So the question asks:
There are N (N > 1) stocks with the same variance $σ^2$ and the same pairwise correlation coefficient γ (i.e. $c_ij$ = γ for all i = j. γ is a given constant such that 0 ≤ γ < 1). Find the minimum variance portfolio.
So far I have: since all the assets play the same role (same variances, same covariances), so the weights have to be equal. As the weights add up to 1, each weight is 1/n.
But is there any way I can "compute” the weight instead of "stating" it? And how can I "show" the minimum variance portfolio in math?
You are right to say that the optimal weight for each stock in the minimum variance portfolio of n stocks = $\frac{1}{n}$. Given this
Variance of the minimum variance portfolio $=n\frac{\sigma^2}{n^2}+ {n\choose2} \frac{2\gamma\sigma^2}{n^2} = \frac{\sigma^2}{n}+\frac{n(n-1)}{2}.\frac{2\gamma \sigma^2}{n^2}$
If you simplify, you get
Variance of this portfolio $$= \frac{\sigma^2}{n}\left(1+(n-1)\gamma\right)$$
If you select n stocks with the same vairnce $\sigma^2$ then you don't need to compute optimal weight, because it is simply $\frac{1}{n}$ and you can compute the varince as stated above.