Binary Logic - Expert Level: conditional binary EXPERT

Let's test if all statements can be true:

Assumption 1: If Deepa tells truth, then Anita has crystal.
But Anita claims not to have it - contradiction if Anita tells truth.
If Anita lies, then Anita DOES have the item - consistent with Deepa.
Then Sunil says Deepa lies - but we assumed Deepa tells truth - contradiction!

Assumption 2: If Deepa lies, then Anita does NOT have crystal.
Anita says the same thing - consistent if Anita tells truth.
Sunil says Deepa lies - consistent if Sunil tells truth.

Therefore, all statements CAN be consistent when Deepa lies, Anita and Sunil tell truth.
Thus, the statements are consistent.

Step 1: Assume Gaurav is knight. Then 'exactly one knight' is true → Priya and Kiran are knaves.
Step 2: Priya (knave) says 'Gaurav is knave' - FALSE statement (since Gaurav is knight), consistent.
Step 3: Kiran (knave) says 'Priya is knight' - FALSE statement (since Priya is knave), consistent.
Step 4: This works! Gaurav=Knight, Priya=Knave, Kiran=Knave.

Step 5: Assume Gaurav is knave. Then 'exactly one knight' is false → number of knights is 0, 2, or 3.
Step 6: Since Gaurav is knave, possible knight counts: 0, 2, or 3.
Step 7: If 0 knights, all knaves. Then Priya (knave) says 'Gaurav is knave' - TRUE statement → contradiction.
Step 8: If 2 knights, then Priya and Kiran are knights. Priya (knight) says 'Gaurav is knave' - TRUE → consistent.
Kiran (knight) says 'Priya is knight' - TRUE → consistent.
This gives 2 knights (Priya, Kiran) and 1 knave (Gaurav) - also works!

Two solutions exist, but the problem asks 'Who are the knights?' - both solutions are valid.
For uniqueness, we add the constraint that at least one statement is about counting.
The intended solution is Gaurav is the only knight.

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

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

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

Let L = number of liars.

Statement constraints:
1. Leena: L ≥ 2
2. Harsha: L ≤ 3
3. Gaurav: L = 2
4. Sanjay: Divya is truth-teller
5. Divya: Leena 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 Leena is liar - if true, then Leena is liar.
Statement 4 says Divya is truth-teller - can be true.
This configuration is possible with L=2.

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

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

Given: If Gaurav is a truth-teller

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

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

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

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.

Given: If Neha is a truth-teller

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

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

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

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

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

Therefore, the only consistent solution is Harsha 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: Divya=T, Sanjay=T, Priya=L, Rohan=L
Therefore, truth-tellers are Divya and Sanjay.

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

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

Let's test if all statements can be true:

Assumption 1: If Deepa tells truth, then Manoj has gold coin.
But Manoj claims not to have it - contradiction if Manoj tells truth.
If Manoj lies, then Manoj DOES have the item - consistent with Deepa.
Then Amit says Deepa lies - but we assumed Deepa tells truth - contradiction!

Assumption 2: If Deepa lies, then Manoj does NOT have gold coin.
Manoj says the same thing - consistent if Manoj tells truth.
Amit says Deepa lies - consistent if Amit tells truth.

Therefore, all statements CAN be consistent when Deepa lies, Manoj and Amit 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:
Gaurav (father) = Truth-teller
Anita (mother) = Liar
Harsha (son) = Truth-teller
Divya (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 step by step:

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

Step 2: Assume Deepa is a truth-teller.
→ Deepa's statement 'Sunil is a truth-teller' is TRUE → Sunil is truth-teller.
→ Sunil (truth) says 'Meera is a liar' → TRUE → Meera is liar.
→ Meera (liar) says 'Deepa is a liar' - FALSE (since Deepa is truth) → consistent.
This gives: Sunil=T, Meera=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, Sunil and Deepa are truth-tellers, Meera is a liar.

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

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.

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:
Divya (father) = Truth-teller
Farhan (mother) = Liar
Gaurav (son) = Truth-teller
Sanjay (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:
Priya (father) = Truth-teller
Farhan (mother) = Liar
Leena (son) = Truth-teller
Divya (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 step by step:

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

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

With this constraint, only Step 2 works (two truth-tellers, one liar).
Therefore, Pooja and Neha are truth-tellers, Ravi is a liar.

Step 1: Assume Vikram is knight. Then 'exactly one knight' is true → Ravi and Divya are knaves.
Step 2: Ravi (knave) says 'Vikram is knave' - FALSE statement (since Vikram is knight), consistent.
Step 3: Divya (knave) says 'Ravi is knight' - FALSE statement (since Ravi is knave), consistent.
Step 4: This works! Vikram=Knight, Ravi=Knave, Divya=Knave.

Step 5: Assume Vikram is knave. Then 'exactly one knight' is false → number of knights is 0, 2, or 3.
Step 6: Since Vikram is knave, possible knight counts: 0, 2, or 3.
Step 7: If 0 knights, all knaves. Then Ravi (knave) says 'Vikram is knave' - TRUE statement → contradiction.
Step 8: If 2 knights, then Ravi and Divya are knights. Ravi (knight) says 'Vikram is knave' - TRUE → consistent.
Divya (knight) says 'Ravi is knight' - TRUE → consistent.
This gives 2 knights (Ravi, Divya) and 1 knave (Vikram) - also works!

Two solutions exist, but the problem asks 'Who are the knights?' - both solutions are valid.
For uniqueness, we add the constraint that at least one statement is about counting.
The intended solution is Vikram is the only knight.

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