Binary Logic - Beginner-Intermediate Level: dual logic BEGINNER-INTERMEDIATE

This is a classic cycle puzzle.

With an even number of people (4) in a cycle of accusations,
the unique solution is that alternating people are truth-tellers.

Therefore:
Gaurav (father) = Truth-teller
Meera (mother) = Liar
Rohan (son) = Truth-teller
Neha (daughter) = Liar

Verification:
Father (T) says 'Mother is liar' - TRUE ✓
Mother (L) says 'Son is liar' - FALSE (son is T) ✓
Son (T) says 'Daughter is liar' - TRUE ✓
Daughter (L) says 'Father is liar' - FALSE (father is T) ✓

This is the unique consistent assignment.

The code represents the truth pattern: 1=truth-teller, 0=liar.

Testing each possible code:
- Code 1010 makes all statements consistent:
* Rohan's statement is true → matches bit 1
* Meera's statement is false → matches bit 0
* Sunil's statement is true → matches bit 1
* Priya's statement is true → matches bit 0

No other code satisfies all constraints.
Therefore, the correct code is 1010.

First, solve the original puzzle:
- If Kiran is truth-teller -> exactly one liar -> p2 and p3 are truth-tellers.
Then p2 (truth) says 'Kiran and Rahul same type' - true (both truth) - consistent.
p3 (truth) says 'at least one truth-teller' - true - consistent.
Solution: Kiran=T, Deepa=T, Rahul=L

Now, hypothetically assume Kiran is liar instead of truth-teller.
Then we need to re-solve:
- Kiran liar -> statement 1 false -> number of liars is NOT exactly one -> 0, 2, or 3 liars.
- If 0 liars, all truth-tellers. Then Kiran truth - contradicts Kiran liar.
- If 2 liars, then Deepa and Rahul are liars. Then Deepa liar says 'Kiran and Rahul same type' - Kiran liar, Rahul liar -> same type -> true statement, but liar can't make true - contradiction.
- If 3 liars, all are liars. Then Deepa liar says 'Kiran and Rahul same type' - both liars -> same -> true statement - contradiction.
- Therefore, no consistent assignment exists when Kiran is liar.
Thus, if we hypothetically assume Kiran is liar, there would be ZERO truth-tellers.

Key insight: The statement 'Sometimes I tell the truth and sometimes I lie' is characteristic of an alternator.
- A truth-teller cannot say 'Sometimes I lie' (would be false).
- A liar cannot say 'Sometimes I tell truth' (would be true, but liars always lie).
- Only an alternator can truthfully acknowledge their alternating nature.
Therefore: Pooja is the alternator.

Identify types by their statements:

- Deepa claims to always tell truth. A truth-teller can say this, a liar cannot (would be true statement). So Deepa is likely truth-teller.
- Kiran claims 'Amit is truth-teller'. Without knowing Amit's type, this is ambiguous.
- Amit admits to sometimes lying and sometimes telling truth - this is the hallmark of an alternator.
- Gaurav and Rohan make contradictory claims about each other, suggesting one is truth-teller, one liar.

The key is Amit's self-description. A truth-teller cannot say 'sometimes I lie' (would be false). A liar cannot say 'sometimes I tell truth' (would be true, but liars always lie). Only an alternator can truthfully describe their alternating nature.

Therefore, Amit is the alternator.

Gaurav says: 'Priya is a liar'. If Gaurav is a truth-teller, then the statement is true, so Priya is a liar. If Gaurav were a liar, the statement would be false, meaning Priya is a truth-teller. Both are possible, but the question asks for Gaurav's type. Since we need a unique answer, consider that truth-tellers can make true statements about others, while liars make false statements. This configuration has a consistent assignment where Gaurav tells truth and Priya lies. Therefore, Gaurav is a truth-teller.

Let's solve step by step:

Step 1: Assume Sanjay is a liar.
→ Sanjay's statement 'Harsha is a truth-teller' is FALSE → Harsha is a liar.
→ Anita says 'Sanjay is a liar' - this is TRUE (since Sanjay is liar).
→ If Anita tells truth, then Anita is truth-teller.
→ Harsha (liar) says 'Anita is a liar' - FALSE (since Anita is truth) → consistent.
This gives: Harsha=L, Anita=T, Sanjay=L (two liars, one truth-teller).

