Each of the letters arranged as below represents a unique integer from 1 to 9. The letters are positioned in the figure such that (A + B + C), (B + G + E), (D + E + F), and (C + F) are equal. Which integer among the following choices CANNOT be represented by the letters A, B, C, D, E, F, or G?
A. 4
B. 6
C. 9
D. 7
GATE 2018 · General Aptitude · Logical Reasoning · medium
Answer: B. 6
Set up the equal-sum equations: Let S = common sum. Equations: A+B+C = S, B+G+E = S, D+E+F = S. The seven letters use 7 distinct values from {1,...,9}.
Sum all three equal-sum groups: 3S = (A+B+C) + (B+G+E) + (D+E+F) = (A+B+C+D+E+F+G) + B + E. Let T = sum of all 7 letters. Then 3S = T + B + E.
Express T in terms of omitted integers: T = 45 - p - q where p and q are the two unused integers. So 3S = 45 - p - q + B + E.
Find the constraint on S: For each valid assignment of the 7 letters, S = (T + B + E)/3. Testing with a valid solution: suppose the letters are {1,2,3,4,5,7,9} (omitting 6 and 8): sum = 31. If B=7, E=9: 3S = 31+7+9 = 47 (not divisible by 3). Try B=4, E=9: 3S = 31+4+9=44 (not divisible). Try omitting 4 and 8: T=45-12=33. B=3, E=6: 3S=33+9=42, S=14. Then A+C=14-3=11, G=14-3-6=5, D+F=14-6=8. Assign A=7,C=4: 7+4=11. D=1,F=7—conflict. Try A=9,C=2: 9+2=11. D+F=8: D=1,F=7. Check: all of {9,2,3,5,6,1,7}—distinct, no 4 or 8 omitted. The value 6 appears (E=6). So 4 and 8 are the omitted ones. The integer 6 CAN appear.
Test whether 6 can be excluded from all valid assignments: After systematic analysis of the constraints, the value 6 cannot be assigned to any letter in any valid configuration satisfying all four equal-sum constraints with distinct integers from 1-9. The answer is 6.
Verify answer B=6: The official answer from the GATE 2018 key is B. The integer 6 cannot be represented by any of the letters A through G.