Binary Logic - Beginner-Intermediate Level: two-option logic BEGINNER-INTERMEDIATE

Given: If Vikram is a truth-teller

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

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

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

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

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

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

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.

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
Farhan (mother) = Liar
Sanjay (son) = Truth-teller
Rohan (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 1001 makes all statements consistent:
* Divya's statement is true → matches bit 1
* Farhan's statement is false → matches bit 0
* Ravi's statement is true → matches bit 0
* Neha's statement is false → matches bit 1

No other code satisfies all constraints.
Therefore, the correct code is 1001.

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

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

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

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

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

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

Therefore, the only consistent solution is Meera took the ruby.

Let's solve step by step:

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

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

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

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
* Rahul's statement is true → matches bit 1
* Anita's statement is true → matches bit 0

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

Let's solve step by step:

Step 1: Assume Leena is a liar.
→ Leena's statement 'Pooja is a truth-teller' is FALSE → Pooja is a liar.
→ Ravi says 'Leena is a liar' - this is TRUE (since Leena 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, Leena=L (two liars, one truth-teller).

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

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

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

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

Let's test if all statements can be true:

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

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

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

Kiran says: 'Amit is a liar'. If Kiran is a truth-teller, then the statement is true, so Amit is a liar. If Kiran were a liar, the statement would be false, meaning Amit is a truth-teller. Both are possible, but the question asks for Kiran'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 Kiran tells truth and Amit lies. Therefore, Kiran is a truth-teller.

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

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

Let L = number of liars.

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

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

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

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: Harsha=T, Kiran=T, Gaurav=L, Divya=L
Therefore, truth-tellers are Harsha and Kiran.

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:
Sunil (father) = Truth-teller
Neha (mother) = Liar
Anita (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.

If this statement is true, then all statements (including itself) are false - contradiction. If false, then not all statements are false, meaning some are true - but which one? This is self-defeating.

Let L = number of liars.

Statement constraints:
1. Meera: L ≥ 2
2. Rahul: L ≤ 3
3. Ravi: L = 2
4. Deepa: Amit is truth-teller
5. Amit: Meera 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 Meera is liar - if true, then Meera is liar.
Statement 4 says Amit is truth-teller - can be true.
This configuration is possible with L=2.

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

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