jmc

Since we are partitioning 81 into sums of perfect squares, we proceed by subtracting out perfect squares and seeing which work: $81-64=17=16+1$. Further, $81-49=32=16+16$. And finally, $81-36=45=36+9$. Although there is more to check through, this sort of method should convince us that these are the only $\boxed{3}$ solutions: $1^2+4^2+8^2=81$, $4^2+4^2+7^2=81$, and $3^2+6^2+6^2=81$.