Secrets from a bathroom floor

An exhaustive algorithm for finding mixed tilings

Step 1 Let $k$ be either 3, 4, 5, or 6. Step 2 Set $c := \frac{k-2}{2} = \frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}.$ Step 3 Set $n$ to be the biggest whole number less than or equal to $\frac{c}{k}$. Note that not all $n_i$ can be bigger than $n$, for otherwise $$\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k} < c.$$ Step 4 Suppose $n_1