The puzzle you uploaded involves placing the digits 1 through 9 in a 3x3 grid. The goal is to make the horizontal and vertical equations true using these digits only once. Here's a breakdown of the puzzle:

Instructions (translated from German):

1. Complete the grid using the nine numbers below (1 to 9).
2. Ensure that each equation (three horizontal and three vertical) is correct.
3. Each number is used only once.

Grid:

• Horizontal equations:

  1. First row: x + x = 8
  2. Second row: + x + = 24
  3. Third row: x - x = 42

• Vertical equations:

  1. First column: x - x = 28
  2. Second column: + 8 - = 2
  3. Third column: + x = 77

Let's solve the puzzle step by step.

Solving Process:

1. Analyze the first row equation:

   • We need two numbers, whose sum gives 8. Potential combinations from the digits 1-9 are (1, 7), (2, 6), (3, 5).

2. Analyze the second row equation:

   • We need to find three numbers that, when added, give 24. Possible candidates are (7, 8, 9) or another combination from the available digits.

3. Analyze the third row equation:

   • The third row involves subtraction to reach 42, so careful selection of higher digits is needed.

4. Vertical Equations: These will give us clues based on what digits remain and fit in the grid.

I will now solve the puzzle mathematically and fill in the numbers for you.It appears there was an issue with defining the solution. Let me rerun the calculations to ensure the correct solution for the puzzle is found.It seems I missed an essential condition while calculating the solution. Let me correct the approach and carefully recalculate the puzzle solution.The solution to the puzzle using the numbers 1 to 9 is as follows:

• $x_1 = 9$, $x_2 = -1$, ( x_3 = … \