Find the pattern and complete the table relating the number of triangles to the number of matches. Express the rule algebraically.

Here are the solutions for parts e) and f):

e) The pattern for the number of matches ($M$) based on the number of triangles ($n$) is given by the rule $M = 2n + 1$.
Using this rule, the completed table is:

$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Number of triangles} & 1 & 2 & 3 & 4 & 5 & 7 & 11 & 13 & 15 & 20 \\ \hline \text{Number of matches} & 3 & 5 & 7 & 9 & 11 & 15 & 23 & 27 & 31 & 41 \\ \hline \end{array}$$

f) If the 'number of triangles' is the input ($n$) and the 'number of matches' is the output ($M$), the rule is:

$$M = \boxed{2n + 1}$$