Al travels for $20$ miles per hour rolling down a hill in his chair for two hours, then four miles per hour climbing a hill for six hours. What is his average speed, in miles per hour?
1. Calculate the total distance Al travels:
   - Rolling down the hill: 
     \[
     20 \text{ miles per hour} \times 2 \text{ hours} = 40 \text{ miles}
     \]
   - Climbing the hill:
     \[
     4 \text{ miles per hour} \times 6 \text{ hours} = 24 \text{ miles}
     \]
   - Total distance:
     \[
     D_{\text{total}} = 40 \text{ miles} + 24 \text{ miles} = 64 \text{ miles}
     \]

2. Calculate the total time Al spends traveling:
   - Rolling down the hill:
     \[
     2 \text{ hours}
     \]
   - Climbing the hill:
     \[
     6 \text{ hours}
     \]
   - Total time:
     \[
     T_{\text{total}} = 2 \text{ hours} + 6 \text{ hours} = 8 \text{ hours}
     \]

3. Calculate the average speed:
   \[
   \text{Average speed} = \frac{D_{\text{total}}}{T_{\text{total}}} = \frac{64 \text{ miles}}{8 \text{ hours}} = 8 \text{ miles per hour}
   \]

The final answer is \(\boxed{8}\) miles per hour.