1. **Problem Statement:** We want to find pairs of numbers from the set $\{1, 2, 2, 2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9\}$ that can produce the numbers in the set $\{4, 8, 9, 10, 11, 12, 13, 14, 15, 16\}$ using addition or multiplication without repeating any number in the pairs. 2. **Approach:** For each target number, we try to find two distinct numbers $a$ and $b$ from the given set such that either $a + b = \text{target}$ or $a \times b = \text{target}$. We must ensure no number is used more times than it appears in the original set. 3. **Example for target 4:** - Addition: $1 + 3 = 4$ - Multiplication: $2 \times 2 = 4$ 4. **Example for target 8:** - Addition: $3 + 5 = 8$ - Multiplication: $2 \times 4 = 8$ 5. **Example for target 9:** - Addition: $4 + 5 = 9$ - Multiplication: $3 \times 3 = 9$ 6. **Example for target 10:** - Addition: $4 + 6 = 10$ - Multiplication: $2 \times 5 = 10$ 7. **Example for target 11:** - Addition: $5 + 6 = 11$ - Multiplication: No integer pair from the set multiplies to 11. 8. **Example for target 12:** - Addition: $5 + 7 = 12$ - Multiplication: $3 \times 4 = 12$ 9. **Example for target 13:** - Addition: $6 + 7 = 13$ - Multiplication: No integer pair from the set multiplies to 13. 10. **Example for target 14:** - Addition: $7 + 7 = 14$ - Multiplication: $2 \times 7 = 14$ 11. **Example for target 15:** - Addition: $7 + 8 = 15$ - Multiplication: $3 \times 5 = 15$ 12. **Example for target 16:** - Addition: $8 + 8 = 16$ - Multiplication: $4 \times 4 = 16$ **Summary:** We can form each target number by selecting pairs from the given set using addition or multiplication without repetition, for example: - 4: $2 \times 2$ - 8: $2 \times 4$ - 9: $3 \times 3$ - 10: $2 \times 5$ - 11: $5 + 6$ - 12: $3 \times 4$ - 13: $6 + 7$ - 14: $2 \times 7$ - 15: $3 \times 5$ - 16: $4 \times 4$ This satisfies the problem's conditions.