Where is Jumbo hiding!

The Jumbo problem : Where is Jumbo ?

There is a jumbo hiding behind one of the doors named A, B, C and D. Only one statement of all statements written on the doors is true. Find out the door behind which the jumbo is hiding.

A (Jumbo is behind B or C)	B Jumbo is behind A or D
C (Jumbo is here)	D (Jumbo is not here)

Note:- The problem appeared in Deccan Herald news paper a few years ago.

Solution

Let Jd mean Jumbo is behind door d  where d ∈ {A, B, C, D}

For example JA indicates Jumbo is behind door A. If Jumbo is not behind true, it is indicated by symbols ~JA

Let Sd mean the statement written on door d  where d ∈ {A, B, C, D}

For example, SA indicates the statement written on door A. It also indicates the statement is TRUE. If the statement is false, we indicate this truth using symbols ~SA.

Case 1: Let us assume SA is true.

SA ==> (JB V JC)  Λ  SB is false  Λ  SC is false  Λ  SD is false
      ==> (JB V JC)  Λ  ~SB  Λ  ~SC  Λ  ~SD
      ==> (JB V JC)  Λ  ~(JA  V  JD)  Λ  ~(JC)   Λ   ~(~JD)
      ==> (JB V JC)  Λ  (~JA  Λ  ~JD)  Λ  ~JC    Λ  JD
      ==> (JB V JC)  Λ ~JA  Λ ~JD Λ ~JC   Λ  JD
      ==>  ~JD  and JD are contradicting  and hence SA must be false.

We started with SA as true and landed with SA as false. Hence. SA must be false.

Case 2: Let us assume SB is true.

SB ==> SB  Λ  ~SA  Λ  ~SC  Λ  ~SD
      ==> (JA  V  JD)  Λ  ~(JB  V  JC)   Λ   ~(JC)  Λ   ~(~JD)
      ==> (JA  V  JD)  Λ  (~JB  Λ  ~JC)   Λ   ~JC   Λ  JD
      ==> (JA  V  JD)  Λ  ~JB  Λ  ~JC  Λ   ~JC   Λ  JD
      ==> (JA  V  JD)  Λ  ~JB  Λ  ~JC   Λ  JD

      ==> (JA   Λ  ~JB  Λ  ~JC   Λ  JD)   V   (JD  Λ  ~JB  Λ  ~JC   Λ  JD)

      ==> (JA   Λ  ~JB  Λ  ~JC   Λ  JD)   V   (~JB  Λ  ~JC   Λ  JD)

The first operand of V  indicates jumbo is behind A as well as behind D. Jumbo can not be behind both doors! Hence, the operand must be  false.

The second operand indicates that jumbo is not in B, not in C and is in D. This makes sense and hence Jumbo is behind D.


SB is true and Jumbo is behind door D.

Case 3: Let us assume SC is true.

SC ==> ~SA  Λ  ~SB   Λ   SC  Λ  ~SD
      ==> (JB  V  JC)    Λ  ~(JA   V  JD)   Λ   (JC)  Λ  ~(~JD)
      ==> (JB  V  JC)    Λ  (~JA  Λ  ~JD)   Λ   JC  Λ  JD

      ==> (JB  V  JC)    Λ  ~JA  Λ  ~JD   Λ   JC  Λ  JD
      ==> (JB  V  JC)    Λ  ~JA  Λ  JC   Λ  ~JD   Λ  JD

The ~JD and JD are contradicting.  Hence SC must be false.

Case 4: Let us assume SD is true.

SD ==> ~SA  Λ  ~SB   Λ   ~SC  Λ  SD

      ==> ~(JB  V  JC)    Λ  ~(JA   V  JD)   Λ   ~(JC)  Λ  (JD)

      ==> (~JB  Λ  ~JC)    Λ  (~JA  Λ  ~JD)   Λ   ~JC  Λ  JD

      ==> ~JB  Λ  ~JC    Λ  ~JA  Λ  ~JD   Λ   ~JC  Λ  JD

      ==> ~JA  Λ  ~JB  Λ  ~JC    Λ   ~JD   Λ  JD

The ~JD and JD are contradicting.  Hence SD must be false.

Result

Statement written on door B is true and the Jumbo is behind door D.