# Ultimate family dart game

A family of 3 - a father, mother and son - are practising on a dart board. It has 3 shapes - a triangle, a circle and a square - on it. Scoring is as follows:

• Intersection of all 3 shapes - 5 points
• Intersection of exactly 2 shapes - 4 points
• Exactly in one shape - 3 points
• Outside all shapes, but on the board - 2 points
• Outside the board - 0 points

After the game, the board looked like this: (Each star is an arrow)

All 3 of them scored the same number of points.

Each of them also made a statement:

• Father: If I was the first to score in the square, then I scored at least one 2.
• Mother: If I was the first to score in the triangle, then my husband and my son did not score the same number of 2s as each other.
• Son: If I scored first in the circle, then I did not score more 2s than my mother.

Exactly one of them told the truth. Who scored the 5?

Note: Remember that, for any

Statement X: If A is true, then B is true.

we can conclude that

Statement X is false if and only if A is true and B is false.

• Can you crop your image so there's not a giant whitespace below it? (or is that relevant somehow?) Also, Mother's statement is unclear. Do you mean "my husband and my son did not score the same number of 2s as each other" or "as I did"? "Neither" in that statement causes an awkward double-negative, making it actually parse as "my husband and my son did score the same number of 2s". – Ian MacDonald Mar 27 '15 at 16:11
• Is there relevance to scoring "first" in a shape? I can't see how it would change if it was "If I scored (at all) in the ..." instead. – Set Big O Mar 27 '15 at 16:32
• Which player only threw two arrows? – Brian Robbins Mar 27 '15 at 16:55
• Your Son and Mother statements don't tell us what you want them to tell us. If Son scored first in the circle, then Mother did not score first in the triangle. This makes Mother's statement meaningless. – Ian MacDonald Mar 27 '15 at 17:35
• @BrianRobbins none- they missed the board – kaine Mar 27 '15 at 17:39

If dad tells the truth, there is a contradiction as everyone who scores in a circle scores in a triangle and vice versa. There is only one first and mom and son would need to have that. This all means the father definitely scored first in the square and did not have any 2's. He scored at least one 3 or the 5.

The dad cannot be telling the truth.

If we assume that the son told the truth, the mother lied (we already know the father must be lying). This means that the mother scored at least once in the triangle so scored a 4 or a 5. Her statement "my husband and my son did not score the same number of 2s as each other" must be false so we know the son must not have scored any 2s just like his Dad. This would require the mother to score at least 12 points ($4*2+1*4=12$) which is impossible as everyone scored 11 points.

This means the son must be lying. It means by process of elimination the mother must be telling the truth.

If mom tells the truth: the father scored in the square first (earned a 3 or 5), the father scored zero 2s, the son scored in the circle first (earned a 4 or 5), and scored more 2s than his mother. If the son scored in the circle first, the mother didn't score in the triangle first.

From this we can determine that the son scored three 2s. If he scored 2 or fewer, he would not score more than his mother. If he scored 4, than he would earn more than 11 points. This means the mother scored one 2. We know he scored a 4 or a 5.

As $3*2+4=10$ and there is no way to score 1 point, the son must have scored the 5. If you wish to continue you can determine that (if at least one person never missed the board) the father scored one 3, two 4s, and one 0. The mother scored one 2 and three 3s. The son scored one 5 and three 2s.

Unlike the previous edit, I've shown that this is the only possible solution.