Find a square, stochastic matrix of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle ...or prove that none exists.  
Note that such a matrix M couldn't be primitive, so there would be at least one entry equal to zero in every power M^k (Perron-Frobenius theory).
Preferably the matrix would have a diagonal that is not all zero, and at the risk of making the problem imprecise, I'd like to find such a matrix with as few zeros and ones as possible.
Thank you.
 A: $$\pmatrix{0&1&.5\cr1&0&.5\cr0&0&0\cr}$$ has $-1$ as an eigenvalue. 
A: The Karpelevič theorem will make it difficult for the matrix not to at least involve a permutation matrix (notice that the upper-left $2$-by-$2$ block of the matrix given by Gerry is a permutation matrix). 
If $C_n$ is the $n$-by-$n$ defined by $C_n = [e_n \mid e_1 \mid \cdots \mid e_{n-1}]$, i.e., $C_n$ is the adjancency matrix of a directed $n$-cycle, where $e_i$ denotes the $i^\text{th}$ canonical basis vector of $\mathbb{C}^n$, then the $n$-by-$n$ (row) stochastic matrix 
$$ M_n :=
\begin{bmatrix}
0 & e_1^\top \\
0 & C_{n-1}
\end{bmatrix}$$
has spectrum $\sigma(M) = \{ 0, 1, \omega,\dots,\omega^{n-1}\}$, where $\omega:=\exp(2\pi i/n)$. Thus, $M_n$ has $n-2$ eigenvalues (distinct from unity) on the unit-circle. 
For instance, the matrix 
$$ M_5=
\begin{bmatrix}
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0  \\
0 & 0 & 0 & 0 & 1  \\
0 & 1 & 0 & 0 & 0  \\
\end{bmatrix} $$
has spectrum $\{0,\pm 1, \pm i\}$.
A: If you want a doubly stochastic matrix,
$$\begin{pmatrix} 
0&1 & 0&0&0 \\
1&0 & 0&0&0 \\
0&0 & 1/3&1/3&1/3 \\
0&0 & 1/3&1/3&1/3 \\
0&0 & 1/3&1/3&1/3 \\
\end{pmatrix}$$
has eigenvalue $-1$.
