In a remote island there are only two types of people that live on it, Knights and Liars. Knights always tell the truth. Liars always speak falsely.

Five suspects are interrogated in connection with a murder. Their statements are as follows:

Al: "Cal and Dan are lying." 
Ben: "Al and Elmer are lying." 
Cal: "Ben and Dan are lying." 
Dan: "Cal and Elmer are lying." 
Elmer: "Al and Ben are lying." 

Which suspect do you know for certain to be a liar?

Logical solution

AL - Means A is lying

AT - Means A is telling the truth

1)A – C & D are lying , 2)B – A & E are lying, 3)C – B & D are lying, 4)D – C & E are lying, 5)E – A & B are lying

Case 1 – A is telling the truth - AT

From 1) CL , DL, From 3) BT, From 4) ET, From 2) AL,EL is false, From 5) AL, BL is false

Hence A is Lying for sure

Case 2 – B is telling the truth - BT

From 2) AL, EL, From 1) either CT or DT or CT&DT, From 5) AL & BT is true, From 3) CT is not possible if CL then DT is only possibility, From 4) CL & EL is confirmed

Hence B could be telling the truth

Case 3 – C is telling the truth - CT

From 3) BL, DL, From 2) either AT or ET or AT&ET, From 4) ET & CT is true, From 1) CT & DL is true & From 5)AL & BL is true

Hence C could be telling the truth

Case 4 – D is telling the truth - DT

From 4) CL, EL, From 3) either BT or DT or BT&DT, From 5) either AT or BT or AT&BT, From 1) DL and AL  hence BT From 2)AL & EL which is true

Hence D could be telling the truth

Case 5 – E is telling the truth - ET

From 5) AL, BL, From 1) either CT or DT or CT&DT, From 2) AL, ET, From 4) DL, CT, ET is solution, From 3) BL, DL and CT is confirmed

Hence E could be telling the truth

Hence Al is the only liar.

Posted by Salil on 2013-02-28 00:49:44