We can try to generate all possible valid solutions, and then calculate the distance from each, and take the minimum.

For the generation process to be effective, there are two key observations: - the value 5 must be in the centre - values must be placed symmetrically against each other (1 against a 9, 2 against 8 etc.)

Main part of C# code, with some bits left ommitted:

public static void GenerateSolutions(Solution solution, List<Solution> found)
{
    if (solution.IsValid)
    {
        found.Add(solution.Clone());
        return;
    }

    if (solution.availableNumbers.Count <= 0)
    {
        return;
    }

    // try to place the first number to all positions
    var num = solution.availableNumbers.First();

    for (int i = 0; i < 3; i++)
    {
        for (int j = 0; j < 3; j++)
        {
            if (solution.numbers[i][j] != -1) continue;

            // place the number
            // also place the number 10-num into the opposite corner
            solution.numbers[i][j] = num;
            solution.numbers[2 - i][2 - j] = 10 - num;

            solution.availableNumbers.Remove(num);
            solution.availableNumbers.Remove(10 - num);

            // recurse
            GenerateSolutions(solution, found);

            // undo the work we have done above
            solution.numbers[i][j] = -1;
            solution.numbers[2 - i][2 - j] = -1;

            solution.availableNumbers.Add(num);
            solution.availableNumbers.Add(10 - num);
        }
    }
}

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