Answer to Question #127027 in Discrete Mathematics for Buhle

Question

Robyn decides to organize a hiking trip and invites her friends - Dylan, Linda and Dominic to join her. On the days leading to the hike, her three friends say the following:Two Days Before the hike:Dylan: Dominic is going to the hike. Dominic: Linda is not going to the hike.Linda: Dylan will go to the hike if, and only if, I do.The Day Before the hike:Dylan: Linda will go to the hike if, and only if, I don't go. Dominic: An even number out of the three of us are going to the hike.Linda: Dylan is going to the hike.The Day of the hike:Dylan: It is not yet 2020. Dominic: Dylan will go to the hike if, and only if, I do.Linda: At least one of the three of us is not going to the hike. Robyn also knows that out of Dylan, Dominic, and Linda:One of them never lies.One lies on days of the month that are divisible by 2, but is otherwise truthful.The remaining one lies on days of the month that are divisible by 3, but is otherwise truthful.Can you figure out who is going to attend?

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c} & Two \ days & One\ day &Day \\ & before & before & the \ hike \\ \hline Dylan & ? & ? & ? \\ \hdashline Dominik & ? & ? & ? \\ \hdashline Linda & ? & ? & ? \end{array}$$

If Dylan never lies:

Dylan: Dominic is going to the hike. True

Dylan: Linda will go to the hike if, and only if, I don't go. True

Dylan: It is not yet 2020. True

Then the participants of the hike will be:

Robin, Dominic, Linda. Or Robin, Dylan, Dominic.

Robin, Dominic, Linda:

Dominic: Linda is not going to the hike. False

Dominic: An even number out of the three of us are going to the hike. True

Dominic: Dylan will go to the hike if, and only if, I do. False

Linda: Dylan will go to the hike if, and only if, I do. False

Linda: Dylan is going to the hike. False

Linda: At least one of the three of us is not going to the hike. True

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c} & Two \ days & One\ day &Day \\ & before & before & the \ hike \\ \hline Dylan & T & T & T \\ \hdashline Dominik & F & T & F \\ \hdashline Linda & F & F &T \end{array}$$

We have a contradiction.

Robin, Dylan, Dominic:

Dominic: Linda is not going to the hike. True

Dominic: An even number out of the three of us are going to the hike. True

Dominic: Dylan will go to the hike if, and only if, I do. True

Dominic never lies.

Linda: Dylan will go to the hike if, and only if, I do. False

Linda: Dylan is going to the hike. True

Linda: At least one of the three of us is not going to the hike. True

Linda lies on days of the month that are divisible by 3, but is otherwise truthful.

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c} & Two \ days & One\ day &Day \\ & before & before & the \ hike \\ \hline Dylan & T & T & T \\ \hdashline Dominik & T & T & T \\ \hdashline Linda & F & T &T \end{array}$$

We have a contradiction.

If Linda never lies:

Linda: Dylan will go to the hike if, and only if, I do. True

Linda: Dylan is going to the hike. True

Linda: At least one of the three of us is not going to the hike. True

Then the participants of the hike will be:

Robin, Dylan, Linda.

Dylan: Dominic is going to the hike. False

Dylan: Linda will go to the hike if, and only if, I don't go. False

Dylan: It is not yet 2020. True

Dominic: Linda is not going to the hike. False

Dominic: An even number out of the three of us are going to the hike. True

Dominic: Dylan will go to the hike if, and only if, I do. False

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c} & Two \ days & One\ day &Day \\ & before & before & the \ hike \\ \hline Dylan & F & F & T \\ \hdashline Dominik & F & T & F \\ \hdashline Linda & T & T &T \end{array}$$

We have a contradiction.

If Dominik never lies:

Dominic: Linda is not going to the hike. True

Dominic: An even number out of the three of us are going to the hike. True

Dominic: Dylan will go to the hike if, and only if, I do. True

Then the participants of the hike will be:

Robin, Dylan, Dominik.

Dylan: Dominic is going to the hike. True

Dylan: Linda will go to the hike if, and only if, I don't go. True

Dylan: It is not yet 2020. False

Dylan lies on days of the month that are divisible by 3, but is otherwise truthful.

Linda: Dylan will go to the hike if, and only if, I do. False

Linda: Dylan is going to the hike. True

Linda: At least one of the three of us is not going to the hike. True

Linda lies on days of the month that are divisible by 3, but is otherwise truthful.

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c} & Two \ days & One\ day &Day \\ & before & before & the \ hike \\ \hline Dylan & T & T & F \\ \hdashline Dominik & T & T & T \\ \hdashline Linda & F & T &T \end{array}$$

We have a contradiction.

I cannot figure out who is going to attend.