You may have seen this:
1/81 = 0.012345679012345679...
But you may not have seen this:
1/243 = 0.004115226337448559...
How does the pattern continue? What is special about the number 243? (Hint: find its factors.) What causes the pattern? Are there analogous numbers in other bases?
Source: I found this in "Surely, You're Joking Mr. Feynmann" by Richard Feynmann.