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I have this problem but I am stuck...

we have three boxes - on each one, there is an inscription

box1 - the magic recipe is here
box2 - the magic recipe is not here
box3 - the magic recipe is not in box1

assuming that $p,q,r$ are true only when the recipe is in the box1, box2, box3, respectively, write the following statements:

a- exact only one box has the recipe, the other two do not
b- only one inscription at most is true
c- exactly two boxes have the recipe, and the third is empty

Q1 - in which box is the recipe if you know that exactly only one box has the recipe, the other does not, and only one inscription at most is true?
Q2 - if two boxes have a recipe and the other is empty while only one inscription is true, which box would you pick?

Solution

The three inscriptions can be written as

$i_1 = p, i_2 = ¬q, i_3 = ¬p$

and then the statements
$a = (p ∧ ¬q ∧ ¬ r) ∨(¬p ∧ q ∧¬ r) ∨ (¬p ∧¬q ∧ r)$
$b = (¬i_1 ∧ ¬i_2 ∧ ¬i_3) ∨ i_1 ∨ i_2 ∨ i_3$
$c = (p ∧ q ∧ ¬r )∨( p ∧ r ∧ ¬q )∨ (q ∧ r ∧ ¬p )$

I would say that $Q_1 = a∧b$ and $Q_2= c ∧ b$, but trying to make the truth table for Q1 does not give me a unique answer - I guess I ve done something wrong with formatting the statements a-c? Any help?

Mauro ALLEGRANZA
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Terma
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  • I think the translation for $b$ is not accurate. At most only 1 true means it's either all false (which you're right about), or 1 is true and 2,3, false, or 2 is true and 1,3, false, or 3 true and 1,2, false. – Giant Ray Dec 12 '22 at 23:22
  • I see... still, if I do include statements for what you say (and you are right), then still not a unique solution... – Terma Dec 12 '22 at 23:40

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