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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?

1. $1$
2. $4$
3. $6$
4. $7$

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Migrated from GO Civil 1 year ago by Arjun

We have to get sum $13$ in $4$ ways.

• $A + B + C = 13$
• $C + D + E = 13$
• $E + F + G = 13$
• $G + H + I = 13$

Adding all $4$ we get

$A + B + D + F + H + I + 2(C + E+ G) = 52$

Since $A \to G$ sum to $45$ we get

$C + E + G = 52-45 =7$

Only option we can add to $7$ with $3$ distinct positive integers is $1 +2+4.$ So, $C,E,G \in \{1,2,4\}$

We also have $C+D+E = 13.$ If $C$ and $E$ are $1,2$ this is not possible as this requires $D$ to be $10$ which is not allowed. So, at least one of $E$ or $C$ must be $4$ also making $G$ to be either $1$ or $2.$

Since, $E + F + G = 13, E + G$ must also be $\geq4$ meaning $E$ and $G$ cannot be $1,2.$ Since, $G \leq 2,$ it means $E > 2.$

So, only option is $E = 4.$

Correct Option: B.

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