Quarter-Hour Offsets

Forty-eight of the offsets in this record sit at a quarter or three quarters of an hour. Thirty-two of them leave some shared working time, and the amounts include values like three and a quarter hours, five and three quarters, and a quarter of an hour on its own.

These are the finest offsets here. Nothing smaller is computed, and nothing smaller is needed, because the pattern is established by this point and a further subdivision would produce more numbers of exactly the same kind.

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What a quarter-hour offset does

The same thing every offset does: it removes its own size from the overlap.

Four even steps from four hours of overlap to three. There is no threshold, no rounding, and no point at which the behaviour changes.

Why a quarter is the awkward one

A half-hour offset shifts the shared stretch to a half-hour mark, which calendars can at least represent — a meeting at half past is an ordinary thing.

A quarter-hour offset shifts it to a quarter past or a quarter to, which almost nothing is organised around. A shared stretch beginning at a quarter past ten does not start at any time that appears on an ordinary schedule, and in practice the first usable hour becomes eleven, which throws away three quarters of an hour of real overlap.

At a quarter-hour offset, the difference between the overlap you have and the overlap you can use is at its widest. Three quarters of an hour of genuinely shared working time is discarded because nothing is built to start at a quarter past.

The small end

A gap of seven and three quarters leaves a quarter of an hour. A gap of seven and a quarter leaves three quarters.

Fifteen minutes of shared working time is, for most purposes, indistinguishable from none — and it is not none, which is the entire reason these offsets are in the record. An arrangement in that position has something, it cannot do much with it, and treating it as zero is wrong by exactly the amount that makes the difference between a short daily exchange and no daily exchange at all.

Twenty-four distinct answers

The quarter-hour and half-hour offsets between them produce twenty-four different non-zero amounts of shared time: every quarter-hour step from a quarter of an hour up to seven and three quarters.

The whole-hour offsets produce eight — the integers from one to eight.

So the part-hour offsets, which are normally treated as a detail, generate three times as many distinct outcomes as the offsets everybody reasons about. That is a strange thing for a record to discover about the part of itself nobody looks at.

What that means for a lookup

It means a part-hour arrangement almost certainly has a figure that nothing else in the record shares. A whole-hour arrangement at a gap of four has the same answer as one at a gap of minus four, and is one of fifteen rows; a part-hour arrangement at a gap of four and three quarters is one of a much larger and finer-grained set.

Practically, this cuts both ways. The finer grain means the answer is more specific and therefore more useful. It also means the answer is less likely to be a number anybody else has quoted, which removes the comfort of recognition — nobody will say "oh, three and a quarter, that is the usual thing" — and that comfort was never worth anything.

A figure that nobody else has is not a worse figure. It is the correct one for one arrangement, which is the only arrangement it has to be correct for.

Why there is nothing finer

The record stops at quarters because the pattern is complete.

Between any two offsets in this record the behaviour is linear, and it is linear for a reason given in the first section: the intersection of two intervals shrinks exactly as fast as one of them moves. Computing offsets of five minutes, or of one minute, would produce a longer table with no new fact in it.

Stopping where the facts stop is a better practice than filling a table because the computer is willing.

Next along

• An Overlap Of Three And A Quarter Hours
  A fractional answer is not an invitation to round. Rounding down discards time that exists; rounding up spends
• The Forty-Eight That Leave Something
  Forty-eight of the seventy-two part-hour offsets leave shared time: thirty-two at quarter marks and sixteen at
• 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.