# Some Kids Competing In Kangaroo Math Are Lying. It’s Puzzling.

## Try this puzzle!

Alexa, Barney, Cara, Dylan, Ethan, and Frances are all planning to compete in the Kangaroo Math Contest, and each week they write Kangaroo Math sample questions and answers for each other so they can practice. Each kid writes one problem for exactly one other kid, and each kid receives one problem from exactly one other kid. You decide to ask every kid who wrote the problem that they are working on. However, some of the kids are sneaky and don’t want you to know who wrote their problem.

A kid who gets a problem from Alexa, Barney, or Cara will always lie when you ask them who wrote their problem.

A kid who gets a problem from Dylan, Ethan, or Frances will always tell the truth when you ask them who wrote their problem.

Alexa: “I received a problem from Cara.”
Barney: “I did not receive a problem from Cara.”
Cara: “I did not receive a problem from Ethan.”
Dylan: “I did not receive a problem from Ethan.”
Frances: “I received a problem from Ethan.”
Ethan: “If Alexa received a problem from Barney, then I received a problem from Alexa.”

Who received a problem from whom?

## Try to solve it, then click to see the answer!

Alexa received a problem from Barney. If she had received her Kangaroo Math sample problem from Dylan, Frances, or Ethan, she would have told the truth, so she wouldn’t have said she received it from Cara. But if she actually received it from Cara, she would have had to lie and say she received it from someone else. She couldn’t have received it from herself, so she has Barney’s problem.

Dylan received a problem from Frances. We can deduce that Dylan is telling the truth because if he is not, it creates an impossibility. If he is lying, he did receive a problem from Ethan, but if he received a problem from Ethan, he must be telling the truth. Since he is telling the truth, the only three people he could have gotten the problem from are himself, Frances, and Ethan. We know he is not lying, so we know didn’t receive a problem from Ethan, and he can’t have received one from himself. Therefore, he got his problem from Frances.

Cara received a problem from Dylan. We start with the same deduction we made with Dylan: Cara has to be telling the truth because if she is lying, it creates a contradiction. So, she could only have received a problem from Dylan or Frances. We already know who received Frances’s problem, so Cara has to have received her problem from Dylan.

Barney received a problem from Ethan. Since each kid only gave one practice
problem to one other kid, the only options for who gave Barney a problem are Ethan and Alexa (those are the only two we haven’t figured out already). But since Ethan received a problem from Cara, Barney must be telling the truth. Therefore, he has to have received his problem from Ethan.

Frances received a problem from Alexa. This is the only option left, but we also can prove that it is correct. Frances claimed that she received a problem from Ethan. If instead, she received a problem from Alexa, this is a lie. In addition, since she received a problem from Alexa, she must be lying. Everything works! We did it!

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