1. **Stating the problem:** We want to find how to get the numbers 4, 30, 36, 52, and 57 from a total of 11 using only ones and twos. 2. **Understanding the problem:** Using ones and twos means we want to express each target number as a sum of 1s and 2s, where the total count of these 1s and 2s used is exactly 11. 3. **Formula and approach:** Let $x$ be the number of ones and $y$ be the number of twos used. Then: $$x + y = 11$$ $$x + 2y = \text{target number}$$ We want to find integer solutions $(x,y)$ satisfying both equations for each target number. 4. **Solving the system:** From the first equation, $x = 11 - y$. Substitute into the second: $$11 - y + 2y = \text{target}$$ $$11 + y = \text{target}$$ $$y = \text{target} - 11$$ Then: $$x = 11 - y = 11 - (\text{target} - 11) = 22 - \text{target}$$ 5. **Check for valid solutions:** Both $x$ and $y$ must be non-negative integers. - For 4: $$y = 4 - 11 = -7 \quad \text{(invalid)}$$ - For 30: $$y = 30 - 11 = 19 \quad \text{(invalid, since } y > 11)$$ - For 36: $$y = 36 - 11 = 25 \quad \text{(invalid)}$$ - For 52: $$y = 52 - 11 = 41 \quad \text{(invalid)}$$ - For 57: $$y = 57 - 11 = 46 \quad \text{(invalid)}$$ 6. **Conclusion:** None of these target numbers can be formed by exactly 11 ones and twos because $y$ is negative or greater than 11, which is impossible since $y$ counts the number of twos and must satisfy $0 \leq y \leq 11$. **Final answer:** It is not possible to get 4, 30, 36, 52, or 57 from a total of 11 ones and twos.