Construct a 5x5 magic square.
1- Numbers from 1 to 21 can be used.
2- Sum should be 55 (Horizontally, Vertically and Diagonally)
3- Any 4 digits from 1 to 21 can be repeated only once.
4- The Numbers placed in center square cannot be repeated.

up vote 10 down vote accepted

This is a valid solution to the given parameters:

enter image description here
The repeated digits are 5, 9, 13, and 17. The number in the center, 11, is not repeated.

How it was constructed:

There is a well-known method of constructing a normal (odd-sized) magic square. First, start with the number 1 in the middle top square. Then, move diagonally up-right, incrementing the number by 1 each time. If you would reach a square that you have visited, instead move down one square. What differs in this construction is that during these downward steps, I also stop incrementing the counter as well. Surprisingly, this continues to keep all of the sums balanced, which gives us the completed grid.

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