Forty-six rings

A ring is a closed chain of dependencies: each step in it waits, directly or at a remove, on a step that waits on it. The map holds forty-six of them.

They were counted by enumeration rather than sampling. Every closed path in the drawing was found, and a ring is counted once however many ways round it can be written.

Six rings, shortest first

• arc above: one requirement
• arc below: the one that closes it
• dashed: a ring with give in it

Six rings drawn, the longest of four steps. Four of them have no give at all: every step in them has to be finished before the next can start. The operational question here can be tested with team productivity reports, provided the team agrees what the information can and cannot establish.

The arc below each row is the one that sends you back to the start. A dashed ring has give in it somewhere; a solid one is shut at every joint. Additional public research relevant to complex systems and work can be found at University of Cambridge.

By length

Forty-six in total, of which seventeen are shut at every joint and twenty-nine have at least one soft link somewhere.

The distribution peaks at six. That is a consequence of the map's shape rather than anything meaningful: the tangle is concentrated in the place and money families, which between them hold twelve steps, and six-step paths through twelve steps are simply the most numerous.

Shortness is not severity

The shortest ring in the map is two steps and shut: the file waits on the fingerprints and the fingerprints wait on the file. Thirty-five days to go round.

The longest ring is nine steps and has give in it. A hundred and seven days to go round, and somewhere in those nine steps there is a provisional version of something that lets a person in.

So the two-step ring is the worse object despite being four times smaller. A short shut ring is a wall with nothing on either side of it; a long ring with give is a long walk.

Counting steps in a ring tells you how far round it is. It tells you nothing about whether you can get in.

Forty-two dependencies, forty-six rings

The forty-six rings are built from only forty-two distinct dependencies, which means the rings overlap heavily. Most of them share most of their edges.

Thirty-four of the forty-six pass through a single step. Forty-two of the forty-six pass through one of two steps. Five steps, taken out of turn, would leave none of the forty-six standing.

That is the most important fact in this part of the record, and it cuts against the alarm that the number forty-six creates. There are not forty-six separate problems here. There is one knot, described forty-six times.

What the rings have in common

Thirty-five of the forty-six pass through the place family or the money family or both. Twenty-three of the thirty-six steps stand in at least one ring, and of the ten most heavily ringed steps, five are places, four are money and one is body and time.

Eleven rings avoid both families. Ten of those eleven sit in the upper half of the map, around cover against illness, the card in hand and the file, and are variations on a single pair.

The eleventh is the strangest object in the record: a four-step ring made entirely of paper, where a translation waits on a stamp, the stamp on a sworn copy, the sworn copy on a certification, and the certification on the translation.

That one is shut, and it is the purest example here of a ring produced by nothing but two offices disagreeing about which document comes first.

Why the count is not the measure

Forty-six is the number of closed paths. It is not the number of problems, and three other figures describe the tangle better.

Each of those counts something once. The count of rings counts the same knot from forty-six directions, which is why it is the largest and least useful of the four.

A reader who takes one number away from this map should take the third one. Five steps, taken out of turn, leave nothing circular anywhere in the map — and one step appears in every one of the forty-eight different five-step sets that would do it.

Forty-six is the frightening number and five is the true one.

What the shortest and longest share

Both are made of steps that take between two and thirty days, both run through parts of the map that nobody would call difficult, and neither contains a single step that anybody would describe as an obstacle in its own right.

The slowest member of the longest ring in this record takes twenty days. There is no hard step anywhere in these forty-six circles; there are only fortnight-long errands arranged so that none of them can be first.

What a ring is not

It is not a mistake in anybody's rules. Each dependency in a ring is individually reasonable: a bank wanting to know where you live, an office wanting to know you have somewhere to live, a landlord wanting to be paid from somewhere local. Nothing in a ring is unreasonable except the ring.

It is also not necessarily a thing anybody experiences. Every ring in this map is got round, constantly, by people producing something provisional or by somebody accepting something they strictly should not. What the map shows is what the rules would do if followed exactly, which is a different object from what happens.

A ring is what a system looks like when nobody has read it end to end. It is nobody's decision and it is in nobody's power to notice.