Questions about normal numbers Hi everyone,
Can you construct a normal sequence in a given base (>1) such that there doesn't exist n such that the n-th suffix of the sequence has n for prefix?
Can you construct a normal sequence such that there are infinitely many such n?
 A: I think the answer to the 2nd question is yes (provided I understand the question). Take any normal sequence in any base $b>1$. Change the 1st symbol (if necessary) so the sequence begins with 1. Change symbols as necessary so the 10th suffix begins with (the base-$b$ expansion of) 10, the 100th suffix with 100, the 1000th with 1000, etc. You're changing a very sparse subset of symbols, so the sequence is still normal, but you have infinitely many $n$ such that the $n$th suffix has $n$ for a prefix. 
EDIT: Here's a possible plan of action for the first question. Start with everybody's favorite normal number, Champernowne's - the one you get by concatenating the naturals, base 10, in order; .123456789101112131415161718192021.... This violates the requested property very badly, as the first 10 suffixes do the wrong thing, but a) maybe the number is sufficiently well-behaved that you can figure out exactly where all the violations are, and, if so, then b) maybe you can prove that the violations are really, really sparse, so sparse that you can change them all to non-violations without affecting normality. 
A: Here's an argument that might work for the first question.
First, take an arbitrary normal number. Given any positive integer k, choose a big chunk of that number so that all sequences of length up to k occur with the correct frequency. Now randomly translate the whole sequence by an amount that's large in absolute terms but much smaller than the length of the sequence. Then on average only a small proportion of suffixes will start in the wrong way (about logarithmic in the length of the sequence, I think, since the nth suffix has a chance of about 1/n of being bad). Change all the digits that start a bad suffix to zero. This makes the number satisfy your property but doesn't change the frequencies of substrings very much. 
Now do the same for a much longer chunk, and so on until you've built a normal number with the desired property.
I haven't checked that this actually works, but it feels reasonably plausible to me.
