# What is the complexity of finding the number (mod 2) of multicolored edges on a loop?

Let $C$ be a circuit that maps $n$-length bitstrings to elements of $\{0, 1, 2\}$. Arrange the $n$-length bitstrings in a giant loop: $0^n$ is connected to $1^n$ and $0^{n-1}1$, $0^{n-1}1$ is connected to $0^n$ and $0^{n-2}10$, etc. An edge of this loop is multicolored if it connects $x$ and $y$ but $C(x) \ne C(y)$.

I want to decide whether there are an even or odd number of multicolored edges on the loop. What is the computational complexity of this problem?

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I can't make head or tail of this. In what way does a "circuit" map things? You also haven't defined your "giant loop". –  Brendan McKay Apr 11 at 8:43