> So how one achieves this magic number 7?

The fact that monoids have to be associative heavily constrains the set of possibilities. [This blog post](https://kavigupta.org/2016/07/20/Counting-Monoids/) goes into some detail about counting monoids and similar structures. They give a formula for "pseudomonoids" (which are really [pointed magmas](https://en.wikipedia.org/wiki/Magma_(algebra))) as \\(n \cdot n^{n \cdot n}\\) (read: \\(n\\) choices of point and \\(n^{n \cdot n}\\) choices of binary operator), which can be adapted to account for the behavior of the (fixed) unit as \\(n^{(n-1) \cdot (n-1)}\\). When \\(n = 3\\), this gives exactly your result of \\(81\\), showing that the objects you're counting need not be associative!

The fact that monoids have to be associative heavily constrains the set of possibilities. [This blog post](https://kavigupta.org/2016/07/20/Counting-Monoids/) goes into some detail about counting monoids and similar structures. They give a formula for "pseudomonoids" (which are really [pointed magmas](https://en.wikipedia.org/wiki/Magma_(algebra))) as \\(n \cdot n^{n \cdot n}\\) (read: \\(n\\) choices of point and \\(n^{n \cdot n}\\) choices of binary operator), which can be adapted to account for the behavior of the (fixed) unit as \\(n^{(n-1) \cdot (n-1)}\\). When \\(n = 3\\), this gives exactly your result of \\(81\\), showing that the objects you're counting need not be associative!