Write out the positive powers of 10 in both base 2 and base 5: Now for any integer n > 1, we’ll find exactly one number of length n somewhere on the two lists. They contain one 3-digit number, one 4-digit number, and so on forever — if n = 100 we find a 100-digit number in the 30th position on the base 2 list. (This result first appeared in the 1994 Asian Pacific Mathematics Olympiad. I found it in Ravi Vakil’s A Mathematical Mosaic.) Two further curious lists: If we write out the triangular numbers, those in positions 3,...