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# notation - can someone explain these strange properties of 10, 11, 12 and 13?

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## notation - can someone explain these strange properties of 10, 11, 12 and 13?

suppose we have a 2 digit number $x$. we can write is in terms of its digits $a$ and $b$. when we attempt to square this number, we get an interesting result.$$x^2=(10a+b)^2=100a^2+10(2a+2b)+b^2$$we can also flip the digits (i'll use $\bar x$ to indicate this) and then square.$$\bar x^2=(10b+a)^2=100b^2+10(2a+2b)+a^2$$this result isn't very useful on its own, but if $a^2$, $b^2$, and $2a+2b$ are all less than $10$, then the three terms above are the three digits of $x^2$ and $\bar x^2$ respectively. it is clear from that that reversing the digits of $x$ reverses the digits of $x^2$ provided it meets those requirements. (if we switch $a$ and $b$, the first and last terms switch while the middle term is unchanged.)note that $10$, $11$, $12$, and $13$ (as well as $20$, $21$, $22$, $30$, ...