A has no information about anybody else, so if A is telling the truth, A is a detective. Otherwise, A must lying and must thus be a mafioso.

Now consider B. If B is an innocent, B has no information about C, D or E; the only way B could know who the detective is would be if A is the detective. But B cannot know from A’s statement whether he is a detective or a mafioso if B is an innocent because either way B would expect A to claim to know B’s identity. This means B, if an innocent, is lying. Thus, B cannot be an innocent. Now, if B is a mafioso and A is a mafioso, then A would be telling the truth, so B cannot be a mafioso if A is a mafioso. Similarly, if A is a detective and B is a mafioso, B would be able to know immediately that A is a detective because B would know if A were a mafioso and since A must be a mafioso or a detective and since B knows A isn’t a mafioso, A would have to be a detective; if this were true, then B would be telling the truth! This means that B must be a detective, which means A must be a mafioso.

Now C must either be a mafioso or an innocent. If C were a mafioso, C would know that A was a mafioso; thus, C would think that B were either an innocent or the detective. If B were innocent, B would not know who the detective is, so C would know that B was a detective. But this would mean C spoke truly! Thus, C must be an innocent.

Finally, we have D. If D is mafioso, then D knows A must be a mafioso. This means, further, that D knows that B must be a detective, since an innocent would not be able to know who the detective is for the reasons stated earlier. Since D know that both he and A are the mafiosos and B is the detective, D would know C and E are innocents, which means D spoke truthfully. This is a contradiction, so C must be an innocent. Thus, we have:

Mafioso, Detective, Innocent, Innocent, Mafioso

]]>1 month with 29 days = 29

4 months with 30 days. 30*4 = 120

7 months with 31 days. 31*7 = 217

Total = 366

so (x,y,z) = (1,4,7) ]]>

I first begin using algebraic manipulations to solve the problems,

and the i notice, 29,30,31 and see that these guys are the number of days in the moths,

so after that I see the Calendar to check the number of days of February in 2019, see 28 days, so the equation can’t be solve in the natural numbers.

I summed 29+30+31 = 90

I made 366-90 = 276

276/3 = 92 = x = y = z

the solution is 121,122,123 ]]>