Squares of a number yields a palindrome?

You can write $11^n = (10^0 + 10^1)^n$ and then you will get that the digit in the $k$th place is the number of ways that you can choose $n-k$ exponents to be $0$ and $k$ exponents to be $1$. This gives you palendromes (at least for $n$ small enough) and identities similar to Pascal's triangle. However, as soon as $n$ is large enough that the "digit" you compute in the $k$th place is greater than $9$, then the palendrome phenomenon will break down. Palendrome phenomena for $111^n$ and $1111^n$ can be analyzed similarly.