Squares from 41-59

There is a quick trick for squaring numbers from 41 to 59. This method uses 50 as a base. Let the number be $(50 \pm k)$. The square is given by $(50 \pm k)^2 = 2500 \pm 100k + k^2 = 100(25 \pm k) + k^2$.

Procedure:

• Last two digits: Square the difference between the number and 50 ($k^2$).
• Preceding digits: Add or subtract this difference ($k$) from 25. Add if the number is greater than 50, subtract if it's less.

For example, $46^2$: $k = 50-46 = 4$. Last two digits: $4^2=16$. Preceding digits: $25-4=21$. Result: 2116.
For example, $57^2$: $k = 57-50 = 7$. Last two digits: $7^2=49$. Preceding digits: $25+7=32$. Result: 3249.