Mental Math Tricks - Finding Two Digit Square Roots

Date: 12/03/2003 at 06:23:59
From: Nat
Subject: how to find a two digit squared number a  root number 

Do you know if there is any trick behind looking at a number which is
the result of a two digit number having been squared and being able to
tell what the number was that was squared? 

I know it is possible as it was shown to the class as a game to
outsmart the teacher.  However, nearly every time he was able to tell
the number and if wrong could correct the error.  

Since the times when the teacher was wrong it was 10 off I have been 
playing around with adding the digits together, dividing the number by
a constant value, then looking for patterns.  But I am yet to find
one.  The information won't help me outsmart the teacher but it would
help me in understanding how he does it each time.

Date: 12/03/2003 at 12:00:36
From: Doctor Rick
Subject: Re: how to find a two digit squared number a  root number 

Hi, Nat.

This is a good question. It's not so much a trick as a skill in 
estimating that is good to know.

I'm surprised that your teacher would be off by 10, because the tens 
digit is the easier digit to be sure of.  In a lot of cases there are 
two possibilities for the units digit, and it would take more effort 
to tell which is correct.

Consider the number 62^2 = 3844.  I can tell the tens digit by looking 
at the thousands and hundreds digits and knowing the squares of 
numbers from 1 to 9.  In this case I see 38.  The nearest perfect
square below 38 is 36, which is 6 squared.  Therefore I know the tens
digit of the square root is 6.

Why does this work?  The square of any number between 60 and 70 will 
be between 3600 and 4900.  The square of a multiple of 10 is a perfect 
square times 100.  If you know the squares of the numbers 1 to 9, you 
know the squares of multiples of 10 up to 90.

The units digit of a perfect square is the square of the units digit 
of the root.  To see this, look at the way we do long multiplication:

    62
  * 62
   ---
   124
  372
  ----
  3844

The only number that gets into the units column is the units digit of 
the product of the units digits.  If you know that 2^2 = 4, then you 
know that the square of any number ending in 2 (not just a two-digit 
number, either!) will end in 4.

Can you work backward, though?  If you know that the square ends in 4, 
can you be sure that the root ends in 2?  No, because 8^2 = 64 also 
ends in 4.  There is an interesting pattern in the units digits of 
squares:

  units  units digit
  digit  of square
  -----  -----------
   0       0
   1       1
   2       4
   3       9
   4       6
   5       5
   6       6
   7       9
   8       4
   9       1

Do you see the pattern?  Only 6 different digits can be the units 
digit of a perfect square: 0, 1, 4, 5, 6, and 9.  Two of these (0 and 
5) only appear once in the table.  The other three appear twice each. 
The two numbers whose square ends in the same digit always add up to 
10.  For instance, 2^2 = 4 and 8^2 = 64; 2 + 8 = 10.

Thus when I see the number 3844 and I am told it is a perfect square, 
I know that it must be the square of either 62 or 68.  Since 3844 is 
much closer to 3600 than to 4900, I guess that it's the square of 62. 
And I'm right!

If your teacher uses my method to guess the roots of perfect squares, 
he or she will be wrong most often when the units digit of the square 
is 6, and will be off by 2 in those cases.  That's what I would try if 
I want to outsmart the teacher: give a number like 56^2 = 3136.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/