Did anyone say *only* a normal number includes all possible finite sequences, that normal was *necessary*? I think it was only offered as being *sufficient*, and that pi, like almost all irrationals, was likely, but not proven to be, normal. I'm pretty sure I did not say it was necessary.

I can imagine many ways to construct an infinite sequence that contained all possible finite sequences, and not be normal: say, just make methodically sure it has all 10 1-digit sequences, a string of 10 zeroes, then all 100 2-digit sequences, 100 zeroes, all 1000 3-digit sequences, as many zeroes, etc. So if you go far enough any given finite sequence is there (if not earlier). Because it contains so many 0 sequences, it is not normal.

Sure. Some infinite sequences include all finite sequences, yet are not normal.

Does that rehabilitate the definition of *normal*?

*0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...*

*Skepticism enables us to distinguish fancy from fact, to test our speculations. --Carl Sagan*