Malvin: What a great idea to have a workshop on the birthday of Alexandru. But I must confess that I do not know how old he gets.
Hans: I also do not know.
Sonja: Of course I know. But instead of just telling you his age, let me give you a puzzle. In fact, on the day of the workshop Alexandru will have an age that is a product of two prime numbers.
Malvin: Oh, nice, an age that can be used for Cryptography!
Hans: But probably the numbers are too low - I guess he must be between 40 and 90?
Sonja: Correct. Let me now tell each of you one of the prime numbers. Malvin will get a prime number between 1 and 10, and Hans will get a prime number between 10 and 20.
[ Sonja secretly tells Malvin and Hans a number. ]
Hans: I do not know the age and I do not know whether you know it.
Malvin: Aha! I did not know it before, but now I know it!
Hans: Oh, good, now I also know it!
Sonja: Great, and because all our readers are perfect logicians, they will also know it now.
Malvin and Hans: Happy Birthday Alexandru!
Written for the occasion of Alexandru Baltag’s birthday workshop. For information about the workshop and many other birthday wishes, see https://tinyurl.com/thebaltagworkshop (spoiler alert!).
The puzzle above can be solved using SMCDEL as follows.
-- Alexandru's Birthday Puzzle in SMCDEL
VARS 2,3,5,7, -- first prime
11,13,17,19 -- second prime
LAW AND (
-- one of each list must be true:
OR ( 2, 3, 5, 7),
OR ( 11, 13, 17, 19 ),
-- but not more than one from each list
AND ( ~(2 & 3), ~(2 & 5), ~(2 & 7)
, ~(3 & 5), ~(3 & 7)
, ~(5 & 7)
),
AND ( ~(11 & 13), ~(11 & 17), ~(11 & 19)
, ~(13 & 17), ~(13 & 19)
, ~(17 & 19)
),
-- manually exclude products below 40 and above 90:
~2,
(3 -> (~11 & ~13)),
(5 -> ~19),
(7 -> ~13 & ~17 & ~19)
)
OBS hans: 11, 13, 17, 19
malvin: 2, 3, 5, 7
-- list all states
WHERE?
Top
-- list all states where the dialogue is true
WHERE?
-- Hans: I do not know the age:
~ (AND ( hans knows whether 2, hans knows whether 3, hans knows whether 5) )
&
-- ... and I do not know whether you know it:
~ (hans knows whether AND (malvin knows whether 11, malvin knows whether 13, malvin knows whether 17))
&
-- Malvin: Aha! I did not know it before ...
~ AND (malvin knows whether 11, malvin knows whether 13, malvin knows whether 17)
&
-- ... but now I know it!
< ! ~ (AND (hans knows whether 2, hans knows whether 3, hans knows whether 5) )
& ~ (hans knows whether (AND ( malvin knows whether 11, malvin knows whether 13, malvin knows whether 17 )))
>
(AND ( malvin knows whether 11, malvin knows whether 13, malvin knows whether 17))