1. **State the problem:** We have 136 boys who play rugby, soccer, or hockey. Given the counts for each sport and their pairwise overlaps, we want to find how many boys played all three games. 2. **Given data:** - Total boys, $N = 136$ - Rugby players, $R = 67$ - Soccer players, $S = 56$ - Hockey players, $H = 40$ - Rugby and Soccer, $R \cap S = 11$ - Soccer and Hockey, $S \cap H = 12$ - Rugby and Hockey, $R \cap H = 9$ - Each boy plays at least one sport. 3. **Let the number of boys who played all three games be $x$**. 4. **Use the principle of inclusion-exclusion:** $$N = R + S + H - (R \cap S + S \cap H + R \cap H) + (R \cap S \cap H)$$ Substitute values: $$136 = 67 + 56 + 40 - (11 + 12 + 9) + x$$ 5. **Calculate the sum and difference:** $$136 = 163 - 32 + x$$ $$136 = 131 + x$$ 6. **Solve for $x$:** $$x = 136 - 131 = 5$$ **Final answer:** The number of boys who played all three games is $5$.