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Question 190374: # 8. Complete the magic square below (Three squares across, three down):
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! # 8
Q: Complete the magic square below (Three squares across, three down):
A:
Let's start with the given magic square:
The unique and defining property of any magic square is that the sum of the elements of any row, column, or diagonal will be the same.
So in this case, the sum of the elements in the 2nd row is: 11+7+3=21
So the sum of the elements in ANY row, column or diagonal will be 21.
Let's find the number in the lower right hand box. First, add up the elements in the 3rd column to get 8+3=11
Since the sum must be 21, this means that we're off by 10 (since 21-11=10). So this means that 10 must go in the lower right hand corner. Let's add this number:
Now let's find the element in the lower left hand corner. Add up the elements along the north-east diagonal (the diagonal from the lower left hand corner to the upper right hand corner) to get: 7+8=15
Since 21 - 15 = 6, this means that 6 must go in the lower left hand box. So let's add it in:
Now let's find the number that belongs in the lower middle box. First, add the elements that we've found so far to get: 6 + 10 = 16
Subtract this from 21 to get 21 - 16 = 5. So the missing number is 5 which means that 5 goes in the lower middle box like so:
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Now let's move up and find the element that goes in the upper left hand corner. First add the numbers in the first column to get: 11 + 6 = 17
Subtract the result from 21 to obtain: 21 - 17 = 4
So the number that belongs in the upper left hand box is 4. Add that in to get:
Finally, let's get the last number that belongs in the upper middle box. So add up the elements in the middle column to get: 7 + 5 = 12
Now take that result away from 21 to get: 21 - 12 = 9
So the number 9 belongs in the upper middle box. So our completed magic square looks like
Check: let's add up the elements in each row, column and diagonal to verify the answer:
Row 1: 4 + 9 + 8 = 21 ... works
Row 2: 11 + 7 + 3 = 21 ... works
Row 3: 6 + 5 + 10 = 21 ... works
Column 1: 4 + 11 + 6 = 21 ... works
Column 2: 9 + 7 + 5 = 21 ... works
Column 3: 8 + 3 + 10 = 21 ... works
North-West Diagonal: 4 + 7 + 10 = 21 ... works
North-East Diagonal: 6 + 7 + 8 = 21 ... works
Since all of the sums are 21, this verifies the answer.
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