$2.(10$ points$)[$Solve using a DLV program$]$

Game II: Orange and Purple $($Source: Alice in Puzzle Land: A Carrollian Tale for Children

Under Eighty by Raymond Smullyan$)$

"Now," Tweedledum continued, "one or both of us will come out of the

house, each carrying either an orange or purple card, and he $($we$)$ will make statements. Then

you are to figure out who is who.$$

"Just a minute," said Alice, "you have not told me the significance

of the colors orange and purple. Does one of them signify lying and

the other truth$-$telling? And if so$,$ which color means which?$$

$"$Ah$,$ that's the most interesting part of the game!" said

Tweedledum. "You see, when I carry an orange card, it means

that I am telling the truth, and when I carry a purple card, it means

I am lying!$$

"Contrariwise," said Tweedledee, "when I carry an orange card, it

means I am lying, and when I carry a purple card, it means I am

telling the truth!$$

On the next round, both brothers came out

and made the following statements:

FIRST ONE: Our cards are of the same color.

SECOND ONE: Our cards are not of the same color.

Did the first or second person tell the truth?

$[$Hint: we need to know if first or second person told the truth. We did not ask what color are

the cards, or WHO the first speaker is$.$ Each speaker is different. Each carries only one card.

Note that compared to Knight$/$Knave problems we $($a$)$ don$$t know the speaker name $($b$)$ both

could either be liars or truth tellers, depending on the card color.

So you need to consider $4$ cases for each speaker $($either Dum or Dee, and either purple or

orange$).$ There may be more than one model of these sentences, but only one answer to the

question.

Use the dlv format used ie an example below

$\%$ There are three students, Dan, Henry, and John.

$\%$ Let d $=$ "Dan got an A and not a B$"$ and ~d $=$ "Dan got a B and not an A$".$

$\%$ Let h $=$ "Henry got an B and not a C$"$ and ~h $=$ "Henry got a C and not an B$".$

$\%$ Let j $=$ "John got an B and not a C$"$ and ~j $=$ "John got a C and not an B$".$

$\%$ Dan could get A or B$,$ and Henry and John could get B or C as their final grade.

d v ~d$.$

h v ~h$.$

j v ~j$.$

$\%$ If Dan gets an A then either Henry has a C or John has a B$.$

~h v j :$-$ d$.$

$\%$ If John doesn$$t get a B then either Dan won$$t get an A or Henry won$$t get a C$.$

~d v h :$-$ ~j$.$

$\%$ If John gets a B and Henry doesn$$t get a C then Dan won$$t get an A$.$

~d :$-$ j$,$ h$.$

$\%$ The output is $\{-$d$,-$h$,-$j$\},\{-$d$,$ h$,-$j$\},\{-$d$,-$h$,$ j$\},\{$d$,-$h$,$ j$\},\{-$d$,$ h$,$ j$\}.$

$\%$ The only outputs where Henry and John receive the same grade are $\{-$d$,-$h$,-$j$\}$ and $\{-$d$,$ h$,$ j$\}.$

$\%$ In both $\{-$d$,-$h$,-$j$\}$ and $\{-$d$,$ h$,$ j$\},$ Dan did not get an A$.$

$\%$ So the answer is if John and Henry received the same final grade, Dan will not get an A$.$