Three ends

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 collapse is steeper than it looks.

Two ends at a gap of four hours share four working hours. Add a third end, also at a gap of four from the first but in the other direction, and the three of them share nothing at all.

That is the shape of the problem. Of the three hundred and twenty-five arrangements of three ends in this record, two hundred and thirty-three leave no hour at which all three are at work. Ninety-two leave at least one.

A transparent implementation lets contributors check the time-recording workflow and discuss what the system records before the pilot starts. A useful external reference for the surrounding topic is CIPD.

Three ends at once

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

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

Both gaps measured from the same end. The grid is a half, because the two gaps are interchangeable. Even within this narrow window — four hours either way — a reversed cell has already appeared.

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

Why it collapses rather than degrades

An intersection of three ranges is the intersection of the first two, intersected again with the third. Each step can only remove, never add.

With two ends, the shared stretch begins at the later start and ends at the earlier end. With three, it begins at the latest start and ends at the earliest end — and the chance that the latest start is before the earliest end falls away quickly as the ends spread out.

Concretely: three ordinary eight-hour days survive only while the widest separation among any two of the three is under eight hours. The two outermost ends are what decides it, and the middle end contributes nothing except by being inside the range already.

A third end does not add a third more difficulty. It replaces "how far apart are these two" with "how far apart are the furthest two", and the second question has a much worse answer.

The numbers

Shared hours	Arrangements
none	233
1	22
2	19
3	16
4	13
5	10
6	7
7	4
8	1

Thirty-five arrangements leave four hours or more. One leaves the full eight, and that is the case where all three ends are at the same gap, which is to say not separated at all.

Where the survivors live

Every arrangement that leaves anything has both of its gaps within seven hours of the reference end, and the sum of the two gaps' sizes never exceeds fourteen. Outside that region the answer is uniformly nothing.

That is a small region inside a large grid, which is the visual form of the same point: the workable three-end arrangements are a tight cluster, and almost everything else is dead.

Why pairwise reasoning gives the wrong answer

The mistake that produces most failed three-end arrangements is not an error of arithmetic. It is an error of composition, and it is extremely natural.

Three ends, each pair of which overlaps comfortably. The first and second share four hours. The second and third share four hours. The obvious inference is that the three of them share something, and the obvious inference is false: if the second end sits exactly between the other two, the first and third are eight hours apart and share nothing, and so do all three together.

Every pairwise relationship is healthy. The group relationship does not exist. Nothing in the pairwise view reveals this, because the pairwise view never compares the first end with the third.

Pair	Gap	Shared
first with second	4	4
second with third	4	4
first with third	8	none
all three	—	none

Where the mistake gets made

It gets made when a group grows one person at a time, which is how groups grow.

Each addition is assessed against the people already there, usually against the one or two who will work with the newcomer most. Those comparisons come back fine. Nobody runs the comparison between the new person and the person at the far end of the group, because those two are not expected to work together directly.

And that is the comparison that decides it. The arithmetic of three ends depends only on the two extremes, and the two extremes are precisely the pair nobody thought to check.

A group can pass every pairwise test it thinks to apply and have no common hour at all. The test it did not apply is the only one that mattered.

The check that takes ten seconds

List every end on one clock. Find the earliest working-day start and the latest working-day end among all of them. If the span between those two is less than the length of the working day, there is shared time, and the amount is the difference. If it is not, there is none.

That check does not get longer as the group grows. It is the same two numbers for three people as for thirty, which is the one genuinely convenient feature of the whole arrangement.

What this does not say

It does not say that three ends cannot work. Ninety-two arrangements leave at least an hour, and thirty-five leave a working morning — those are real arrangements with real shared time.

It says that the odds change character when the third end is added, and that the question to ask is no longer "is this gap manageable" but "what is the widest separation among all three". Anybody reasoning pairwise about a three-end arrangement will get the wrong answer, and will get it in the optimistic direction.