Thank you BWL. This is now getting a bit out of my league: but I note that the number you produce is the square of 'the number I first thought of', as the catch-problems used to say.. Is that why the phenomenon of the added digits repeats. And will the same thing happen by adding the digits of 6561²? No -- their sum appeared to be 27: the cube of the square root of 18/2 Now, where does that fit into the relationship?

See what I mean about not being any sort of mathematician. One of those would presumably have extrapolated some general principle from all this.

But not me tho... LoL