### Brain Teasers

# Playing Darts

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Isabelle, Judy, and Kate played darts. On the dartboard there was a circle, a square, and a triangle, which all overlapped each other. Any dart that landed outside the shapes scored 2 points. Any dart that landed within one shape scored 3 points. Any dart that landed within two shapes scored 4 points. Any dart that landed within all three shapes scored 5 points.

There were four 2-pointers, four 3-pointers (all of which landed within the square), two 4-pointers (both of which landed within the circle and the triangle), and one 5-pointer. All three women scored the same number of points.

Isabelle said, "If I was the first to score in the square, then I scored at least one 2-pointer." Judy said, "If I was the first to score in the triangle, then Isabelle and Kate did not score the same number of 2-pointers." Kate said, "If I was the first to score in the circle, then I did not score more 2-pointers than Judy." If only one is telling the truth, who scored the 5-pointer?

There were four 2-pointers, four 3-pointers (all of which landed within the square), two 4-pointers (both of which landed within the circle and the triangle), and one 5-pointer. All three women scored the same number of points.

Isabelle said, "If I was the first to score in the square, then I scored at least one 2-pointer." Judy said, "If I was the first to score in the triangle, then Isabelle and Kate did not score the same number of 2-pointers." Kate said, "If I was the first to score in the circle, then I did not score more 2-pointers than Judy." If only one is telling the truth, who scored the 5-pointer?

### Hint

All three women made "hypothesis and conclusion" statements. If the hypothesis of such a statement is false, then the whole statement is true. If the conclusion is true, then the whole statement is true. However, if the hypothesis is true and the conclusion is false, the statement is false.### Answer

Kate scored the 5-pointer.If you add up the total amount of points scored, you get 33 points. Divide this by three women and get 11 points for each woman.

Since all the 3-pointers were within the square, all points scored within the circle were also within the triangle, and vice versa. So Judy could not have been the first to score in the triangle if Kate was the first to score in the circle. This means that at least one of Judy and Kate has a false hypothesis, and since there is only one truthteller, exactly one of Judy and Kate must have a true hypothesis.

Suppose Isabelle and Judy lied, while Kate told the truth. Isabelle's false conclusion tells us that Isabelle did not score any 2-pointers, and Judy's false conclusion tells us that Kate did not score any 2-pointers, either. Judy must have scored all four 2-pointers, plus one 3-pointer to get 11 points. Judy did not score any 4-pointers or 5-pointers, which means her hypothesis must be false, making her a...truthteller. So Isabelle and Judy did not both lie.

This means that Isabelle and Kate both lied. From their false conclusions, we know that Judy and Kate have four 2-pointers between them, and Kate has more 2-pointers, giving her either three or four 2-pointers. If Kate scored four 2-pointers, she must have also scored one 3-pointer to get 11 points, but this would make her hypothesis false since she scored no 4-pointers or 5-pointers. So Kate must have scored three 2-pointers and one 5-pointer.

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