Stretching

What One Extra Hour Is Worth

Lengthening one end's working day by an hour brings exactly two more gaps into range. Not three, not a vague improvement — two.

Without any lengthening, fifteen of the twenty-five whole-hour gaps leave at least an hour of shared time. Lengthen one end's working day by a single hour and that rises to seventeen.

Two more gaps. Specifically, the gap of eight hours in each direction, which previously left nothing and now leaves one hour.

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Why exactly two

The arrangement fails when the gap exceeds the length of the shorter working day. With two eight-hour days, that happens at a gap of eight. Lengthen one of them to nine and the shorter day is still eight, so the threshold does not move — except that the extra hour sits at one end of the longer day and catches a gap pushing in that direction.

One hour at one end, in one direction. The mirror case gains the other direction. Two gaps, and no more, because there is no third direction for an hour to point in.

Lengthening	Gaps leaving at least an hour	Gained
none	15	—
1 hour	17	2

What it does to gaps already in range

It also adds an hour of shared time to several of the gaps that already worked, in the direction the extra hour points. A pair at a gap of three in the favourable direction gains an hour; a pair at a gap of three in the unfavourable direction gains nothing at all.

That asymmetry is the part people miss when they agree to the change. The hour is only worth something against gaps pushing one way, and which way that is depends on whether the extra hour is added at the start of the day or the end — a decision usually made for entirely unrelated reasons.

An extra hour added at the wrong end of the day is worth nothing whatsoever. It is the same hour, the same tiredness, and no additional overlap at all.

What it costs

A nine-hour working day instead of an eight-hour one, every day, for as long as the arrangement lasts.

This record counts the hour as a gain because it counts shared hours and nothing else. It cannot see that the person working the longer day is working longer, that the hour comes out of their evening, and that the gain is collected by the arrangement while the cost is paid by one person.

Any use of this table as an argument for longer hours is using it for something it does not support. It measures what the hour buys. It has no view on whether buying it is reasonable.

The hour inside the range

The two newly reachable gaps are the headline, and they are not the whole effect. The extra hour also adds an hour of shared time to every gap already in range, in the direction it points.

Gap	eight-hour day both ends	one end on nine hours
1	7	8
3	5	6
5	3	4
7	1	2

An hour at every row, in one direction. In the other direction, the same column reads seven, five, three, one — unchanged, because the extra hour is on the wrong side to catch anything.

So the lever is worth one hour of shared time if the gap runs the right way and nothing at all if it does not, and whether it runs the right way is a fact about the arrangement rather than a choice.

Who should lengthen

If lengthening is going to happen, it should happen at the end that is behind, with the hour added to the end of the day.

A person whose clock is behind the other end finishes earlier in the other end's terms. An extra hour at the end of their day reaches forward towards the other end's afternoon, which is exactly where the shared time is being lost.

The mirror case: a person whose clock is ahead should add their hour at the start of the day, reaching back towards the other end's morning.

Lengthening at the wrong end of the wrong day is a common outcome because the decision is usually made by whoever is willing, in whichever direction suits their evening. It costs the same hour of work and returns nothing.

There are four ways to add an hour to a two-end arrangement and three of them are worth nothing. The one that works is determined by the sign of the gap.

The honest summary

One extra hour, at the correct end of the day, extends the workable range from a gap of seven to a gap of eight in one direction. That is the whole of what it does, it is a real improvement, and it is smaller than it is usually assumed to be.