This problem is a challenging puzzle where each letter represents a unique value from 1 to 9. We need to determine the value of each letter so that all the given equations are satisfied simultaneously.

Let's denote the letters as:

• $a, b, f, g, l, m, p, s, z$

We need to find the values of $a, b, f, g, l, m, p, s, z$ such that each of them is a unique integer between 1 and 9.

The equations are:

1. & \ g + (f \times s) = 45 \\ 2. & \ b + a + m + f + g + s = 32 \\ 3. & \ m + s + b + z + p + l = 33 \\ 4. & \ p + z + a + g + b + f = 27 \\ 5. & \ g + m + l + f + a + s = 30 \\ 6. & \ l + f + s + b + p + z = 31 \\ 7. & \ z + g + m + a + l + p = 27 \\ 8. & \ b + m + p + g + l + z = 30 \\ 9. & \ m + b + a + l + s + m = 33 \\ 10. & \ f + z + g + f + b + a = 24 \\ 11. & \ g + p + b + a + p + l = 32 \\ 12. & \ s + l + f + s + z + p = 33 \\ \end{align*}$$ ### Step 1: Analyzing the First Equation The first equation is unique as it involves multiplication: $$g + (f \times s) = 45$$ Given that $$g$$, $$f$$, and $$s$$ are digits from 1 to 9, we can try to find combinations of $$f$$ and $$s$$ such that $$g = 45 - (f \times s)$$ is also a digit between 1 and 9. ### Step 2: Testing Combinations Let's test some possible values of $$f$$ and $$s$$ to see which combinations fit this equation. - For $$f = 5$$ and $$s = 8$$: $$f \times s = 5 \times 8 = 40 \quad \Rightarrow \quad g = 45 - 40 = 5$$ - For $$f = 9$$ and $$s = 5$$: $$f \times s = 9 \times 5 = 45 \quad \Rightarrow \quad g = 45 - 45 = 0 \quad (\text{but 0 is not valid since we need digits from 1 to 9})$$ - For $$f = 7$$ and $$s = 6$$: $$f \times s = 7 \times 6 = 42 \quad \Rightarrow \quad g = 45 - 42 = 3$$ This testing approach helps us narrow down possible values. ### Step 3: Using a Systematic Approach or Programming Given the complexity of solving this by hand, a systematic approach such as creating a program to iterate through possible values for each letter might be more efficient. Let me write a simple code to solve this efficiently.It seems that the process of finding the solution encountered an issue. I will rerun the calculation to identify the correct values for each letter. Let's proceed again.It appears the solution is not being identified. Let me correct this and provide you with the correct values for each letter.It appears that there is no valid combination of values for $$a, b, f, g, l, m, p, s, z$$ that satisfies all the equations given. This could indicate an issue with the constraints or that the problem may not have a solution within the given parameters (digits 1-9). If you want, I can assist in checking the constraints or exploring possible relaxations to the conditions. Would you like to explore further or receive some advice on how to approach such puzzles?