# Squaring Numbers between 40 and 60

Here’s a quick way to square numbers between 40 and 60. You know the squares of 40, 50, and 60, right?

So let’s show what to do for N=51 through 59. Write N as (50+R) for R between 1 and 9. Then N2 will have first two digits (25+R) and last two digits R2. Thus:

522= 2704,
582= 3364,

since 27=25+2 and 33=25+8.

The rule for N=41 through 49 is similar. Write N as (40+R) for R between 1 and 9. Then N2 will have first two digits (15+R) and last two digits (10-R)2. So:

422= 1764.
482= 2304.

Presentation Suggestions:
This Fun Fact goes nicely with other lightning arithmetic tricks, such as Squaring Quickly and Squares Ending in 5. The reference gives many more secrets of mental math.

The Math Behind the Fact:
These “rules” are easy to check by noting, for example, that

(50+R)2 = 2500+100*R+R2.

And since (40+R)2 = (50 – (10-R) )2, we have

(50 – (10-R) )2 = 2500 -100*(10-R)+(10-R)2 = 1500 +100*R+(10-R)2.

Su, Francis E., et al. “Squaring Numbers between 40 and 60.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
A. Benjamin and M. Shermer, Secrets of Mental Math, Three Rivers Press, 2006

Fun Fact suggested by:
Francis Su

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