 # Problem of the Week Problem B and Solution Who Hit the Middle?

## Problem 1. What was the total number of points scored by the two players?

2. If Ebony’s total score was $$1$$ more than Serena’s, what was each person’s score?

3. What individual shots could each player have had to get their scores?

4. Whose dart landed in the inner circle?

## Solution

1. Since one dart landed in the band worth $$1$$ point, two darts landed in the band worth $$2$$ points, four darts landed in the band worth $$5$$ points, and one dart landed in the inner circle worth $$10$$ points, the total number of points scored by the two players was $1+2+2+5+5+5+5+10=35$

2. Since $$35=17+18$$, Ebony scored $$18$$ points and Serena scored $$17$$ points.

3. Trying all combinations of four shots, we can see that the only way to get a score of $$18$$ is as $$1+2+5+10$$.

Therefore, Ebony made shots worth $$1$$, $$2$$, $$5$$, and $$10$$ points. This means that Serena made shots worth $$2$$, $$5$$, $$5$$, and $$5$$ points.

4. Since the inner circle is worth $$10$$ points, then one of Ebony’s darts landed in the inner circle.