How a row of seats fills
A row of eleven seats — a waiting room, the top deck of a bus, the bench outside a changing room. The rule has not been posted and will not be broken lightly: do not sit beside an occupied seat while an unoccupied one is free. Below is the rule being obeyed. This entry is drawn rather than written, which is a first here; the words underneath it are a transcript, not an argument.
In words, for anyone the drawing does not reach. The same row of eleven seats, numbered left to right, at five stages. One: all empty. Two: seat 1 taken; seat 2 now unusable. Three: seats 1 and 11 taken; 2 and 10 unusable. Four: seats 1, 6 and 11 taken; 2, 5, 7 and 10 unusable; only 3, 4, 8 and 9 can still be chosen. Five: seats 1, 4, 6, 9 and 11 taken — and every one of the six seats left over sits beside somebody. Five people have filled a row of eleven.
The arithmetic is worse than it looks, and it has been worked out properly by people who found it interesting enough to name: let a long row fill this way, each arrival choosing at random among the seats the rule still allows, and it jams at just under 44 per cent occupied. Choosing well rescues very little. Take the ends, then halve the gaps, and eleven seats still stop at five. The rule itself permits six — seats 1, 3, 5, 7, 9, 11, every other one, nobody adjacent to anybody. Six requires that somebody walk into an empty-ish room and sit down at seat 3 while seat 11 is free. No one has ever done that. The unposted rule is obeyed so exactly that it costs the room half its furniture, and nobody has ever seen the bill.
What the drawing cannot show is the moment it ends. The sixth person stops in the doorway and takes in the whole row at a glance — five occupied, six empty, no legal move — and the row knows it, because the five look up, or very carefully do not. Then something that has been pure geometry for four stages turns into a small public statement about one particular stranger. That is why people stand when there are seats. Standing is the last move the rule still allows.
Thread: the unwritten law