Binary Logic - Intermediate Level: binary chains INTERMEDIATE

This creates a circular reference. If the first is true, the second must be true, but the second says the first is false - contradiction. If the first is false, the second must be false, but the second says the first is false (true statement) - contradiction.

Identify types by their statements:

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

The key is Kiran'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, Kiran is the alternator.

Key insight: The statement 'On some days I lie, on others I tell truth' 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: Manoj is the alternator.

Given: If Amit is a truth-teller

Step 1: Amit tells truth → 'Rahul is liar' is true → Rahul is liar.
Step 2: Rahul (liar) says 'Sunil is truth-teller' → this statement is false → Sunil is liar.
Step 3: Sunil (liar) says 'Deepa and I are different types' → this statement is false → Deepa is SAME type as Sunil → Deepa is liar.
Step 4: Deepa (liar) says 'Amit is truth-teller' → this statement is false → Amit is liar → CONTRADICTION with our assumption!

This shows that Amit CANNOT be a truth-teller under these statements.
However, the conditional question asks: IF we assume Amit is truth-teller, who MUST be a liar? From step 1, Rahul must be a liar.

Therefore, under the given condition, Rahul must be a liar.

Gaurav says: 'Amit is a truth-teller'. If Gaurav is a liar, then the statement is false, meaning Amit is a liar. This creates a consistent assignment where both are liars. If Gaurav were a truth-teller, the statement would be true, making Amit a truth-teller. Both assignments are possible, but the question asks for Gaurav's type. The configuration has a consistent assignment where Gaurav lies, so Gaurav is a liar.

Key insight: The statement 'On some days I lie, on others I tell truth' 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: Sanjay is the alternator.

Let L = number of liars.

Statement constraints:
1. Amit: L ≥ 2
2. Rahul: L ≤ 3
3. Anita: L = 2
4. Meera: Leena is truth-teller
5. Leena: Amit is liar

From statement 3, L must be exactly 2 for that statement to be true.
But statements 1 and 2 are consistent with L=2 as well.
Now check statements 4 and 5:
If L=2, then 3 truth-tellers exist.
Statement 5 says Amit is liar - if true, then Amit is liar.
Statement 4 says Leena is truth-teller - can be true.
This configuration is possible with L=2.

Can L=1? Statement 1 would be false, so Amit would be liar.
Then statement 5 (Leena says 'Amit is liar') would be TRUE.
So Leena would be truth-teller. Then statement 4 (Meera says 'Leena is truth-teller') would be TRUE.
So Meera would be truth-teller. That gives at least 2 truth-tellers (Leena, Meera) plus possibly others, contradicting L=1.

Therefore L cannot be 1.
The minimum L is 2.

Meera tells truth on: Saturday, Tuesday, Monday, Thursday.
Meera lies on: Wednesday, Friday, Sunday.
On Saturday (a truth day), Meera says: 'The capital of France is Paris'.
Since this is a factual true statement, and Meera tells truth on this day, the statement is a truth.

Identify types by their statements:

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

The key is Priya'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, Priya is the alternator.

Let's solve by cases:

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

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

Therefore, the only consistent assignment is Deepa came first.

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: Ravi=T, Priya=T, Kiran=L, Neha=L
Therefore, truth-tellers are Ravi and Priya.

Let L = number of liars.

Statement constraints:
1. Priya: L ≥ 2
2. Kiran: L ≤ 3
3. Anita: L = 2
4. Divya: Deepa is truth-teller
5. Deepa: Priya is liar

From statement 3, L must be exactly 2 for that statement to be true.
But statements 1 and 2 are consistent with L=2 as well.
Now check statements 4 and 5:
If L=2, then 3 truth-tellers exist.
Statement 5 says Priya is liar - if true, then Priya is liar.
Statement 4 says Deepa is truth-teller - can be true.
This configuration is possible with L=2.

Can L=1? Statement 1 would be false, so Priya would be liar.
Then statement 5 (Deepa says 'Priya is liar') would be TRUE.
So Deepa would be truth-teller. Then statement 4 (Divya says 'Deepa is truth-teller') would be TRUE.
So Divya would be truth-teller. That gives at least 2 truth-tellers (Deepa, Divya) plus possibly others, contradicting L=1.

Therefore L cannot be 1.
The minimum L is 2.

Let's solve step by step:

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

Step 2: Assume Deepa is a truth-teller.
→ Deepa's statement 'Amit is a truth-teller' is TRUE → Amit is truth-teller.
→ Amit (truth) says 'Rahul is a liar' → TRUE → Rahul is liar.
→ Rahul (liar) says 'Deepa is a liar' - FALSE (since Deepa is truth) → consistent.
This gives: Amit=T, Rahul=L, Deepa=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:
Deepa also says: 'Exactly one of us is a liar'

With this constraint, only Step 2 works (two truth-tellers, one liar).
Therefore, Amit and Deepa are truth-tellers, Rahul is a liar.

Let's test if all statements can be true:

Assumption 1: If Meera tells truth, then Kiran has emerald.
But Kiran claims not to have it - contradiction if Kiran tells truth.
If Kiran lies, then Kiran DOES have the item - consistent with Meera.
Then Leena says Meera lies - but we assumed Meera tells truth - contradiction!

Assumption 2: If Meera lies, then Kiran does NOT have emerald.
Kiran says the same thing - consistent if Kiran tells truth.
Leena says Meera lies - consistent if Leena tells truth.

Therefore, all statements CAN be consistent when Meera lies, Kiran and Leena tell truth.
Thus, the statements are consistent.

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:
Rohan (father) = Truth-teller
Ravi (mother) = Liar
Amit (son) = Truth-teller
Deepa (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.

Let L = number of liars.

Statement constraints:
1. Deepa: L ≥ 2
2. Rahul: L ≤ 3
3. Vikram: L = 2
4. Sanjay: Leena is truth-teller
5. Leena: Deepa is liar

From statement 3, L must be exactly 2 for that statement to be true.
But statements 1 and 2 are consistent with L=2 as well.
Now check statements 4 and 5:
If L=2, then 3 truth-tellers exist.
Statement 5 says Deepa is liar - if true, then Deepa is liar.
Statement 4 says Leena is truth-teller - can be true.
This configuration is possible with L=2.

Can L=1? Statement 1 would be false, so Deepa would be liar.
Then statement 5 (Leena says 'Deepa is liar') would be TRUE.
So Leena would be truth-teller. Then statement 4 (Sanjay says 'Leena is truth-teller') would be TRUE.
So Sanjay would be truth-teller. That gives at least 2 truth-tellers (Leena, Sanjay) plus possibly others, contradicting L=1.

Therefore L cannot be 1.
The minimum L is 2.

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

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

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:
Meera (father) = Truth-teller
Pooja (mother) = Liar
Vikram (son) = Truth-teller
Ravi (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.

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: Anita=T, Rahul=T, Ravi=L, Leena=L
Therefore, truth-tellers are Anita and Rahul.

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

Testing each possible code:
- Code 1100 makes all statements consistent:
* Farhan's statement is true → matches bit 1
* Deepa's statement is false → matches bit 1
* Gaurav's statement is true → matches bit 0
* Sunil's statement is true → matches bit 0

No other code satisfies all constraints.
Therefore, the correct code is 1100.