\( \begin{array}{cc}\text { Alphametic } \\ G E T & \sqrt{E T}=A \\ +\quad \sqrt{F R E} & \sqrt{T G}=C \\ \frac{C U E}{} & \end{array} \)

To solve the alphametic puzzle given, we need to assign digits to the letters \( G, E, T, A, F, R, C, U \) such that the equations hold true. The equations we have are: 1. \( \sqrt{E T} = A \) 2. \( \sqrt{F R E} + \sqrt{T G} = C \) 3. \( C U E \) is a three-digit number. Let's break down the equations step by step. ### Step 1: Analyze the first equation From the first equation \( \sqrt{E T} = A \), we can deduce that \( E T \) must be a perfect square. Therefore, \( E T = A^2 \). ### Step 2: Analyze the second equation The second equation \( \sqrt{F R E} + \sqrt{T G} = C \) suggests that both \( F R E \) and \( T G \) must also be perfect squares. Let’s denote: - \( F R E = B^2 \) - \( T G = D^2 \) Thus, we can rewrite the equation as: \[ B + D = C \] ### Step 3: Determine possible values Since \( A, C, G, E, T, F, R, U \) are digits (0-9), we need to find suitable values for \( A, E, T, F, R, C, U \) that satisfy the above conditions. 1. **Finding \( A \)**: - Since \( A = \sqrt{E T} \), \( A \) must be a digit from 0 to 9. The possible values for \( A^2 \) (which is \( E T \)) are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81. 2. **Finding \( E \) and \( T \)**: - \( E T \) must be a two-digit number that is a perfect square. The possible perfect squares that are two-digit numbers are 16, 25, 36, 49, 64, 81. - This gives us the pairs: - \( 16 \rightarrow (1, 6) \) - \( 25 \rightarrow (2, 5) \) - \( 36 \rightarrow (3, 6) \) - \( 49 \rightarrow (4, 9) \) - \( 64 \rightarrow (6, 4) \) - \( 81 \rightarrow (8, 1) \) ### Step 4: Testing pairs Let’s test the pairs to see if they can lead to valid values for \( C \) and \( U \). 1. **Testing \( E = 6, T = 4 \) (from \( 64 \))**: - \( A = 8 \) (since \( \sqrt{64} = 8 \)) - Now we need to find \( F, R, C, U \) such that \( \sqrt{F R E} + \sqrt{T G} = C \). 2. **Finding \( F, R \)**: - Let’s try \( F R E = 36 \) (which is \( 6^2 \)), so \( F = 3, R = 6, E = 6 \). - \( T G = 4G \) must also be a perfect square. The possible values for \( G \) can be 0, 1, 4, 9 (since \( 4G \) must be a perfect square). - If \( G = 0 \), \( T G = 0 \) (not valid). - If \( G = 1 \), \( T G = 4 \) (valid, \( \sqrt{4} = 2 \)). - If \( G = 4 \), \( T G = 16 \) (valid, \( \sqrt{16} = 4 \)). - If \( G = 9 \), \( T G = 36 \) (valid, \( \sqrt{36} = 6 \)). 3. **Calculating \( C \)**: - If \( G = 1 \): \( C = 6 + 2 = 8 \) (valid). - If \( G = 4 \): \( C = 6 + 4 = 10 \) (not valid). - If \( G = 9 \): \( C = 6 + 6 = 12 \) (not valid). ### Conclusion The only valid