the wall / 0016
Five thousand and forty
To ring a peal you must ring at least five thousand changes without
repeating one and without stopping, which takes about three hours. The five thousand is not a
round number somebody liked. It is a rounding-down of 5040, which is 7! —
every order of seven bells, each of them once — and it is the largest extent that fits inside
an afternoon. This page starts from the thing that makes the puzzle hard: a bell weighs as
much as a car engine and swings full circle, so between one row and the next it can move
one place. That single constraint decides how many changes exist (a Fibonacci number),
whether an extent is reachable at all (a Hamiltonian cycle), and — the part I did not expect —
which half of the extent an ordinary method can never touch.
The number itself
A peal on seven bells is the extent: 7! = 5040 rows, the tenor
covering behind them, nothing repeated. On eight or more the extent is out of reach —
8! is 40320 rows and about eighteen hours — so the rule asks only for 5000, and
what gets rung is 5040 anyway, because compositions are built out of whole leads and the
old number stuck.
5040 is also 7 × 8 × 9 × 10, and it has 60 divisors: it
divides by every number up to twelve except eleven. Plato wanted exactly this many
citizens in the Laws, for exactly that reason. The ringers arrived at it from the
other end, by asking how many ways seven bells can be put in order, and the same number came
out.
What half a tonne of bronze will let you do
A tower bell is rung mouth-upward, swinging through nearly a full circle and sounding once each way. You cannot hurry it and you cannot hold it: all you can do is nudge it a little earlier or a little later. So from one row to the next, a bell may stay where it is or swap with one neighbour — never move two places, never jump. Several disjoint pairs swap at once; that is a single change.
Count them and the answer is a surprise the first time. A change is a set of
non-touching swaps along a row of n positions — a matching on a path —
and those are counted by the Fibonacci numbers. Take the last bell: it either stays, leaving
the problem for n−1, or swaps with its neighbour, leaving n−2.
That is the whole recurrence, and it is why a tower of twelve has 232 changes rather than
some number nobody has met before.
changes on n bells = F(n+1) − 1
The extent exists, and here it is
Getting every order exactly once, moving one place at a time, is a
Hamiltonian cycle in a graph whose vertices are the n! rows. Nothing guarantees
such a cycle; here it exists, and ringers had it in print in 1668, in Fabian Stedman's
Tintinnalogia, as plain changes — three centuries before it was rediscovered
as an algorithm for generating permutations. Each row below differs from the one above by a
single swap, and the last row differs from the first by a single swap too, so it closes.
The traced path is what ringers call the blue line: not the order of the bells but the movement of one bell through the rows, which is the only thing a ringer can memorise and the only thing they need. Plain changes are honest and nobody rings them for long, because one bell hardly moves for stretches at a time and the shape is dull to hold in the head. Methods exist to make the line memorable, not to make the maths work.
A method, in one line of notation
A method is written as its place notation: for each change, which
positions stay put. x means every pair swaps; 16 means the bells in
first and sixth place stand still and the rest swap around them. Plain Bob on six is
x16 five times over and then x12, and that is the entire
method — twelve changes, after which the pattern repeats with the bells shuffled.
The half you cannot reach
Plain Bob Minor gives 60 true rows in a plain course and the extent is 720,
so you call bobs: at chosen lead ends the last change becomes 14 instead
of 12, which sends the bells round a different way and starts a new course. Call
bobs wherever you like, in any pattern, for as long as you like, and you will never ring the
extent. Not because the search is hard — because half of it is not there.
Every change is a permutation, and permutations are odd or even.
x on six bells is three swaps, odd. 16, 12 and
14 are two swaps each, even. A lead is six xs and six even
changes, so every lead — plain or bobbed — is an even permutation, and composing even
things can never give you an odd one. A single is the escape: 1234 leaves
four bells standing and swaps one pair, which is odd.
Parity says one more thing, and it is why no composition anywhere has an odd number of singles: coming home to rounds means the leads you rang, multiplied together, come to nothing. Plains and bobs are even and contribute nothing; each single contributes one. So singles come in pairs. One of them can open the far half of the extent — it can never shut the door behind you.
The page will try, exhaustively, and tell you what it found — a depth-first search over the sixty leads, backtracking whenever a lead would repeat a row that has already been rung.
Why it takes three hours
Bells strike about four to a second, and a row of n bells is
n strikes. But rows come in whole pulls — a handstroke row, then a
backstroke row — and after the backstroke there is an extra beat of silence before the next
handstroke. The handstroke gap is not decoration; it is how a band hears where it is.
It costs one beat in every 2n+1, and over a peal that is ten and a half minutes.
5040 rows ÷ 2 × (2n+1) beats × beat
And the beat is not free either. The gap between a bell's handstroke and its
next handstroke is one whole swing, so (2n+1) beats must equal the bell's
period as a pendulum — which is set by its size and nothing else. A bell swinging near the
balance takes far longer than the small-swing formula says: at 175° of amplitude the period
is — times the textbook one. Run the arithmetic backwards from a
three-hour peal and the bell you get has an equivalent pendulum length of about
—, which is the size of a real bell and not something this page put in.
Heavier bells swing slower, but only just: geometrically similar bells have
period ∝ √L ∝ M1/6, so a tenor sixteen times the weight of a
light one is slower by a factor of —, not sixteen. Ringing a heavy
tower is a matter of forty minutes more, not a different sport.
What this page checked while you were reading
Every number above is computed here, in this file, from the rules at the top: one place at a time, and no row twice. The tests that live beside this page in the repo re-derive the same numbers by other routes — counting involutions instead of matchings, ranking permutations instead of generating them — and shout if the two disagree.