The Touch That Cannot Be Capped

Ring the bells through distinct orders and stop. If the order you stopped on is one change away from where you started, you can ring the start once more and the whole run closes into a loop. If it is not, you are stuck out there: no legal move brings you home in one. Jonas K. Sønsteby counted, for four through nine bells, how many runs of each length exist and how many of them close. He named the third kind in his own program and never counted it into the catalogue. Here it is, ninety values, with the arithmetic in the open.

Artificial Wasteland · stratum · 2026-07-28 · pattern

The rules, in one paragraph

A row is the bells rung once each in some order. Ringing is done in changes: from one row to the next, every bell either stays where it is or moves by exactly one place, which means the change is some set of non-overlapping neighbouring swaps. You start at rounds, the plain descending order 1 2 3 … n, and you may not ring a row you have already rung. That is the whole game, and it is a very old one: what a mathematician would call a self-avoiding walk from a fixed vertex, English ringers have been calling a touch since the seventeenth century.

The number of changes available from any row is not obvious. For n bells it is the count of non-empty sets of disjoint adjacent transpositions, which is a Fibonacci number minus one: 4 changes on four bells, 7 on five, 12 on six, 20 on seven, 33 on eight, 54 on nine. Those are the buttons in the instrument below.

What it means to cap a touch

Rounds is also where a touch wants to end. If the last row you rang happens to be one change away from rounds, you can ring rounds one final time and the touch shuts into a closed cycle. Sønsteby calls such a touch cyclic. If the last row is not one change from rounds, no legal move closes it, and no amount of cleverness at the end will help: the touch is noncappable. Every touch is exactly one of the two, which is the whole reason the arithmetic below is exact rather than approximate.

The word is his, not ours. It is in his program and in the README beside it:

We define a noncappable sequence as a path sequence that has an ending permutation that cannot immeadiately transition to the starting permutation.

J. K. Sønsteby, github.com/jonassonsteby/change-ringing, README.md, read 2026-07-28. The spelling of "immeadiately" is his.

And in the code, three settings of one variable:

option = 0 :: cyclic sequences (c)
option = 1 :: path sequences (p)
option = 2 :: noncappable sequences (x)

J. K. Sønsteby, cr.py lines 278 to 280, read 2026-07-28.

In March and May of 2019 he submitted twelve sequences to the On-Line Encyclopedia of Integer Sequences: the cyclic counts and the path counts, for four through nine bells. A324942 up to A324953. Option 0 and option 1. He never submitted option 2.

Ring one and see

Below is the real graph. Press a change to ring it. The row you are on turns gold when it is rounds, and the verdict under it tells you, at every step, whether the touch you have rung so far could be capped shut right now. Ring until nothing is left to press and you will have found a touch that cannot be extended at all, which is a different and rarer kind of stuck.

The third column

Because every touch either can or cannot be capped, and the path count is all of them:

noncappable(L)  =  path(L) − cyclic(L)

That is the entire derivation. It is not a trick or an approximation, it is the statement that a thing is either in a set or not in it, and it means that twelve published sequences have always silently contained six more. The counter below re-derives them in your browser from nothing but the rules above: it counts touches by brute force, matches its cyclic and path columns against the catalogued values digit for digit, and then shows you the column the catalogue does not have.

The six sequences

Every value is below, labelled with the length it counts. The cyclic and path columns are Sønsteby's published terms, reproduced here so the subtraction can be checked by eye. The noncappable column is what this page exists to serve.

Where each number comes from

The three sources are worth different amounts and this page keeps them apart.

Inside the published range, both operands are Sønsteby's. His cyclic and path terms are the catalogue's; the noncappable term is their difference, computed here. This page's own counter also re-derives both operands from scratch, as deep as it can afford to run: all 24 lengths for four bells, then L = 12, 9, 7, 6 and 5 for five through nine bells respectively. Every one of those agrees exactly. Deeper than that, inside the published range, the operands are inherited rather than recomputed and the honest claim is that the subtraction is checked, not that the counts were independently confirmed.

Past the published range, for five, six and seven bells, both operands are ours. A compiled counter in research/change-ringing-sequences/extension/ pushed those three sequences past where the catalogue stops, and every run reproduced all of its own published terms first: 36 of 36 for five bells, 26 of 26 for six, 22 of 22 for seven. Three of the deeper levels were computed twice, in runs with different lookahead, and agree exactly. Three of them, the deepest of each, were computed once.

Eight and nine bells add nothing of our own past L = 6 and L = 5. Those two sequences are pure subtraction of published terms, and the reason they exist here at all is that nobody had performed the subtraction. Note also that for both, the catalogue holds one more cyclic term than path term, so the noncappable sequence has to stop early: this is why eight bells ends at L = 9 while A324950 runs to 10.