Squaring Palindromes to Make other Palindromes

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How many single digit palindromes when squared produce a square palindrome?

First, we must understand square numbers. These are numbers that when we multiply a number by itself such as 1,4,9,16,25,36,.....

Surprisingly, only three one digit palindromes square to make a palindrome.

1x1=1

2x2=4

3x3=9

How many two and three-digit palindromes when squared produce a square palindrome?

11x11=121

22x22=484

101x101=10201

111x111=12321

121x121=14641

202x202=40804

212x212=44944

What do they all have in common?

There are an infinite number of palindromes with only the digits 0, 1, and/or 2 that when squared produce a palindrome.

Can you find the patterns of the original palindromes and their resulting squares?

11x11=121

101x101=10201

1001x1001=1002001

10001x10001=100020001

100001x100001=10000200001, etc.

22x22=484

202x202=40804

2002x2002=4008004

20002x20002=400080004

200002x200002=40000800004, etc.

Don't you love this infinite series?

How far will palindromes with only ones as digits continue to have palindrome squares? And what happens to the resulting squares after they are no longer palindromes? (hint: use graph paper)

1x1=1^2=

11x11=11^2=  

111x111=111^2=              

1111x1111=1111^2=       

11111^2=

111111^2=      

1111111^2=    

11111111^2=  

111111111^2=

1111111111^2=

11111111111^2=

111111111111^2=

1111111111111^2=

11111111111111^2=

111111111111111^2=

1111111111111111^2=

11111111111111111^2=

111111111111111111^2=

The children very quickly see this pattern and claim to understand when it stops. The challenge is to find out what happens when the square is no longer a palindrome but is only "palindromic." Close to a palindrome. I gave answers below that go to 18 ones squared. The pattern of numbers "in between" is quite beautiful.

1x1=1^2= 1

11x11=11^2=   121

111x111=111^2=               12321

1111x1111=1111^2=        1234321

11111^2= 123454321

111111^2=       12345654321

1111111^2=     1234567654321

11111111^2=   123456787654321

111111111^2= 12345678987654321

1111111111^2=       1234567900987654321

         (from 11 to 18 ones squared)

11111111111^2=123456790120987654321

111111111111^2=12345679012320987654321

1111111111111^2=1234567901234320987654321

11111111111111^2=123456790123454320987654321

111111111111111^2=12345679012345654320987654321

1111111111111111^2=1234567901234567654320987654321

11111111111111111^2=123456790123456787654320987654321

11111111111^2=12345679012345678987654320987654321

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