Step 2: Assume Sanjay is a truth-teller.
→ Sanjay's statement 'Harsha is a truth-teller' is TRUE → Harsha is truth-teller.
→ Harsha (truth) says 'Anita is a liar' → TRUE → Anita is liar.
→ Anita (liar) says 'Sanjay is a liar' - FALSE (since Sanjay is truth) → consistent.
This gives: Harsha=T, Anita=L, Sanjay=T (two truth-tellers, one liar).

Both assignments are valid! This puzzle has two solutions.

To guarantee a unique solution, we add a fourth person:
Sanjay also says: 'Exactly one of us is a liar'

With this constraint, only Step 2 works (two truth-tellers, one liar).
Therefore, Harsha and Sanjay are truth-tellers, Anita is a liar.

Let's test if all statements can be true:

Assumption 1: If Farhan tells truth, then Rohan has sapphire.
But Rohan claims not to have it - contradiction if Rohan tells truth.
If Rohan lies, then Rohan DOES have the item - consistent with Farhan.
Then Neha says Farhan lies - but we assumed Farhan tells truth - contradiction!

Assumption 2: If Farhan lies, then Rohan does NOT have sapphire.
Rohan says the same thing - consistent if Rohan tells truth.
Neha says Farhan lies - consistent if Neha tells truth.

Therefore, all statements CAN be consistent when Farhan lies, Rohan and Neha tell truth.
Thus, the statements are consistent.

First, solve the original puzzle:
- If Deepa is truth-teller -> exactly one liar -> p2 and p3 are truth-tellers.
Then p2 (truth) says 'Deepa and Divya same type' - true (both truth) - consistent.
p3 (truth) says 'at least one truth-teller' - true - consistent.
Solution: Deepa=T, Rohan=T, Divya=L

Now, hypothetically assume Deepa is liar instead of truth-teller.
Then we need to re-solve:
- Deepa liar -> statement 1 false -> number of liars is NOT exactly one -> 0, 2, or 3 liars.
- If 0 liars, all truth-tellers. Then Deepa truth - contradicts Deepa liar.
- If 2 liars, then Rohan and Divya are liars. Then Rohan liar says 'Deepa and Divya same type' - Deepa liar, Divya liar -> same type -> true statement, but liar can't make true - contradiction.
- If 3 liars, all are liars. Then Rohan liar says 'Deepa and Divya same type' - both liars -> same -> true statement - contradiction.
- Therefore, no consistent assignment exists when Deepa is liar.
Thus, if we hypothetically assume Deepa is liar, there would be ZERO truth-tellers.

Priya says: 'Leena is a liar'. If Priya is a truth-teller, then the statement is true, so Leena is a liar. If Priya were a liar, the statement would be false, meaning Leena is a truth-teller. Both are possible, but the question asks for Priya's type. Since we need a unique answer, consider that truth-tellers can make true statements about others, while liars make false statements. This configuration has a consistent assignment where Priya tells truth and Leena lies. Therefore, Priya is a truth-teller.

Let's solve by cases:

Case 1: Neha came first.
Then statement 1 is true, statement 2 is false.
If statement 2 is false, then Kiran is liar.
Statement 3: Harsha says 'I came second' - unknown.
Statement 4: Rohan says 'Harsha is lying'.
This leads to multiple possibilities.

Case 2: Kiran came first.
Then statement 1 is false → Neha is liar.
Statement 2 is true → Kiran is truth-teller.
If Harsha came second, statement 3 is true → Harsha is truth-teller.
Then statement 4 says 'Harsha is lying' - false → Rohan is liar.
This gives 2 truth-tellers (Kiran, Harsha) and 2 liars, consistent.

Therefore, the only consistent assignment is Kiran came first.

Let's solve by cases:

Case 1: Meera came first.
Then statement 1 is true, statement 2 is false.
If statement 2 is false, then Priya is liar.
Statement 3: Neha says 'I came second' - unknown.
Statement 4: Sanjay says 'Neha is lying'.
This leads to multiple possibilities.

Case 2: Priya came first.
Then statement 1 is false → Meera is liar.
Statement 2 is true → Priya is truth-teller.
If Neha came second, statement 3 is true → Neha is truth-teller.
Then statement 4 says 'Neha is lying' - false → Sanjay is liar.
This gives 2 truth-tellers (Priya, Neha) and 2 liars, consistent.

Therefore, the only consistent assignment is Priya came first.

As an alternator, the person alternates between truth and lies.
Starting with truth on the first statement:
Statement 1: True
Statement 2: False
Statement 3: True
Statement 4: False

Therefore, statements 1 and 3 are true.

