Find the value of $\alpha+2 \beta$ angle, if $\operatorname{tg} \alpha=\frac{1}{7}, \operatorname{tg} \beta=\frac{1}{3}$?
If $\operatorname{tg} \beta=\frac{1}{3}$, then $\operatorname{tg} 2 \beta=\frac{3}{4}$, so

$$
\operatorname{tg}(\alpha+2 \beta)=\frac{\frac{1}{7}+\frac{3}{4}}{1-\frac{3}{28}}=1
$$

thus

$$
\alpha+2 \beta=45^{\circ}
$$

Number of solutions: 24.