Square 35, then compute 34 × 36

Question

Solve both in your head. What is 35^2, and then 34 	imes 36? Aim: under 10 seconds total.

Accepted Answer

Squaring a number ending in 5. For n5, the square is n(n+1) followed by 25. For 35, take n = 3, so n(n+1) = 3 	imes 4 = 12, giving 35^2 = 1225. Now the neighbors. 34 and 36 straddle 35 at distance 1, so difference of squares gives 34 	imes 36 = 35^2 - 1^2 = 1225 - 1 = 1224. The two tricks chain: square the "ends in 5" midpoint once, then the product of its neighbors is just one less. (Why the rule works: (10n + 5)^2 = 100n^2 + 100n + 25 = 100\,n(n+1) + 25.) Takeaway. Any number ending in 5 squares to n(n+1) with 25 tacked on. Pair that with difference of squares and you get its neighbors, l…

Square 35, then compute 34 × 36

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