Let's solve using truth table method:

Let A,B,C,D represent if each person tells truth (1) or lies (0).

Statement 1: A says 'Exactly two truth-tellers' → A = 1 iff (A+B+C+D = 2)
Statement 2: B says 'C is liar' → B = 1 iff C = 0
Statement 3: C says 'D is truth-teller' → C = 1 iff D = 1
Statement 4: D says 'A is liar' → D = 1 iff A = 0

From statement 4: D = 1 - A
From statement 3: C = D = 1 - A
From statement 2: B = 1 - C = 1 - (1 - A) = A
From statement 1: A = 1 iff (A + B + C + D = 2)

Substitute: A + B + C + D = A + A + (1-A) + (1-A) = 2
The sum is ALWAYS 2! So statement 1 is TRUE regardless.
Therefore A = 1 (truth-teller).

Then:
A = 1 (truth-teller)
B = A = 1 (truth-teller)
C = 1 - A = 0 (liar)
D = 1 - A = 0 (liar)

Final assignment: Meera=T, Deepa=T, Sunil=L, Priya=L
Therefore, truth-tellers are Meera and Deepa.

This is the classic liar paradox. If the statement is true, then it must be false. If false, then it must be true. This creates a logical contradiction that cannot be resolved in classical binary logic.

Step 1: If Anita is truth-teller, then:
- Priya took the emerald (from statement 1).
- Exactly one person took the item (from statement 4).
- Priya says 'I did not take it' - FALSE, so Priya is liar (consistent).
- Manoj says 'Anita is truth-teller' - TRUE, so Manoj is truth-teller.
This gives: Anita=T, Priya=L, Manoj=T with Priya as thief.

Step 2: If Anita is liar, then:
- Priya did NOT take the item (statement 1 false).
- 'Exactly one person took it' is FALSE → either 0 or 2+ people took it.
- Since Priya didn't take it, someone else must have.
- Manoj says 'Anita is truth-teller' - FALSE, so Manoj is liar.
- Priya says 'I did not take it' - TRUE, so Priya is truth-teller.
- This gives Anita=L, Priya=T, Manoj=L with no thief identified - INCONSISTENT.

Therefore, the only consistent solution is Priya took the emerald.

Key insight: The statement 'I am not consistent with my statements' is characteristic of an alternator.
- A truth-teller cannot say 'Sometimes I lie' (would be false).
- A liar cannot say 'Sometimes I tell truth' (would be true, but liars always lie).
- Only an alternator can truthfully acknowledge their alternating nature.
Therefore: Farhan is the alternator.

Key insight: The statement 'Sometimes I tell the truth and sometimes I lie' is characteristic of an alternator.
- A truth-teller cannot say 'Sometimes I lie' (would be false).
- A liar cannot say 'Sometimes I tell truth' (would be true, but liars always lie).
- Only an alternator can truthfully acknowledge their alternating nature.
Therefore: Kiran is the alternator.

Sanjay says: 'Divya is a liar'. If Sanjay is a truth-teller, then the statement is true, so Divya is a liar. If Sanjay were a liar, the statement would be false, meaning Divya is a truth-teller. Both are possible, but the question asks for Sanjay's type. Since we need a unique answer, consider that truth-tellers can make true statements about others, while liars make false statements. This configuration has a consistent assignment where Sanjay tells truth and Divya lies. Therefore, Sanjay is a truth-teller.

Step 1: If Gaurav is truth-teller, then:
- Anita took the sapphire (from statement 1).
- Exactly one person took the item (from statement 4).
- Anita says 'I did not take it' - FALSE, so Anita is liar (consistent).
- Ravi says 'Gaurav is truth-teller' - TRUE, so Ravi is truth-teller.
This gives: Gaurav=T, Anita=L, Ravi=T with Anita as thief.

Step 2: If Gaurav is liar, then:
- Anita did NOT take the item (statement 1 false).
- 'Exactly one person took it' is FALSE → either 0 or 2+ people took it.
- Since Anita didn't take it, someone else must have.
- Ravi says 'Gaurav is truth-teller' - FALSE, so Ravi is liar.
- Anita says 'I did not take it' - TRUE, so Anita is truth-teller.
- This gives Gaurav=L, Anita=T, Ravi=L with no thief identified - INCONSISTENT.

Therefore, the only consistent solution is Anita took the sapphire.