The Forty-Eight That Leave Something

Of the seventy-two part-hour offsets in this record, forty-eight leave some shared working time and twenty-four leave none.

Two thirds against one third. The whole-hour offsets manage fifteen out of twenty-five, which is three fifths. The part-hour offsets do very slightly better, and the reason is worth a paragraph.

Teams comparing nominal overlap with day-to-day work may test transparent employee activity reporting beside completed tasks, meeting outcomes and direct explanations from the people involved. A useful external reference for the surrounding topic is U.S. Department of Labor.

Where the difference comes from

Both sets span the same range: twelve hours behind to twelve hours ahead. Both die at the same point, a gap of eight hours. The question is how many of each set falls inside the surviving range.

The whole-hour set has twenty-five members and fifteen of them lie strictly inside eight hours of zero. The part-hour set has seventy-two members, and forty-eight of them do. The second proportion is marginally higher because the part-hour offsets include values such as seven and three quarters, which survives, while there is no whole-hour offset between seven and eight to fall in that band.

Both part-hour rows split two to one, exactly. That is not a coincidence either: within eight hours of zero there are twice as many quarter positions as outside it, across the twelve-hour reach of the record.

The proportions here are a property of where the record's boundaries were drawn, not a finding about the world. They are reported because leaving them out would invite somebody to compute them wrongly.

What the forty-eight actually contain

Twenty-four distinct non-zero amounts of shared time, running from a quarter of an hour to seven and three quarters in quarter-hour steps. Every value in that sequence appears.

The whole-hour offsets, by contrast, produce only eight distinct non-zero answers: the integers from one to eight. So the part-hour offsets, which are usually treated as a rounding detail, generate three times as many distinct outcomes as the offsets everybody reasons about.

Why the boundary is at the same place

Both sets die at a gap of eight hours, and it is worth being explicit about why, because it is the one thing that could plausibly have been otherwise.

The arrangement fails when the gap reaches the length of the shorter working day. That length is eight hours in this part of the record, and it does not know or care whether the gap approaching it is a whole number.

So a gap of seven and three quarters leaves a quarter of an hour, a gap of seven and seven eighths would leave an eighth, and a gap of exactly eight leaves nothing. The approach to the boundary is continuous and the boundary itself is sharp.

What differs between the two sets is only how densely each samples the approach. The whole-hour set takes one sample on the way in — the gap of seven, leaving one hour — and then lands on the boundary. The quarter-hour set takes four, and so has three arrangements sitting in territory the whole-hour view does not represent at all.

Three of those five rows exist only in this section. An arrangement sitting on one of them is, as far as the whole-hour table is concerned, either a one-hour arrangement or a dead one, and it is neither.

The caution that belongs with the count

Forty-eight sounds like a large number of workable arrangements, and the count on its own is misleading in the usual way: it treats a quarter of an hour of shared time and seven and three quarters as equally successful.

Twelve of the forty-eight leave less than two hours. At the bottom of that range are the arrangements leaving fifteen or thirty minutes, which are counted here as leaving something because they do, and which support almost nothing.

The honest summary is the same as everywhere else in this record: say how many leave nothing, then say the distribution of what the rest leave, and never compress the two into a single proportion.

Next along

• Why A Part-Hour Gap Is Harder To Schedule
  The shared stretch no longer starts on the hour, and everything built on hourly slots loses a slot. The loss e
• The Law Does Not Care About Fractions
  The one-hour-per-hour rule is exact for any offset, whole or not, while both working days are the same length.
• Why These Get Left Out
  Part-hour offsets are omitted from almost every discussion of these arrangements. The reason is not a good one
• Gaps That Do Not Divide Into Hours
  Seventy-two offsets here are not whole numbers of hours, and forty-eight of them leave shared time. They get l