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A 3x3 magic square has a diagonal sum of 21. What are the missing values for a, b, and c?

Mathematics

ScanSolve AI Answer

Step-by-step solution

5 steps
  1. a)

  1. Identify a row, column, or diagonal with all known numbers to find the magic sum.

    The diagonal from the top-right corner to the bottom-left corner has the numbers 8, 7, and 6.

  2. Calculate the sum of these numbers.

    Magic Sum=8+7+6\text{Magic Sum} = 8 + 7 + 6 Magic Sum=21\text{Magic Sum} = 21

    The magic sum is 21\boxed{21}.

    b)

  3. Find the value of aa.

    Using the middle row, the sum of the numbers must be equal to the magic sum (21).

    5+7+a=215 + 7 + a = 21 12+a=2112 + a = 21 a=21−12a = 21 - 12 a=9a = \boxed{9}

  4. Find the value of bb.

    Using the rightmost column, the sum of the numbers must be equal to the magic sum (21).

    8+a+b=218 + a + b = 21

    Substitute the value of a=9a=9:

    8+9+b=218 + 9 + b = 21 17+b=2117 + b = 21 b=21−17b = 21 - 17 b=4b = \boxed{4}

  5. Find the value of cc.

    Using the bottom row, the sum of the numbers must be equal to the magic sum (21).

    6+c+b=216 + c + b = 21

    Substitute the value of b=4b=4:

    6+c+4=216 + c + 4 = 21 10+c=2110 + c = 21 c=21−10c = 21 - 10 c=11c = \boxed{11}
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