Binary Logic - Intermediate Level: two-state logic INTERMEDIATE

Identify types by their statements:

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

The key is Sanjay'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, Sanjay is the alternator.

Step 1: If Kiran 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).
- Neha says 'Kiran is truth-teller' - TRUE, so Neha is truth-teller.
This gives: Kiran=T, Priya=L, Neha=T with Priya as thief.

Step 2: If Kiran 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.
- Neha says 'Kiran is truth-teller' - FALSE, so Neha is liar.
- Priya says 'I did not take it' - TRUE, so Priya is truth-teller.
- This gives Kiran=L, Priya=T, Neha=L with no thief identified - INCONSISTENT.

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

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

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

Let's solve by cases:

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

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

Therefore, the only consistent assignment is Anita came first.

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

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

Therefore, the only consistent solution is Leena took the crystal.

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: Neha=T, Leena=T, Divya=L, Gaurav=L
Therefore, truth-tellers are Neha and Leena.

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.

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.

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.

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

Testing each possible code:
- Code 1100 makes all statements consistent:
* Neha's statement is true → matches bit 1
* Rahul's statement is false → matches bit 1
* Rohan's statement is true → matches bit 0
* Ravi's statement is true → matches bit 0

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

Identify types by their statements:

- Harsha claims to always tell truth. A truth-teller can say this, a liar cannot (would be true statement). So Harsha is likely truth-teller.
- Pooja 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.
- Priya and Neha 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.

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

Therefore, statements 2 and 4 are true.

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:
Leena (father) = Truth-teller
Farhan (mother) = Liar
Divya (son) = Truth-teller
Vikram (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: Divya=T, Deepa=T, Meera=L, Pooja=L
Therefore, truth-tellers are Divya and Deepa.

Let L = number of liars.

Statement constraints:
1. Farhan: L ≥ 2
2. Amit: L ≤ 3
3. Leena: L = 2
4. Manoj: Sunil is truth-teller
5. Sunil: Farhan 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 Farhan is liar - if true, then Farhan is liar.
Statement 4 says Sunil is truth-teller - can be true.
This configuration is possible with L=2.

Can L=1? Statement 1 would be false, so Farhan would be liar.
Then statement 5 (Sunil says 'Farhan is liar') would be TRUE.
So Sunil would be truth-teller. Then statement 4 (Manoj says 'Sunil is truth-teller') would be TRUE.
So Manoj would be truth-teller. That gives at least 2 truth-tellers (Sunil, Manoj) 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 Divya is a liar.
→ Divya's statement 'Priya is a truth-teller' is FALSE → Priya is a liar.
→ Vikram says 'Divya is a liar' - this is TRUE (since Divya is liar).
→ If Vikram tells truth, then Vikram is truth-teller.
→ Priya (liar) says 'Vikram is a liar' - FALSE (since Vikram is truth) → consistent.
This gives: Priya=L, Vikram=T, Divya=L (two liars, one truth-teller).

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

With this constraint, only Step 2 works (two truth-tellers, one liar).
Therefore, Priya and Divya are truth-tellers, Vikram is a liar.

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

Now, hypothetically assume Meera is liar instead of truth-teller.
Then we need to re-solve:
- Meera liar -> statement 1 false -> number of liars is NOT exactly one -> 0, 2, or 3 liars.
- If 0 liars, all truth-tellers. Then Meera truth - contradicts Meera liar.
- If 2 liars, then Rohan and Sanjay are liars. Then Rohan liar says 'Meera and Sanjay same type' - Meera liar, Sanjay liar -> same type -> true statement, but liar can't make true - contradiction.
- If 3 liars, all are liars. Then Rohan liar says 'Meera and Sanjay same type' - both liars -> same -> true statement - contradiction.
- Therefore, no consistent assignment exists when Meera is liar.
Thus, if we hypothetically assume Meera 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:
Pooja (father) = Truth-teller
Amit (mother) = Liar
Priya (son) = Truth-teller
Vikram (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.

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:
Neha (father) = Truth-teller
Sanjay (mother) = Liar
Divya (son) = Truth-teller
Manoj (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.

Identify types by their statements:

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

The key is Rahul'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, Rahul is the alternator.