The numbers 1 through 9 are arranged in a 3 x 3 grid so that each number is in the grid exactly once. They are arranged such that the top row plus the middle row gives the bottom row. If the grid forms another such addition when it is rotated 90 degrees to the left, what is its composition?
(Note: The numbers don't flip, for example 6 doesn't turn into 9.)
(In reply to
first part of full solution by pcbouhid)
Case IVa : G = 8, (B+C+E+F) = 17 and (A+D+H+I) = 20.
In (2), (H+I) = 8 or 18. If (H+I) = 18, (A+D) = 2, no way. If (H+I) = 8, (A+D) = 12. But in (1), (A+D) or (A+D+1) = 8. Eliminated.
Case IVb: G = 8, (B+C+E+F) = 26 and (A+D+H+I) = 11.
In (2), (B+C) = A or (B+C+1) = A. If (B+C) = A, and A is at most 7, (E+F) >= 19, no way. And if (B+C+1) = A, (E+F) >=18, no way too. Eliminated.

Case V: G = 9, (B+C+E+F) = 18 and (A+D+H+I) = 18.
In (2), (H+I) = 9 or 19. (H+I) canīt be 19, so (H+I) = 9, and (A+D) = 9.
Since (A+D) = 9, we have in (1), (B+E) = H (which means (C+F) = I) or (B+E+1) = H (which means (C+F) = 10+I).
(B+E) = H and (C+F) = I......(B+E+C+F) = (H+I). But (B+C+E+F) = 18 and (H+I) = 9. No way.
So, (B+E+1) = H and (C+F) = (10 + I) [or (C+F) = (19H)].
Since A >= 3, we have (A,D) = (3,6), (4,5), (5,4), (6,3), (7,2) and (8,1).
A is greater than B and than C, and (B+C) = A or (B+C+1) = A.

(A,D) = (3,6)..(B,C) = (1,2) or (2,1).
(B,C) = (1,2)... H = (E+2) = (17F)...(E+F) = 15 so (E,F,H) = (8,7,10) or (7,8,9). Eliminated.
(B,C) = (2,1)...H = (E+3) = (18F)...(E+F) = 15 impossible because E must be less than 6 (see H).

(A,D) = (4,5)...(B,C) = (1,3) or (3,1) or (1,2) or (2,1)
(B,C) = (1,3)...H = (E+2) = (16F)...(E+F) = 14 so (E,F,H) = (8,6,10) or (6,8,8). Eliminated.
(B,C) = (3,1)...H = (E+4) = (18F)....(E+F) = 14 so (E,F,H) = (8,6,12) or (6,8,10). Eliminated.
(B,C) = (1,2) and (2,1) eliminated as above.

(A,D) = (5,4)..(B,C) = (2,3) or (3,2) or (1,3) or (3,1)
(B,C) = (2,3)...H = (E+3) = (16F)...(E+F) = 13 so (E,F,H) = (7,6,10) or (6,7,9). Eliminated.
(B,C) = (3,2)...H = (E+4) = (17F)...(E+F) = 13 so (E,F,H) = (7,6,11) or (6,7,10). Eliminated.
(B,C) = (1,3) and (3,1) eliminated .

(A,D) = (6,3)...(B,C) = (5,1) or (1,5) or (4,2) or (2,4) or (4,1) or (1,4)
(B,C) = (5,1)...H = (E+6) = (18F)...(E+F) = 12 impossible because E must be less than 4.
(B,C) = (1,5)...H = (E+2) = (14F)...(E+F) = 12 so (E,F,H) = (8,4,10) or (7,5,9) or (5,7,7) or (4,8,6). Eliminated.
(B,C) = (4,2)...H = (E+5) = (17F)...(E+F) = 12 impossible because E must be less than 4 (see H).
(B,C) = (2,4)...H = (E+3) = (15F)...(E+F) = 12 so (E,F,H) = (5,7,8)
(B,C) = (4,1)...H = (E+5) = (18F)...(E+F) = 13 impossible because E must be less than 4 (see H).
(B,C) = (1,4)...H = (E+2) = (15F)...(E+F) = 13 so (E,F,H) = (8,5,10) or (7,6,9) or (6,7,8) or (5,8,7)

(A,D) = (7,2)...(B,C) = (6,1) or (1,6) or (4,3) or (3,4) or (5,1) or (1,5)
(B,C) = (6,1)...H = (E+7) = (18F)...(E+F) = 11 impossible because E must be 1 (to make H = 8).
(B,C) = (1,6)...H = (E+2) = (13F)...(E+F) = 11 so (E,F,H) = (8,3,10) or (6,5,8) or (5,6,7) or (3,8,5). All but one eliminated.
(B,C) = (4,3)...H = (E+5) = (16F)...(E+F) = 11, all possibilities eliminateds.
(B,C) = (3,4)...H = (E+4) = (15F)...(E+F) = 11, all possibilities eliminated.
(B,C) = (5,1)...H = (E+6) = (18F)...(E+F) = 12, no possibilities for (E,F,H).
(B,C) = (1,5)...H = (E+2) = (14F)...(E+F) = 12, so (E,F,H) = (4,8,6).

(A,D) = (8,1)...(B,C) = (6,2) or (2,6) or (5,3) or (3,5) or (5,2) or (2,5) or (4,3) or (3,4)
(B,C) = (6,2)...H = (E+7) impossible (E canīt be 1 or 2).
(B,C) = (2,6)...H = (E+3) = (13F)...(E+F) = 10, all possibilities eliminated.
(B,C) = (5,3)...H = (E+6) = (16F)...(E+F) = 10, no possibilities.
(B,C) = (3,5)...H = (E+4) = (14F)...(E+F) = 10, no possibilities.
(B,C) = (5,2)...H = (E+6)....no values for E.
(B,C) = (2,5)...H = (E+3) = (14F)...(E+F) = 11 no possibilities.
(B,C) = (4,3)...H = (E+5) = (16F)...(E+F) = 11 no possibilities.
(B,C) = (3,4)...H = (E+4) = (15F)...(E+F) = 11 no possibilities.
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A = 6, B = 2, C = 4, D = 3, E = 5, F = 7, G = 9, H = 8, I = 1.
A = 6, B = 1, C = 4, D = 3, E = 5, F = 8, G = 9, H = 7, I = 2.
A = 7, B = 1, C = 6, D = 2, E = 3, F = 8, G = 9, H = 5, I = 4.
A = 7, B = 1, C = 5, D = 2, E = 4, F = 8, G = 9, H = 6, I = 3.
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624.......471 614......482 716....684 715....583
357.......258 358......157 238....135 248....146
   
981 639 972.....639 954....729 963.....729
WRONG RIGHT WRONG RIGHT
FOR A RAINNY DAY, THATīS ENOUGH!! I DONīT KNOW WHERE I MADE A MISTAKE. TWO SOLUTIONS ONLY.

Posted by pcbouhid
on 20050812 20:18:04 