Properties of 3x3 Magic Squares

  1. For each item below, if possible, find a magic square (all rows, columns and diagonals have the same sum S) with nine unique positive integers. For ease in comparing solutions, minimize S, then A, then B. If it's not possible, show why.
    1. A through I, arranged in increasing order, do not form an arithmetic series. (In an arithmetic series, the difference between successive terms is always the same value.)
    2. The average value of A through I is not one of the nine values.
    3. A through I are all odd integers.
    4. Exactly 2 of A through I are odd integers.
    5. (Are there other instructive examples to add here?)
  2. Ignoring symmetry, if you are told the values of any three of the nine squares in a 3x3 magic square, you can almost always determine the remaining values. For which combinations of three squares can you not solve the rest of the square? (Thanks to Helen Warman for showing there are 16 different combinations of three squares.)
There are many properties of a 3x3 magic square (relationships among the numbers.) I'll include a list with the solution. If you'd like to contribute, please send them along. For example:

Source: Original.

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