jmc

We begin by converting $233_{10}$ into base-4. Since $4^3=64$ is the largest power of 4 that is less than 233, and $3\cdot 64=192$ is the largest multiple of 64 that is less than 233, the coefficient of the $4^3$ term will be 3. From here, we are left with a remainder of $233-192=41$. The largest power of 4 that is less than this number is $4^2=16$, and the largest multiple of 16 that is less than 41 is $2\cdot 16=32$. This leaves us with $41-32=9$, which we can express as $2\cdot 4^1+1\cdot 4^0$. So, we find that $233_{10}=3\cdot 4^3+2\cdot 4^2+2\cdot 4^1+1\cdot 4^0=3221_4$, which has $\boxed{2}$ odd digits.