Numbers in Reverse

I saw a note recently, pointing out that pairs of 2-digit numbers each squared could be chosen so that the original numbers and the results are each other backwards:

12² = 144 and 21² = 441

Another example:

13² = 169 and 31² = 961

Question: How can we generate more examples?

Solution:

A number such as 89 can be represented as (10×8+9). So, in general, a 2-digit number can be written as (10A+B), where A=8 and B=9 in our example.

The advantage is that we can now make algebraic manipulations. Moreover, the “10” simply means we are going to work in base 10, as usual.

Now, for numbers A and B:

(10A + B)² = (10A)² + B² + 2(10AB)

= 10²A² + 10(2AB) + B²

Also:

(10B + A)² = (10B)² + A² + 2(10BA)

= 10²B² + 10(2AB) + A²

Now A², 2AB and B² are our 3 digits and therefore each digit must be a number from {0,1,2,3,4,5,6,7,8,9} because we are working in base 10.

Remember that a 2-digit number squared will be at most a 4-digit number (99² = 9801, according to my calculator). On the other hand, we should eliminate A=0 and B=0 from consideration because our numbers should not start with “0.”

Also, A² is 1, 2, or 3 because 3² = 9 (but 4²=16, which is more than 1 digit). B² has the same limitation: B² must be 1, 2, or 3.

Finally, 2AB must be less than 10. The product AB must be less than 5, so if A=2, B is 1 or 2. To summarize:

If A=1, then B=1, B=2, or B=3

If A=2, then B=1, or B=2

If A=3, then B=1

The complete list of possibilities is therefore: 11, 12, 13, 21, 22, and 31.

(31² = 961, which is only 3 digits. That justifies why I used 3 terms (instead of 4) in my formula above.

The “reverse digits” phenomena would simply not occur for numbers bigger than 31². The number 32²=1024, but that 4-digit number (and larger) was excluded by my reasoning above.

The obvious extensions to these results is to use a different base (8 or 16 would be fun), or work with 3-digits, 4-digits, squared. The latter case seems messy, unless you can find a pattern — and prove it.

Also, a number or word that is the same forwards and backwards is called a “palindrome.” Popular examples of words are “racecar” and “kayak.”

Notice that 11² = 121, 22² = 484, 33²= 1089. Except for the last result, we have number palindromes. You can tell from our reasoning above why we get palindromes and why it doesn’t work for 33². Are there an infinite number of palindromes that are perfect squares?

(A “perfect square” is a counting number that is another counting number times itself. For example, in 5²=25, 8²=64, 9²=81, the numbers 25, 64, and 81 are perfect squares.)