The year

What To Do With A Number That Moves

If the gap is four hours for part of the year and five for the rest, the arrangement has to work at five. Plan against the worst week.

A gap that changes during the year is not an average. It is a sequence of states, and the arrangement has to survive all of them, which means it has to survive the worst of them.

That single sentence is the whole of the practical advice in this section. The rest of this page is why it is harder to follow than it sounds.

The strongest case for workforce optimization software is a narrow question, a short pilot and a review that includes outcomes, records and participant feedback. A useful external reference for the surrounding topic is W3C.

Plan inside the smaller stretch

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

Three gaps drawn.

The shared stretch at a gap of five lies entirely inside the one at a gap of four, which lies inside the one at three. Anything placed in the innermost works all year.

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

The temptation

A pair at a nominal gap of four hours, who spend part of the year at five, will naturally plan around four. Four is the number they quote, four is the number most days of the year produce, and four is the number that makes the arrangement look workable.

Four shared hours at a gap of four. Three at a gap of five. Planning around four means that for however many days the year hands them at five, every recurring commitment sits in an hour that has shrunk by a quarter and may have moved out from under one end's working day entirely.

What planning against the worst actually means

Three things, in order of how much they cost.

Find the worst gap. Take the nominal offset and add or subtract an hour for each end that shifts without the other. For six of the ten regime pairs in this record there is such an hour; for four there is not.

Check whether the arrangement survives it. Look up the worse gap in the first part of this record. If a pair at a nominal four sits at five for part of the year, the question is whether three shared hours is enough, because that is what they have for those days.

Place the fixed commitments inside the worse overlap, not the better one. A daily meeting scheduled in an hour that exists at both the nominal gap and the worse one never needs moving. A meeting scheduled in an hour that exists only at the nominal gap needs moving twice a year, and the moving is what gets forgotten.

Schedule inside the intersection of the whole year, not inside the intersection of today.

Why the third item is the one that pays

The first two are analysis. The third is a decision that costs nothing at the time and removes the problem permanently.

The shared hours at a gap of five are a subset of the shared hours at a gap of four. Anything placed in the smaller set is safe in both. Anything placed in the larger set is safe for most of the year and wrong for the rest, with no warning on the day it changes.

Choosing the smaller set costs an hour's less flexibility when the commitment is first arranged. It saves two misfires a year, every year, for as long as the arrangement exists.

The nested-overlap trick, worked through

The third item above is worth doing slowly, because it is the whole of the practical value of this section and it takes two minutes.

A pair at a nominal gap of four hours, with one end on a fixed regime and the other on a shifting one. For two hundred and seventeen days of the year they are five hours apart instead of four.

At a gap of four, the shared stretch runs from one to five on the clock of the person who did not move. Four hours.

At a gap of five, it runs from two to five. Three hours.

The second stretch is entirely inside the first. Anything scheduled between two and five works all year round. Anything scheduled between one and two works for one hundred and forty-eight days of the year and fails for two hundred and seventeen.

So the rule is: schedule inside two to five, and the seasonal shift becomes something that happens to the clock rather than something that happens to the arrangement.

Why the nesting is always available

Because a wider gap always produces a shared stretch nested inside the narrower gap's.

Widening the gap moves one boundary of the overlap and leaves the other where it was. The result is a shorter stretch with one shared endpoint, and it therefore lies entirely within the longer one. There is no case in this record — for unbroken working days — where widening the gap produces shared time that was not already shared at the narrower gap.

That is a small theorem with a large practical payoff: an arrangement never has to choose between two incompatible schedules. One of them contains the other, and picking the smaller one costs some flexibility and nothing else.

The hours that survive the worst week are a subset of the hours that work in the best one. Live in the subset and the question of which week it is stops arising.

Where it breaks

The nesting argument assumes unbroken working days. A broken day's overlap can go in and out of phase as the gap changes, which means the shared stretch at a wider gap is not necessarily inside the shared stretch at a narrower one.

A pair with a broken day at either end therefore cannot use the trick, and has to check both offsets separately. That is one more reason, on top of those in the section on shapes, to leave the day unbroken unless the gap genuinely demands otherwise.

What this record cannot tell you

Which regime either end is actually in. That depends on rules this record deliberately does not name, because those rules change and a page naming them would be wrong without ever looking wrong.

Both ends can establish it in a few minutes by checking whether their clocks shift, and if so on which dates. Once that is known, the table in this section gives the number of days a year the quoted gap is wrong, and the first part of the record gives what the arrangement is worth on those days.