SOLUTION: In the following figure:A B C D E F G H

Question 2252: 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.
Which digit does E represent?

Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
Sol: 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.

Kenny

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