what you're seeing
A physical constraint: a ringer can only swap their bell with a neighbour — one place at a time.
So the only legal move is a single adjacent transposition. The surprise: under that one rule you can ring
every order of n bells exactly once and return home.
That is a Hamiltonian cycle on the permutohedron — the Cayley graph of the symmetric group
Sₙ generated by adjacent swaps. We ring plain hunt, which closes the loop
(the last row returns to the first by one more swap).
Every order has a unique address. A second, independent counter — the Lehmer code (factorial base) — gives each
ordering a number in 0…n!−1. The proof: the hunting walk and the counting address are
two strangers who share no code, yet they agree — the walk visits each address once.
Why 7 is magic: 7! = 5040 — a full peal on seven bells rings all 5040 changes in ~3 hours, no order twice.
The one legal move is an adjacent transposition — the same atom as the
best-rational continued
fraction and the
abacus carry: change by neighbours, one place at a time.