Square 995 in your head

Question

Solve in your head. What is 995^2? Aim: under 6 seconds.

Accepted Answer

Apply (10n+5)^2 = 100\,n(n+1) + 25 with 995 = 10 	imes 99 + 5, so n = 99: 99 	imes 100 = 9900 \quad\Rightarrow\quad 995^2 = 990025. The multiplication 99 	imes 100 is trivial because n+1 is a round 100, so this large-looking square is actually one of the easiest. Append the 25 and finish. A cross-check with the "close to a round number" method: 995 = 1000 - 5, so 995^2 = 1000^2 - 2\cdot 1000 \cdot 5 + 25 = 1000000 - 10000 + 25 = 990025. Same answer, two ways. Takeaway. Ending-in-5 squaring is size-independent; pick whichever method makes n(n+1) easiest. When the number sits just below a ro…

Square 995 in your head

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