet, and the second one meeting both of them does not help.

The practical form

If three ends must have four hours together, they must all sit inside a four-hour span. That is a strong constraint and it is worth stating before anybody starts arranging things, rather than after.

It also means the question of where to put a third end has exactly one sensible answer: inside the span the existing two already occupy. A third end placed outside it widens the span and costs an hour of shared time for every hour it sticks out, regardless of how convenient its own position is for anything else.

Where to put the third person

The constraint has an immediate practical form, and it is the only piece of advice in this section that can be acted on before anything is decided rather than after.

If two ends already exist and a third is being added, the third end's position changes the group's shared time by exactly one rule: nothing at all if it lands inside the existing span, an hour for every hour by which it lands outside.

So the right question when considering a third position is not "does this position overlap enough with the two we have". It is "is this position inside the window the two existing ends already occupy".

Four rows, and the first one is free. A third person anywhere inside the existing window costs the group nothing, which means the set of acceptable positions is both precisely defined and often larger than people assume.

The compounding version

A group that adds people one at a time, each assessed on its own merits, will drift outward. Every addition either leaves the span alone or widens it; none of them narrows it.

That is a ratchet, and it has a predictable end state: a group whose span exceeds a working day and which therefore has no common hour. Nobody decided to arrive there, and no individual decision along the way was unreasonable.

The only defence is to know the span and to treat widening it as a decision with a price attached, rather than as a side effect of a decision about somebody's suitability.

A span widens by accident and narrows only by someone moving. Groups should know which number they are about to change.

And the honest qualifier

Four hours in common is not four hours of collaboration. It is four hours during which collaboration is possible, which is a weaker and much more useful claim.

Nothing in this record measures what happens inside the shared hours. It measures only whether they exist.

Next along

• The End In The Middle
  When one of three ends sits between the other two it contributes nothing to the arithmetic and everything to t
• 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.