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, 33, etc. show a pattern:

T(3) = 6

T(33) = 561

T(333) = 55611

T(3,333) = 5556111

T(33,333) = 555561111

T(333,333) = 55555611111

Similarly:

T(6) = 21

T(66) = 2211

T(666) = 222111

T(6,666) = 22221111

T(66,666) = 2222211111

T(666,666) = 222222111111

(Thanks, Larry.)