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Extreme Entireties (Posted on 2009-08-25) Difficulty: 3 of 5
Each of the capital letters are substituted by a different digit from 0 to 9 in this 3x3 square such that the six sums formed by the leftmost column, rightmost column, top row, bottom row and the two main diagonals are each equal to x. Each of the remaining row and the remaining column may or may not sum to x.

A    B    C
D    E    F
G    H    I

Determine the respective minimum value and the maximum value of x.

See The Solution Submitted by K Sengupta    
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Spreadsheet offering Comment 2 of 2 |
Before reading the comment below, note that I have only used the digits from 1 to 9!  I only realised that when I looked back at Charlie's offering. 

I take some satisfaction that my two responses occur within Charlie's results for which there is only one result for 12 and 18 using those same digits (and by coincidence the digits have the same layout).

In each of the offered solutions the tables A-I offer the values to be placed within the 3x3 array.
The bold 12 and 18 numbers are the respective diagonal sums.

Max X                              
A B C D E F G H I
6 5 7 4 3 2 8 1 9                         
A B C    6  5  7 18  
D E F    4  3  2     
G H I    8  1  9 18
   18 18 18 18  
                               
Min X 
A B C D E F G H I
1 9 2 8 7 6 3 5 4

A B C     1  9  2 12
D E F    8  7  6
G H I    3  5  4 12
      12 12 12 12

Note that there is a very distinct similarity to the 3x3 15 "magic square".  With that realisation, if one determines either the Max or Min solution it is realtively easy to determine the other since both of the needed sums are 3 different to 15.
                                

Edited on August 25, 2009, 10:25 pm
  Posted by brianjn on 2009-08-25 22:13:48

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