The gap

The One Law This Whole Record Rests On

Two working days of the same length lose exactly one hour of overlap for every hour of gap. Almost every figure here follows from that sentence.

Overlap falls by exactly one hour for every hour of gap. Not roughly, not on average — exactly, for as long as both working days are the same length, and the proof fits in a paragraph.

Take two days of eight hours. Lay them on one clock. Shift one of them by some amount. The shared stretch begins when the later of the two days begins and ends when the earlier of them ends, so every hour of shift pushes the start one hour later without moving the end. The shared stretch therefore shortens by exactly the shift, until the start passes the end and there is no shared stretch at all.

For work spread across several clocks, remote team time tracking is most useful when the team first agrees which scheduling question the evidence should answer. A useful external reference for the surrounding topic is World Bank.

What a whole-hour gap leaves

outlined: the day where the person is
hatched: the other end
solid: the hours they share
a reversed cell counts none

Nine gaps drawn; one of them leaves nothing at all.

The outlined bar is the day where the person sits; the hatched bar is the other end on the same scale; the solid block is what they share. The loss is perfectly even until there is nothing left to lose.

Computed from the gap. The working day is a stated convention, not a measurement; everything else here is arithmetic.

What the law does and does not cover

It covers any two working days of equal length, whatever that length is and wherever they sit on the clock. A pair of seven-hour days loses an hour per hour of gap just as a pair of nine-hour days does; they simply start from a different number and run out sooner or later.

It does not cover days of unequal length, which is why the long shape gets its own treatment, and it does not cover a day with a break in it, which is why the split shape gets its own treatment too. Those two exceptions account for every figure in this record that does not fall straight out of the sentence above.

A rule that holds exactly is worth more than a rule that holds usually, because an exact rule tells you where it stops. This one stops in two places and nowhere else.

Why the evenness matters

Most reasoning about arrangements like these assumes diminishing returns in one direction or the other: that the first hour of gap is the expensive one and later hours cost less, or that things are fine until some threshold and then collapse. Neither is true here.

The first hour of gap costs an hour. So does the seventh. The cost is flat, which means the arrangement does not have a comfortable zone followed by a difficult one. It has a straight slope and then a floor.

The consequence for anybody deciding is that there is no clever point on the curve to aim for. Every hour of gap removed is worth exactly the same as every other, and the only question is how many hours of shared time the work actually needs.

Where the floor is

At a gap of eight hours, two eight-hour days have nothing in common. At nine, ten, eleven and twelve the answer is still nothing — not less than nothing, not worse, just the same nothing. This is worth stating because the instinct is to treat a wider gap as a worse problem, and past the point of no overlap it simply is not a different problem.

Gap	Shared
0	8
4	4
7	1
8	0
12	0

That flatness at the bottom is the reason this record counts ten dead gaps rather than ranking them. Among gaps that leave nothing there is nothing to rank.

Working it on a clock face

The proof above is general, which makes it easy to believe and hard to feel. Here it is with actual hours in it.

One end works nine to five. The other end is three hours ahead, so when it is nine where the first person sits, it is already noon where the second one does. The second person's day began three hours ago, at six by the first person's clock, and it will end at two by the first person's clock.

So the two of them are both at work from nine — when the first one arrives — until two, when the second one leaves. Five hours. Eight minus three.

Now move the second end one hour further ahead. Their day now runs from five to one by the first person's clock. The shared stretch still begins at nine, because that is when the first person arrives and nothing about the second end changes that. It now ends at one. Four hours.

The end of the shared stretch moved and the beginning did not. That is the entire mechanism, and it is why the loss is exactly one hour per hour rather than some fraction of one: only one of the two boundaries is in motion.

Two things that look like exceptions and are not

Breaks and lunches. A working day with an hour off in the middle of it is still a working day eight hours long on this record's accounting, and the law applies to the hours worked. What a shared lunch hour does to a working relationship is real and is not an intersection.

Overtime on one side. Somebody staying late occasionally does not change the shape of their working day; it changes one day. The record describes the arrangement, which is what recurs. An arrangement that depends on somebody regularly staying late is an arrangement with a longer working day at one end, and that case has a page of its own.

An exception that happens sometimes is not a different law. It is the same law applied to a different day, and the honest way to account for it is to say how often.

Why an exact law is worth more than a good estimate

A rule that holds approximately tells you roughly what to expect. A rule that holds exactly tells you something better: where it stops.

This one stops in two places and nowhere else. It stops when the two working days are of different lengths, and it stops when one of them is broken into pieces. Both of those exceptions are named, both are measured in the section on shapes, and between them they account for every figure in this record that cannot be produced by subtracting the gap from eight.

That is a short list of exceptions, and knowing it is short is what makes the rule usable. Anybody who meets a figure here that does not fit the law can check which of the two exceptions is in play, and will always find one.

The one number to carry away

For two ordinary eight-hour days, shared hours equal eight minus the gap, floored at zero. Everything in the first part of this record is a consequence of that, and anybody who remembers it can reconstruct the whole table without looking.