In the following figure:A B C D E F G H I Each of the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 is:
a)Represented by a different letter in the figure above. b)Positioned in the figure above so that each of A + B + C,C + D +E,E + F + G, and G + H + I is equal to 13.
By given condition, we have A+B+C = 13...(1) C+D+E = 13...(2) E+F+G = 13...(3) G+H+I = 13...(4) Add up (1)+(2)+(3)+(4), we get 52=13*4 = A + B + C+ C + D + E+ E + F + G+ G + H + I = C+E+G+(A + B + C+ D +E + F + G + H + I) = C+E+G+(1+2+ 3+ 4+ 5+ 6+ 7+ 8+ 9) = C+E+G+(1+9)*9/2 = C+E+G + 45
Hence,C+E+G = 52 - 45 = 7...(5)
By (5), we have E = 7-C-G <= 7-3 = 4 (since, the smallest C,G are 1 or 2) By (2)-(5)., we get D-G = 6, or D = G+6...(6) Since D,G are between 1 and 9, so D= 7, 8,or 9, and we have G=1,2 or 3
By (3)-(5)., we get F-C = 6, or F = C+6...(7) Since F,C are between 1 and 9, so F= 7, 8,or 9, and we have C=1,2 or 3
By (6) ,(7) & (5): Case(i) When G =1, D = G+6 =7, and then C = 2, F = C+6=8, E = 7-C-G= 4 Or C = 3, F = 9, E = 3 (invalid,since the same C,E value)
Case(ii) When G =2, D = G+6= 8, and then C = 1, F = C+6=7, E= 7-C-G=4 Or C = 3, F = 9, E = 2(invalid,since the same C,G value)
Case(iii) When G =3, D = G+6=9, and then C = 1, F = C+6=7, E =7-C-G= 3(invalid,since the same E,G value) Or C = 2, F = C+6=8, E=7-C-G = 2(invalid,since the same C,E value)
In these three possible cases of the values of G, we obtain that the only valid value of E is 4
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By given condition we have A+B+C 13...(1) C+D+E 13...(2) E+F+G 13...(3) G+H+I 13...(4) Add up (1)+(2)+(3)+(4) we get 52 13*4 A + B + C+ C + D + E+ E + F + G+ G + H + I C+E+G+(A + B + C+ D +E + F + G + H + I) C+E+G+(1+2+ 3+ 4+ 5+ 6+ 7+ 8+ 9) C+E+G+(1+9)*9/2 C+E+G + 45
Hence C+E+G 52 - 45 7...(5)
By (5) we have E 7-C-G < 7-3 4 (since the smallest C G are 1 or 2) By (2)-(5). we get D-G 6 or D G+6...(6) Since D G are between 1 and 9 so D 7 8 or 9 and we have G 1 2 or 3
By (3)-(5). we get F-C 6 or F C+6...(7) Since F C are between 1 and 9 so F 7 8 or 9 and we have C 1 2 or 3
By (6) (7) & (5): Case(i) When G 1 D G+6 7 and then C 2 F C+6 8 E 7-C-G 4 Or C 3 F 9 E 3 (invalid since the same C E value)
Case(ii) When G 2 D G+6 8 and then C 1 F C+6 7 E 7-C-G 4 Or C 3 F 9 E 2(invalid since the same C G value)
Case(iii) When G 3 D G+6 9 and then C 1 F C+6 7 E 7-C-G 3(invalid since the same E G value) Or C 2 F C+6 8 E 7-C-G 2(invalid since the same C E value)
In these three possible cases of the values of G we obtain that the only valid value of E is 4
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