Three ends

The Ninety-Two That Survive

Ninety-two three-end arrangements leave at least one shared hour. They are tightly clustered and easy to characterise.

Ninety-two of the three hundred and twenty-five three-end arrangements leave at least one hour at which all three are at work. Every one of them satisfies the same condition: the two outermost ends are seven hours apart or less.

That condition is not approximately right. Checked against all three hundred and twenty-five arrangements, it predicts every single one, with no exceptions in either direction.

A practical use of team activity context for managers is tracing time to projects and tasks while keeping the limits of every aggregate visible to reviewers. A useful external reference for the surrounding topic is Buffer.

The strongest corner of the grid

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

15 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.

Three ends all within two hours of each other. Not one reversed cell appears; this is the whole of the comfortable region.

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

How the ninety-two are distributed

Shared hours	Arrangements
1	22
2	19
3	16
4	13
5	10
6	7
7	4
8	1

The distribution falls away steeply. Nearly a quarter of the survivors leave only one hour, and a single arrangement leaves the full eight.

Thirty-five leave four hours or more — a working morning or afternoon. That is thirty-five out of three hundred and twenty-five, which is a little over one in ten of all possible three-end arrangements.

The single best case

One arrangement leaves eight hours: all three ends at the same gap. Which is to say, all three in the same hours, which is to say, no separation at all.

It is worth stating because it establishes the ceiling. Three ends can do no better than two ends do, ever. Adding people never increases the intersection; it can only leave it alone or shrink it.

The best possible three-end arrangement is exactly as good as the best possible two-end arrangement. Everything else is worse. There is no configuration in which a third person helps.

The shape of the surviving region

Drawn as a grid, the survivors form a band around the diagonal. The diagonal itself is the case where the second and third ends are at the same gap — effectively two ends with an extra person at one of them — and those are the strongest arrangements at any given distance.

Moving away from the diagonal in either direction spreads the two outer ends apart, and the shared time falls by an hour for each hour of spread, in exactly the way the two-end law predicts. The three-end problem is the two-end problem applied to the extremes.

Reading the grid

Each cell holds the number of hours all three share, printed rather than coloured, so the figure is available without reference to a key. A reversed cell counts none at all.

The grid is drawn as a half, because the two gaps are interchangeable: an arrangement with one end at a gap of two and another at a gap of five is the same arrangement as one with those two swapped. Drawing both halves would have doubled the size and added nothing.

Why the distribution falls away so steeply

Twenty-two arrangements leave one hour. One leaves eight. The counts decline by three at each step, which is a clean pattern and has a clean cause.

An arrangement leaves a given number of hours when its span is exactly eight minus that number. So the count of arrangements leaving one hour is the count of three-end configurations with a span of seven, the count leaving two is those with a span of six, and so on down to a span of zero.

There are more ways to arrange three points inside a wide window than a narrow one. A span of seven admits many configurations; a span of zero admits exactly one, with all three ends together. The declining counts are a geometric fact about placing points in intervals and say nothing about the world.

Span	Shared	Arrangements
0	8	1
1	7	4
4	4	13
7	1	22

The distribution is a property of the counting, not a finding. More arrangements leave one hour than eight because there are more ways to be spread out than to be together, which is true of any three points on any line.

What a three-end arrangement is worth, in practice

Of the ninety-two survivors, twenty-two leave one hour and nineteen leave two. That is forty-one — nearly half the survivors — leaving two hours or less.

Two hours of time shared by three people is not two hours of three-way work. Each of the three has their own commitments inside it, and the chance that a given hour is free for all three simultaneously is considerably lower than for a pair.

So the honest reading of the ninety-two is that about a third of them support a genuine shared working period, another third support a daily meeting if it is defended, and the last third support a daily meeting in principle and will lose it to the first scheduling conflict.

The record counts all ninety-two as leaving something, because they do. The grading above is a judgement and is offered as one.

What to do with the ninety-two

Find the two ends furthest apart. If they are eight hours apart or more, stop — the arrangement leaves nothing and no rearrangement of the middle end changes that. If they are seven apart or less, the shared time is eight minus their separation, and the third end does not enter the calculation at all.