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Several discrete geometry problems are closely related to the arithmetictheory of elliptic curves defined on the rational fields $\mathbb{Q}$. In thispaper we consider the $\theta$-congruent number for $\theta=\frac{\pi}{3}$ and$\frac{2\pi}{3}$ and tiling number n. For the case that $n\geqslant 2$ issquare-free odd integer, we determine all $n$ such that the Selmer rank ofelliptic curve $E_{n,\frac{\pi}{3}}:\ y^2=x(x-n)(x+3n)$ or/and$E_{n,\frac{2\pi}{3}}:\ y^2=x(x+n)(x-3n)$ is zero. From this, we provideseveral series of non $\theta$-congruent numbers for $\theta=\frac{\pi}{3}$ and$\frac{2\pi}{3}$, and non tiling numbers n with arbitrary many of primedivisors.

$\theta$-Congruent Numbers, Tiling Numbers and the Selmer Rank of Related Elliptic Curves: odd n