Symmetric number

Take a integer n between 10 and 99. We write n=10x+y with x, y between 0 and 9.  We define the symmetric number \bar n=10y+x. For example, if n=74 then \bar n=47.

 

It is then easy to show that the difference of the square n^2 -\bar n ^2 can be express as the sum of differences between 3 digits symmetric numbers.

n^2 -\bar n ^2=10(m-\bar m)+(m'-\bar m') with

m=100a_x+a_y and m'=100b_x+b_y, with x^2=10a_x+b_x and y^2=10a_y+b_y.

 

For example with n=74 we have 7^2=4\times 10+9 and 4^2=16, leads to

74^2-47^2=10\times(401-104)+(906-609)

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