jmc

Question

Each of the points A,B,C,D,E, and F in the figure below represents a different digit from 1 to 6. Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is 47. What is the digit represented by B? FIGURE

Accepted Answer

We can form the following expressions for the sum along each line: dcasesA+B+C+E+F+D+E+D+Fdcases Adding these together, we must have 2A+3B+2C+2D+2E+2F=47, i.e. 2(A+B+C+D+E+F)+B=47. Since A,B,C,D,E,F are unique integers between 1 and 6, we obtain A+B+C+D+E+F=1+2+3+4+5+6=21 (where the order doesn't matter as addition is commutative), so our equation simplifies to 42 + B = 47. This means B = 5.