[tex]\begin{bmatrix}\frac{1}{6} & \frac{1}{2} & \frac{1}{3} \\ \\ \frac{3}{6}& \frac{2}{6} & \frac{1}{6} \\ \\ \frac{1}{3} & \frac{4}{24} & \frac{1}{2} \\ \end{bmatrix} [/tex]
Explanation:
The magic square is a n×n square which contains different number and sum of all the number of any row or column gives same value.
in any magic square the sum values of all row or column is three time of the number in centre of the square.
in above magic square the middle number is 2/6
so we can calculate the final value by multiplying with 3,
[tex] \frac{2}{6} \times 3 = \frac{2}{ \cancel6}_2 \times \cancel3 = \cancel{\frac{2}{2}} = 1[/tex]
The magic number or sum of any row or column is 1,
using this we can find the missing,
in column first,
1/6 and 1/3 are given and middle term is missing
let the middle term be x now,
[tex]x + \frac{1}{6} + \frac{1}{3} = 1 \\ x = 1 - \frac{1}{6} - \frac{1}{3} \\ x = 1 - ( \frac{3 + 6}{18}) \\ x = 1 - \frac{9}{18} \\ x = \frac{18 - 9}{18} \\ x = \cancel \frac{9}{18} = \frac{1}{2} [/tex]
similarly we can solve for the 3rd fraction of second column,
let the third number be y,
[tex] \frac{3}{ 6} + \frac{2}{6} + y = 1 \\ y = 1 - \frac{3}{ 6} - \frac{2}{6} \\ y = 1 - \frac{5}{6} = \frac{1}{6} [/tex]
Now in first and third row, the third number is missing which can be calculated in same way.
The rough work and final calculation is attached!
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