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Square Search (Posted on 2006-12-21) Difficulty: 3 of 5
A 3x3 magic square is an array of nine distinct positive integers such that the sum of the numbers in each row, each column, and each of the two diagonals is the same. In the following magic square:

9

?

14

What are the possible values of the lower left corner? And what is the maximum possible value of any number in the array?

  Submitted by Dennis    
Rating: 4.2500 (4 votes)
Solution: (Hide)
Let x and y represent the numbers in the upper right and lower left corners respectively. This forces the array ...

          x+y-14     2y-9      x 
          37-2y      23-y      9 
            y        x-y+9    14 

So (x+y-14)+(2y-9)+(x) = x+23 --> x=46-3y. The array can be written solely in terms of y as follows ...

          32-2y     2y-9     46-3y 
          37-2y     23-y       9
            y       55-4y     14  

Now (2y-9)>0 and (55-4y)>0 --> 4 < y < 14. Since array values are distinct, y cannot equal 9 and 11. So y = 5,6,7,8,10,12, and 13 only. Also, since the maximum value in a 3 by 3 magic square with distinct numbers cannot be in the center nor the corners, we need only consider maximizing (2y-9), (37-2y), and (55-4y). Substituting y=5 into (55-4y) yields a maximum value of 35 for any number in the array.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some ThoughtsPuzzle ThoughtsK Sengupta2022-12-08 02:50:14
No SubjectJan Barnett2020-09-15 02:38:36
Center CellGamer2006-12-21 16:55:44
Solutionre: heuristic computer exploration and SolutionEric2006-12-21 15:55:49
Hints/Tipsheuristic computer exploration (spoiler)Charlie2006-12-21 15:37:28
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