# construction of Y gate from X,Z and H gates

As a part of textbook exercise, Y gate is to be constructed using H,Z and X-gates, just like we have $$X = HZH$$. is there some way/process/intuition to find such combinations or it is just like we need to use hit and trial method to find such combination?

First, you got a mistake there. The correct decomposition is $$X = HZH.$$ The way you find the decomposition is by understanding that $$HZH$$ performs the change of basis. The right-most $$H$$ brings a vector from the $$Z$$ basis into the $$X$$ basis. Then $$Z$$ applies the $$NOT$$ operation in the $$X$$ basis. Finally, the left-most $$H$$ brings a vector into the $$Z$$ basis again.
Another way of looking at it is realizing that $$X = HZH= H Z H^{\dagger}$$ is an eigendecomposition of $$X$$ where $$H$$ is a matrix of eigenvectors of $$X$$ and $$Z$$ is a diagonal matrix of eigenvalues of $$X$$.
Following this logic, we can quickly decompose $$Y$$ by just knowing its eigenvectors and eigenvalues. So we get $$Y = (SH) Z (SH)^{\dagger}.$$