I recently convinced myself that I was right and a published paper was wrong. I did all the calculus by hand and decided they must have missed something. I resolved it by calculation.
I played some very basic examples in my head and thought I was right, but I couldn't be sure. So I quickly implemented a concrete example case in Python using numpy. This helped me resolve pretty quickly that I was wrong; some searching produced a counterexample. This was a much more efficent way to reslove this rather than reading the whole paper in detail. But now I need to work out what I did wrong.