In this problem, I gave a partially completed magic square and had people figure out what the rest of the information is.
That led me to think about the minimum amount of information that we need to provide, in order to determine the magic square.
I approach the problem in the following way:
- There are 10 unknowns - 9 values of the squares, 1 of the magic sum
- We have 8 equations - 3 horizontal, 3 vertical, 2 diagonal
- The system of equations isn't linear independent because the 3 horizontal sum up to the same as the 3 vertical. But I believe that dropping one of these equations is sufficient for linear independence, so we have 7 equations.
- If so, we then have a unique solution when there are 7 unknowns, which means that we need to provide information on 3 values.
- To show that this is optimal, in the above image, if we're given any 3 of those 4 values, we can uniquely determine the magic square.
And how about the \( 4 \times 4 \) case?