Square 497, a six-digit answer in seconds

Question

497 is only 3 below 500, and 500 is a round base whose square you know. Exploit the gap. Solve in your head. What is 497^2? Aim: under 20 seconds.

Accepted Answer

Use the slide n^2 = (n-d)(n+d) + d^2 with d = 3 to land on the round number 500: 497^2 = (497 - 3)(497 + 3) + 3^2 = 494 	imes 500 + 9. Now 494 	imes 500 = 494 	imes 5 	imes 100 = 2470 	imes 100 = 247000, so 497^2 = 247000 + 9 = 247009. So 497^2 = \mathbf{247009}. The only real work is 494 	imes 5 = 2470. Takeaway. The slide n^2 = (n-d)(n+d) + d^2 converts any square into a round-number product plus a tiny correction. Aim d so one factor becomes a multiple of 500 or 1000.

Square 497, a six-digit answer in seconds

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