Square 205 in your head

Question

Solve in your head. What is 205^2? Aim: under 6 seconds.

Accepted Answer

The rule (10n+5)^2 = 100\,n(n+1) + 25 does not care how many digits n has. Here 205 = 10 	imes 20 + 5, so n = 20: 20 	imes 21 = 420 \quad\Rightarrow\quad 205^2 = 42025. Append the 25 to 420 and you are done. A quick sanity check: 200^2 = 40000 and 205^2 should be a bit more, 42025 fits. Takeaway. The ending-in-5 shortcut scales to three or more digits as long as you can form n(n+1): 305^2 	o 30 	imes 31 = 930 	o 93025; 255^2 	o 25 	imes 26 = 650 	o 65025. Big round-ish n often makes the multiplication easier, not harder.

Square 205 in your head

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