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 3x3 Grid Near Magic (Posted on 2009-12-05)
N is a 3x3 square grid which is constituted by using each of the digits from 1 to 9 exactly once.

Determine the probability that the first digit minus the second digit plus the third digit in each row (reading left to right), each column (reading top to bottom), and each main diagonal (reading top to bottom) of N is the same.

 No Solution Yet Submitted by K Sengupta No Rating

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 computer solution | Comment 1 of 6

using qbasic I found the following 8 solutions

Solution # 1: constant=  5
2  1  4
3  5  7
6  9  8
+++++++++++++
Solution # 2: constant=  5
2  3  6
1  5  9
4  7  8
+++++++++++++
Solution # 3: constant=  5
4  1  2
7  5  3
8  9  6
+++++++++++++
Solution # 4: constant=  5
4  7  8
1  5  9
2  3  6
+++++++++++++
Solution # 5: constant=  5
6  3  2
9  5  1
8  7  4
+++++++++++++
Solution # 6: constant=  5
6  9  8
3  5  7
2  1  4
+++++++++++++
Solution # 7: constant=  5
8  7  4
9  5  1
6  3  2
+++++++++++++
Solution # 8: constant=  5
8  9  6
7  5  3
4  1  2

and since there are 9!=362880 possible grids then then
the probability is 8/9!=1/45360=0.0000220459

the solutions were generated with the following code
OPEN "output.txt" FOR OUTPUT AS #1
CLS 0
cnt = 0
FOR g1 = 1 TO 9
FOR g2 = 1 TO 9
IF g2 <> g1 THEN
FOR g3 = 1 TO 9
IF g3 <> g1 AND g3 <> g2 THEN
c = g1 - g2 + g3
FOR g4 = 1 TO 9
IF g4 <> g1 AND g4 <> g2 AND g4 <> g3 THEN
FOR g5 = 1 TO 9
IF g5 <> g1 AND g5 <> g2 AND g5 <> g3 AND g5 <> g4 THEN
FOR g6 = 1 TO 9
IF g6 <> g1 AND g6 <> g2 AND g6 <> g3 AND g6 <> g4 AND g6 <> g5 THEN
FOR g7 = 1 TO 9
IF g7 <> g1 AND g7 <> g2 AND g7 <> g3 AND g7 <> g4 AND g7 <> g5 AND g7 <> g6 THEN
FOR g8 = 1 TO 9
IF g8 <> g1 AND g8 <> g2 AND g8 <> g3 AND g8 <> g4 AND g8 <> g5 AND g8 <> g6 AND g8 <> g7 THEN
FOR g9 = 1 TO 9
IF g9 <> g1 AND g9 <> g2 AND g9 <> g3 AND g9 <> g4 AND g9 <> g5 AND g9 <> g6 AND g9 <> g7 AND g9 <> g8 THEN
IF g4 - g5 + g6 = c AND g7 - g8 + g9 = c AND g1 - g4 + g7 = c AND g2 - g5 + g8 = c AND g3 - g6 + g9 = c AND g1 - g5 + g9 = c AND g3 - g5 + g7 = c THEN
cnt = cnt + 1
PRINT #1, "Solution #" + STR\$(cnt) + ": constant= " + STR\$(c)
PRINT #1, g1; g2; g3
PRINT #1, g4; g5; g6
PRINT #1, g7; g8; g9
PRINT #1, "+++++++++++++"
END IF
END IF
NEXT g9
END IF
NEXT g8
END IF
NEXT g7
END IF
NEXT g6
END IF
NEXT g5
END IF
NEXT g4
END IF
NEXT g3
END IF
NEXT g2
NEXT g1
CLOSE #1

 Posted by Daniel on 2009-12-05 13:12:45
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