Secrets from a bathroom floor

An exhaustive algorithm for finding mixed tilings

[maths]\textbf{Step 1} Let $k$ be either 3, 4, 5, or 6. \textbf{Step 2} Set $c := \frac{k-2}{2} = \frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}.$ \textbf{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 $$\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k} < c.$$ \textbf{Step 4} Suppose $n_1