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Lost in Translation:
This was the problem as originally set:
primus: Spartacus sum!
secundus: Spartacus sum!
tertius: Uterque mentitur, Spartacus sum!
quartus: Tertius mentitur - Spartacus sum!
(Spartacus semper veridicus est; semper socii mentiuntur)
Quis Spartacus est?
Translation:
A: I'm Spartacus!
B: I'm Spartacus!
C: They both lie, I'm Spartacus!
D: C lies, I'm Spartacus!
(Spartacus always tells the truth, while his companions always lie)
Who is Spartacus?
Note: ''Uterque mentitur' can mean 'both of them'; or 'each of them'; or 'either of them'
Solution
C is Spartacus.
C really makes either 2 or 3 statements, depending whether 'both' is construed conjunctively or disjunctively:
(a) that C is Spartacus
(b)(i) that at least one of (A or B) is lying (Case 1)
(b)(ii) that (A is lying and B is lying) (Case 2)
(b)(iii)(1) that (A is lying) and b(iii)(2) that (B is lying)(Case 3)
C can only be telling the truth if C is Spartacus.
1. Disjunctive 'both' = either (A lies OR B lies); at least one of (A or B) is lying.
Assume that A or B is Spartacus.
Then C (a) tells the truth that one of A or B is lying, but (b) lies that he, C is Spartacus, a contradiction.
So either C or D is Spartacus.
D also makes 2 statements:(a) C is lying and (b) he, D is Spartacus.
Assume that D is Spartacus.
Then C is lying, but C truthfully said that at least one of (A or B) was lying, a contradiction.
So D is a liar.
It follows that C is Spartacus.
2. Conjunctive 'both'= Both (A lies AND B lies)
3. Distributive: (1) that (A is lying) and (2) that (B is lying) (i.e. If C is lying, it is because he says that 'both' are lying, when only one of them is.)
The same reasoning applies in both cases:
Assume that A or B is Spartacus.
Then C is lying, because one of A or B is telling the truth; but then D is telling the truth that C is lying.
So D must be Spartacus, a contradiction since we have assumed that A or B is Spartacus.
Assume that D is Spartacus.
Then C was telling the truth that A lies AND B lies, but lying that he, C, was Spartacus, a contradiction.
It again follows that C must be Spartacus.
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