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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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Solution computer solution | Comment 1 of 2

There are 12 arrangements, not counting rotations/reflections.

The minimum x is 9 and the maximum x is 18.

The solutions shown below have A as the smallest corner and G > C so as to avoid the trivial variations. In the array below, the value of x is shown below the array. An * appears to the right of the bottom row of the array in the cases where it actually forms a magic square, where the middle row and the middle column each total to x.


081  091  092  093  183  192  285  294  385  546  647  657
765  876  753  852  642  876  741  753  642  321  321  432
243  253  416  417  507* 354  609  618* 709  708  809  819
 9    10   11   12   12   12   15   15   16   15   17   18

DECLARE SUB place (row!, col!)
DIM SHARED used(9), bd(3, 3), x, minX, maxX, solCt

CLS
minX = 999
place 1, 1
PRINT : PRINT : PRINT minX; maxX, solCt

END

SUB place (row, col)
 FOR i = 0 TO 9
   IF used(i) = 0 THEN
     good = 1
     used(i) = 1

     bd(row, col) = i
     seq = 3 * (row - 1) + col
     SELECT CASE seq
      CASE 3
       x = bd(1, 1) + bd(1, 2) + bd(1, 3)
       IF i < bd(1, 1) THEN good = 0
      CASE 7
       IF x <> bd(1, 1) + bd(1, 2) + bd(1, 3) THEN good = 0
       IF x <> bd(1, 3) + bd(2, 2) + bd(3, 1) THEN good = 0
       IF x <> bd(1, 1) + bd(2, 1) + bd(3, 1) THEN good = 0
       IF i < bd(1, 3) THEN good = 0
      CASE 9
       IF x <> bd(3, 1) + bd(3, 2) + bd(3, 3) THEN good = 0
       IF x <> bd(1, 1) + bd(2, 2) + bd(3, 3) THEN good = 0
       IF x <> bd(1, 3) + bd(2, 3) + bd(3, 3) THEN good = 0
       IF i < bd(1, 1) THEN good = 0
     END SELECT
     IF good THEN
       IF seq < 9 THEN
         c = col + 1: r = row
         IF c > 3 THEN c = 1: r = r + 1
         place r, c
       ELSE
         FOR j = 1 TO 3
           LOCATE j, solCt * 5 + 1
           PRINT LTRIM$(STR$(bd(j, 1))); LTRIM$(STR$(bd(j, 2))); LTRIM$(STR$(bd(j, 3)));
         NEXT
         IF x <> bd(2, 1) + bd(2, 2) + bd(2, 3) THEN good = 0
         IF x <> bd(1, 2) + bd(2, 2) + bd(3, 2) THEN good = 0
         IF good THEN PRINT "*";
         LOCATE j, solCt * 5 + 1
         PRINT x;
         solCt = solCt + 1
         IF x < minX THEN minX = x
         IF x > maxX THEN maxX = x
       END IF
     END IF

     used(i) = 0
   END IF
 NEXT
END SUB

 


  Posted by Charlie on 2009-08-25 13:57:41
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