I was stunned when I first saw the article Counterexample to Euler's conjecture on sums of like powers by L. J. Lander and T. R. Parkin:.
How was it possible in 1966 to go through the sheer astronomical space of possibilities, on a CDC 6600 computer?
1) Did Lander and Parkin reveal their strategy?
2) How would you, using all the knowledge that was accessable until 1965, go to search for counter-examples, if you have access to a computer with 3 MegaFLOPS?
(As a comparison, todays home computers can have beyond 100 GigaFLOPS, using GPUs even TeraFLOPs)