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Fill in the missing numbers in the magic square so that the sum of each row, column, and diagonal is the same.

Mathematics

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Step-by-step solution

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To complete a magic square, the sum of the numbers in each row, each column, and both main diagonals must be the same. This sum is called the magic constant.

The given square is:

[9_7_6_510_]\begin{bmatrix} 9 & \_ & 7 \\ \_ & 6 & \_ \\ 5 & 10 & \_ \end{bmatrix}
  1. Find the magic constant.

    We can use the diagonal from the top-right to the bottom-left, as it has three known numbers: 7, 6, and 5.
    Magic Constant (S) = 7+6+5=187 + 6 + 5 = 18.
    So, the sum of every row, column, and diagonal must be 18.

  2. Fill in the missing values.

    Let's denote the cells as:

    [abcdefghi]\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}

    Given: a=9,c=7,e=6,g=5,h=10a=9, c=7, e=6, g=5, h=10. We need to find b,d,f,ib, d, f, i.

    • Find ii (bottom-right cell): Using the bottom row: g+h+i=Sg + h + i = S 5+10+i=185 + 10 + i = 18 15+i=1815 + i = 18 i=18−15i = 18 - 15 i=3i = 3

    • Find bb (top-middle cell): Using the top row: a+b+c=Sa + b + c = S 9+b+7=189 + b + 7 = 18 16+b=1816 + b = 18 b=18−16b = 18 - 16 b=2b = 2

    • Find dd (middle-left cell): Using the left column: a+d+g=Sa + d + g = S 9+d+5=189 + d + 5 = 18 14+d=1814 + d = 18 d=18−14d = 18 - 14 d=4d = 4

    • Find ff (middle-right cell): Using the middle row: d+e+f=Sd + e + f = S 4+6+f=184 + 6 + f = 18 10+f=1810 + f = 18 f=18−10f = 18 - 10 f=8f = 8

  3. Write the completed magic square.

    [9274685103]\begin{bmatrix} 9 & 2 & 7 \\ 4 & 6 & 8 \\ 5 & 10 & 3 \end{bmatrix}

    The completed magic square is:

    [9274685103]\boxed{ \begin{bmatrix} 9 & 2 & 7 \\ 4 & 6 & 8 \\ 5 & 10 & 3 \end{bmatrix} }
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