However, I knew that not all of those 1680 solutions would be valid. That was when it dawned on me, that in order to make every sum crossing the center the pairs would be, which means I just had to place 1-4 and the rest was given. I thought of checking all combinations but on the outset there was 9! = 362.880 possible solutions. What did I know about forming a magic square? I know that the so called magic constant has to be 15, so 5 have to be placed in the middle, which reduces the combinations to 8! = 40.320 a lot better. Checking all possibilities for forming a magic squareīack to the drawing board. But I realized from the second example that in some of the cases multiple numbers needed to be changed in order to which meant that I couldn’t just evaluate the individual digits to see if it should be changed. I started out with the idea that I could go through the square and change the number one by one based on an evaluation of the individual digits. Note: The resulting magic square must contain distinct integers in the inclusive range .
Given , convert it into a magic square at minimal cost by changing zero or more of its digits. We can convert any digit, a, to any other digit, b, in the range at cost.
We define a magic square to be an matrix of distinct positive integers from 1 to n 2 where the sum of any row, column, or diagonal (of length n) is always equal to the same number (i.e., the magic constant).Ĭonsider a matrix, s, of integers in the inclusive range . Anyway, before rambling on lets get to the actual problem. Once you have the right insight on forming a magic Square it is really straight forward. I must admit that this problem actually took me a good while to solve.
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