Why is this not just a test of your knowledge of the base positional system?

456 (b) = 2201 (7)

so

4*b*b + 5*b + 6 = 2*7*7*7 + 2*7*7 + 0*7 + 1

Solve for 'b'. In this case, it's only quadratic so you should be able to solve it. Of course, since we need a nice whole number, you should be able to factor it. Further, since you just made this one up, or it's a really odd example, you'll need to find some other way to solve the quadratic.