Each of the letters in the figure below represents a unique integer from 1 to 9

Q. Each of the letters in the figure below represents a unique integer from 1 to 9. The letters are positioned in the figure such that each of (A+B+C), (C+D+E), (E+F+G) and (G+H+K) is equal to 13. Which integer does E represent?

(A) 1                           (B) 4                           (C) 6                           (D) 7

Ans: 4

Sol:

(A + B + C) + (C + D + E) + (E + F + G) + (G + H + K) = A + B + 2C + D + 2E + F + 2G + H + K = 13 + 13 + 13 + 13 = 52  ……(i)

2. A + B + C + D + E + F + G + H = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45   ……(ii)

Subtracting (ii) from (i) we get, C + E + G = 7

We have C + D + E = 13

(C + D + E) – (C + E + G) = 6

⇒ D – G = 6

Also, (E + F + G) – (C + E + G) = 6

⇒ F – C = 6

∴ D – G = F – C = 6

Possible values for (F, C) and (D, G) are (9, 3), (8, 2) and (7, 1).

If  (F, C) = (9, 3), then E + F + G = 13 would mean either E or G must be equal to 3 (only possibility for (E, G) = (1, 3) or (3, 1)) which is not possible as C is equal to 3 for the selected set of conditions.

Similarly, (D, G) cannot be (9, 3).

Thus, the possibility of (9, 3) is ruled out.

Now, Case 1: (D, G) = (7, 1) and (F, C) = (8, 2)

Then, C + D + E =  2 + 7 + E = 13

Case 2: (F, C) = (7, 1) and (D,G) = (8, 2)

Then, C + D + E = 1 + 8 + E = 13

Both cases give us E = 4.

Hence, 4 is the correct answer.