The Importance of being Odd
  In this letter I consider mainly a finite XXZ spin chain with periodic boundary conditions and \bf{odd} \rm number of sites. This system is described by the Hamiltonian $H_{xxz}=-\sum_{j=1}^{N}\{\sigma_j^{x}\sigma_{j+1}^{x}   +\sigma_j^{y}\sigma_{j+1}^{y}   +\Delta \sigma_j^z\sigma_{j+1}^z\}$. As it turned out, its ground state energy is exactly proportional to the number of sites $E=-3N/2$ for a special value of the asymmetry parameter $\Delta=-1/2$. The trigonometric polynomial $q(u)$, zeroes of which being the parameters of the ground state Bethe eigenvector is explicitly constructed. This polynomial of degree $n=(N-1)/2$ satisfy the Baxter T-Q equation. Using the second independent solution of this equation corresponding to the same eigenvalue of the transfer matrix, it is possible to find a derivative of the ground state energy w.r.t. the asymmetry parameter. This derivative is closely connected with the correlation function $<\sigma_j^z\sigma_{j+1}^z> =-1/2+3/2N^2$. In its turn this correlation function is related to an average number of spin strings for the ground state of the system under consideration: $<N_{string}> = {3/8}(N-1/N)$. I would like to stress once more that all these simple formulas are \bf wrong \rm in the case of even number of sites. Exactly this case is usually considered. 
Yu. Stroganov