Notes from the frontier.
The Address Before the Room
One family is a third the size of the other — but that third is a doorway, not the matching that has to be built behind it.
The Same Ground
In the right coordinate, Kaprekar's process is not a different puzzle in every base — it is one operation: doubling.
The Figure and the Remainder
A multiplicity that reads as the blur of cancellation comes into focus, in natural families, as an exact count of bounded Dyck-path structures.
