I have to solve the following system of equations (which appeared while obtaining limiting distribution of a Markov chain) manually without using software. Any suggestion which way to think or how to start, as it will take a lot of time in solving them in usual manner.
Further I would like to state that the value of only $\pi_1 $ has been asked to find, and it must equal $\frac{1}{23}$, so if that can be obtained without calculating rest of the values i.e if there is some other idea or concept that I am not able to think of, please let me know.
\begin{align*} \pi_1 &= \frac {1}{2} \pi_{23} + \frac {1}{2} \pi_2 \\ \pi_2 &= \frac {1}{2} \pi_1 + \frac {1}{2} \pi_3\\ \pi_3 &= \frac {1}{2} \pi_2 + \frac {1}{2} \pi_4\\ \pi_4 &= \frac {1}{2} \pi_3 + \frac {1}{2} \pi_5\\ &\ \ \,\vdots\qquad \ \ \ \, \vdots\\ \pi_{21} &= \frac {1}{2} \pi_{20} + \frac {1}{2} \pi_{22}\\ \pi_{22} &= \frac {1}{2} \pi_{21} + \frac {1}{2} \pi_{23}\\ \pi_{23} &= \frac {1}{2} \pi_{22} + \frac {1}{2} \pi_1.\\ \end{align*}
I think it is easy to see that having all the $\pi_i$ equal to each other gives a solution.
I assume you also want $\sum \pi_i$ to be $1$. This would give the solution $\pi_i = \frac{1}{23}$.
I didn't explain why that is the unique solution.