Is there anything similar to this (page written by Field Medalist Timothy Gowers) for quadratic reciprocity ?
I mean, the link there explains how you can figure out the solution of cubic equation by yourself without having a suddent flash of inspiration/ genius genes. Is there some similar guide for quadratic reciprocity ?
There is a book by David Marshall, Edward Odell, and Michael Starbird called "Number Theory Through Inquiry". Chapter 7 is about quadratic reciprocity. It's sort of guided inquiry. The authors suggest questions for the reader, such as looking for patterns in certain tables. So the reader has some guidance and is not just completely on her own. But on the other hand there are a lot of exercises and theorems to be proven; the authors give the reader an opportunity to develop everything. Perhaps this book might fit your needs.