Solve the magic square

Question

My friend gave me the following magic square to solve
$$begin{bmatrix}frac23&5&?frac19&?&??&?&?end{bmatrix}$$
I can solve it. Can you?

You must provide logical reasoning in your answer to get the green checkmark.

Jaap Scherphuis · Accepted Answer

First I'll prove a property of $3times3$ magic squares.

Using this property you can use a similar proof to find the central cell in this case:

The rest of the magic square then follows:

I originally used a less elegant more general method by finding a generic solution:

Now it is just a matter of applying that to this particular problem.

happystar · Answer

The most elegant solution I could find was this one: let the matrix be
begin{equation*}
begin{pmatrix}
A & B & C 
D & E & F 
G & H & I 
end{pmatrix}
end{equation*}
Let the sum of each row/column/diagonal be $S$. Then
begin{eqnarray}
A+B+C + D+E+F = A+E+I + C+F+I = 2S &to& I = frac{B+D}{2} 
A+D+G = G+H+I + S  &to& H = A+D-I
A+B+C = C+F+I = S &to& F = A+B-I
end{eqnarray}
This immediately gives us values for $F,H,I$. We know the sum of $C+E$ and also the difference $C-E$ because
begin{eqnarray}
A+D+G=C+E+G &to& C+E = A+D 
A+B+C=B+E+H &to& C-E = H-A
end{eqnarray}
Therefore we know the values of C,E and hence G. This yields the same as Jaap's solution.

Solve the magic square

2 Answers

Add your own answers!

Ask a Question