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Solve the magic square

Puzzling Asked on October 3, 2021

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.

2 Answers

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.

Correct answer by Jaap Scherphuis on October 3, 2021

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.

Answered by happystar on October 3, 2021

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