1. **Problem Statement:** They can play a maximum of 8 games and want to guarantee winning the prize. We need to find the minimum number of board games they should play to ensure this. 2. **Understanding the problem:** To guarantee the prize, they must win more than half of the games played. This means they need to win at least $\lceil \frac{n}{2} + 1 \rceil$ games if they play $n$ games. 3. **Given:** Maximum games played $n = 8$. 4. **Goal:** Find the minimum number of games they must play to guarantee the prize. 5. **Step-by-step solution:** - If they play 4 games, they must win at least $\lceil \frac{4}{2} + 1 \rceil = 3$ games. - If they play 5 games, they must win at least $\lceil \frac{5}{2} + 1 \rceil = 4$ games. - If they play 6 games, they must win at least $\lceil \frac{6}{2} + 1 \rceil = 4$ games. - If they play 7 games, they must win at least $\lceil \frac{7}{2} + 1 \rceil = 5$ games. - If they play 8 games, they must win at least $\lceil \frac{8}{2} + 1 \rceil = 5$ games. 6. **Interpretation:** To guarantee the prize, they need to play enough games so that winning the minimum required games is possible. 7. **Answer:** The minimum number of games they should play to guarantee the prize is **6**. This corresponds to option b) 6.