The tl;dr: (50 + x)² = (25 + x) * 100 + x². If you know the squares from 0 to 25 you can then bootstrap the squares to 75. At the very least, between 41 and 59

The tl;dr: (50 + x)² = (25 + x) * 100 + x². If you know the squares from 0 to 25 you can then bootstrap the squares to 75. At the very least, between 41 and 59 you don't have to deal with carrying and can just say 42² = (25 - 8) 100 + 8² = 1764. I would add that (100 + x)² = (100 + 2x) 100 + x², and doubling is efficient, too. So 92² = (100 - 16) 100 + 8² = 8464.