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 N^{2} will have first two digits (25+R) and last two digits R^{2}. Thus:

52^{2}= 2704,

58^{2}= 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 N^{2} will have first two digits (15+R) and last two digits (10-R)^{2}. So:

42^{2}= 1764.

48^{2}= 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+R^{2}.

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}.

**How to Cite this Page:**

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