Regular expression for strings with an even number of zeroes and ones

Question

What is a regular expression for strings of 0 and 1 with an even number of zeros and an even number of ones?

I have something like (1*01*01*)*(0*10*10*)*

Answer 1

A counterexample for your given regular expression is 01010101.

You may find that writing a regular expression for this particular problem is not going to be possible (unless you use some non-regular extensions to the usual regular expression language).

As mentioned by Jim Lewis below, this should indeed be a solvable problem.