The number $7^2 = 49$ is a square which has has all of it's digits being squares $2^2 = 4$ and $3^2=9$. The number $13^2 = 169$ where $4^2=16$ and $3^2=9$ can be concatenated to make $169$. So now to the question.
How often does this happen, that numbers which are squares have a decimal representation where the string of decimal digits is separable (not necessarily unique) into squares?
Own work So far I have mostly thought about probability and combinatorics. How many squares are in $[10^k,10^{k+1}]$ and how to combine and permute them and what distribution for digits do the squares follow.