The Day With A Hole In The Middle

The split shape runs nine to one, breaks for two hours, and runs three to seven. Eight working hours, the same as early, standard and late, arranged in two pieces instead of one.

Everything strange in this record comes from that hole.

It breaks the one law

Two equal days lose an hour of overlap per hour of gap. Two split days do not. Here is what they actually do:

A manager can review the time-tracking approach, run one complete cycle and invite the team to challenge any misleading interpretation. A useful external reference for the surrounding topic is When2meet.

Read that column twice. The shared time falls to two hours at a gap of four, then climbs back to four hours at a gap of six, then falls again. It is the only sequence in the record that goes up.

The reason is straightforward once seen. At a gap of four, the first block of one day has slid onto the other day's break, and so has the second. At a gap of six, the first block of the moved day has slid all the way onto the other day's second block. The pieces have stopped overlapping their own counterparts and started overlapping the next one along.

A day with a hole in it does not degrade. It goes in and out of phase, and there are gaps at which being further apart is better than being closer.

It reaches further than anything else here

Two split days still share an hour at a gap of nine. No other pairing in this record survives that far: two long days give out at nine, and every eight-hour pair of a single block gives out at eight or sooner.

Counted across the full range, the split-with-split pairing leaves nothing at only six of the twenty-five gaps. Every pairing of early, standard or late with each other leaves nothing at ten of them.

So on the single measure of how many arrangements survive at all, the broken day is the best shape in the record.

And it is worse where most people are

At a gap of two, two split days share four hours. Two standard days share six. At a gap of four they share two against four.

The split shape wins at the edges and loses badly in the middle, which is where most arrangements actually sit. A pair at a gap of two who both adopt a split day to improve matters would lose two hours by doing it.

Against an unbroken day the split shape never does better than six hours even at no gap, because its own break removes two hours from any single-block day it meets.

Three pieces at a gap of three

The phase behaviour produces one arrangement stranger than the rest. At a gap of three hours, two split days share three hours — delivered in three separate pieces.

Each day consists of two blocks. Offset by three hours, the first block of the moved day catches the tail of the other's first block, the second block catches the tail of the other's second block, and a third small window opens where the moved day's first block meets the other day's break-end.

The total in the table reads three hours, which is a perfectly respectable figure for a gap of three. What it represents is three separate windows, none of them long, scattered across an eleven-hour stretch of clock.

For anything requiring a continuous run — a conversation that develops, a piece of work done together, a meeting that might overrun — three hours in three pieces is worth very much less than three hours in one, and the table cannot say so. That is why the section on broken overlaps exists separately.

Why the dip and the rise are not symmetric

Reading the sequence again: eight, six, four, three, two, three, four, three, two, one, nothing.

The fall from eight to two is steep — six hours lost across four hours of gap, which is faster than the one-hour-per-hour law, because both blocks are losing their counterparts simultaneously. The recovery from two to four is slower and shorter, and the second fall is slower still.

The shape is not a simple wave. It is the sum of several overlaps appearing and disappearing at different rates, and there is no simple sentence that reproduces it. This is the one place in the record where the answer genuinely has to be looked up rather than derived, and that is itself worth knowing: a broken day is the point at which the arithmetic stops being memorable.

Every other figure in this record can be reconstructed from one sentence. The split-with-split column cannot, and anybody relying on intuition there will be wrong in both directions.

What it is actually for

Not comfort. A working day with a two-hour hole in the middle is harder to live than one without, and the record has nothing to say about that cost.

Its use is narrow and real: it is the only shape here that makes a very wide gap workable at all. If the alternative is a gap of nine hours and no shared time whatsoever, one shared hour bought by breaking both days is a genuine option. At a gap of two it is not an option, it is a mistake.

Next along

• When The Shared Time Arrives In Two Pieces
  Forty-nine arrangements deliver their shared time in more than one piece. Every one of them involves the split
• All Six Hundred And Twenty-Five Pairs
  Five shapes at each end and twenty-five gaps: six hundred and twenty-five arrangements, all computed. The extr
• The Pairing That Fails Soonest
  An early day against a late one runs out two hours sooner than an identical pair. The two shapes have added a
• The Pairing That Survives Longest
  Two broken days still share an hour at a gap of nine. They pay for it by sharing least where most arrangements