Create 3 input AND from 2 input NANDs
I think you are overthinking this. You probably know that \$\mathsf{and}(x,y) = \mathsf{not}(\mathsf{nand}(x,y))\$, and negation can be implemented using \$\mathsf{not}(x) = \mathsf{nand}(x,x)\$ or \$\mathsf{not}(x) = \mathsf{nand}(x, \mathbf{1})\$. 

Here,
$$ r = \mathsf{and}_3(x,y,z) = \mathsf{and}(\mathsf{and}(x,y), z),$$
and you can do no better than using 4 nand-gates:

$$ w = \mathsf{and}(x, y) = \mathsf{not}(\mathsf{nand}(x,y)) \\
r = \mathsf{and}(w, z) = \mathsf{not}(\mathsf{nand}(z,w))
$$