Here’s what was in the rabbit hole from last time (I had been almost there):
I had way too much data to exploit, so I started to think about culling it out, using the length of the “mumbers” to cut off all the items too big to care about. That led to the key missing insight: my method of mapping mumbers mapped the first digit of each item to the same position – that is, 9, 90, 900, 9000 all had the same angle, just further out. This distance was already a logarithm of the number, but once I dropped my resistance to taking the logarithm twice…
… then I could create a transition plot function which worked for almost any mumber in the sets of mumbers I was playing with …
Then I could easily visualize the small set of transitions – “mumbers” with 3 digits – that yielded the graph above; for reference these are:
The actual samples I wanted to play with were larger, like this up to 4 digits:
This yields a still visible graph:
And this, while it doesn’t let me visualize the whole space that I wanted, does provide the insight I wanted. The “mumbers” up to 10000 do indeed “produce” most of the space of the smaller “mumbers” (not surprising, as the “mumber” rule 2XYZ produces XYZ, and 52XY produces XYXY … meaning most numbers in the first 10,000 will be produced by one in that first set). But this shows that sequences of 52 rule transitions on the left produce a few very, very large mumbers – probably because 552552 produces 552552552552 which produces 552552552552552552552552552552552552 which quickly zooms away to the “mumberOverflow” value at the top of my chart.
And now the next lesson: finishing up this insight, which more or less closes out what I wanted to explore here, took 45 minutes. I had 15 allotted to do various computer tasks before leaving Aqui, and I’m already 30 minutes over that … which suggests again that you be careful going down rabbit holes; unlike leprechaun trails, there isn’t likely to be a pot of gold down there, and who knows how far down it can go?
P.S. I am not suggesting this time spent was not worthwhile; I’m just trying to understand the option cost of various different problem solving strategies so I can become more efficient.