Binary Logic - Intermediate-Advanced Level: binary statements INTERMEDIATE-ADVANCED

Let's test if all statements can be true:

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

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

Therefore, all statements CAN be consistent when Amit lies, Kiran and Manoj 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:
Farhan (father) = Truth-teller
Rahul (mother) = Liar
Sanjay (son) = Truth-teller
Anita (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.

Given: If Divya is a truth-teller

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

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

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

Sanjay says: 'Ravi is a liar'. If Sanjay is a truth-teller, then the statement is true, so Ravi is a liar. If Sanjay were a liar, the statement would be false, meaning Ravi 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 Ravi lies. Therefore, Sanjay is a truth-teller.

Identify types by their statements:

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

The key is Farhan'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, Farhan is the alternator.

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: Rahul=T, Neha=T, Deepa=L, Ravi=L
Therefore, truth-tellers are Rahul and Neha.

Let's test if all statements can be true:

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

Assumption 2: If Priya lies, then Leena does NOT have crystal.
Leena says the same thing - consistent if Leena tells truth.
Manoj says Priya lies - consistent if Manoj tells truth.

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

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: Rohan is the alternator.

Let's solve step by step:

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

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

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

Harsha tells truth on: Tuesday, Friday, Sunday, Saturday.
Harsha lies on: Monday, Wednesday, Thursday.
On Saturday (a truth day), Harsha says: 'Humans need oxygen to survive'.
Since this is a factual true statement, and Harsha tells truth on this day, the statement is a truth.

Identify types by their statements:

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

The key is Pooja'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, Pooja is the alternator.

Let's solve step by step:

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

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

Step 1: Assume Vikram is knight. Then 'exactly one knight' is true → Sanjay and Manoj are knaves.
Step 2: Sanjay (knave) says 'Vikram is knave' - FALSE statement (since Vikram is knight), consistent.
Step 3: Manoj (knave) says 'Sanjay is knight' - FALSE statement (since Sanjay is knave), consistent.
Step 4: This works! Vikram=Knight, Sanjay=Knave, Manoj=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 Sanjay (knave) says 'Vikram is knave' - TRUE statement → contradiction.
Step 8: If 2 knights, then Sanjay and Manoj are knights. Sanjay (knight) says 'Vikram is knave' - TRUE → consistent.
Manoj (knight) says 'Sanjay is knight' - TRUE → consistent.
This gives 2 knights (Sanjay, Manoj) 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.

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

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

Identify types by their statements:

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

The key is Gaurav'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, Gaurav is the alternator.

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 L = number of liars.

Statement constraints:
1. Farhan: L ≥ 2
2. Sanjay: L ≤ 3
3. Neha: L = 2
4. Anita: Rahul is truth-teller
5. Rahul: 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 Rahul 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 (Rahul says 'Farhan is liar') would be TRUE.
So Rahul would be truth-teller. Then statement 4 (Anita says 'Rahul is truth-teller') would be TRUE.
So Anita would be truth-teller. That gives at least 2 truth-tellers (Rahul, Anita) 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 Harsha is a liar.
→ Harsha's statement 'Amit is a truth-teller' is FALSE → Amit is a liar.
→ Anita says 'Harsha is a liar' - this is TRUE (since Harsha is liar).
→ If Anita tells truth, then Anita is truth-teller.
→ Amit (liar) says 'Anita is a liar' - FALSE (since Anita is truth) → consistent.
This gives: Amit=L, Anita=T, Harsha=L (two liars, one truth-teller).

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

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

Given: If Kiran is a truth-teller

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

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

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

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

Testing each possible code:
- Code 0110 makes all statements consistent:
* Vikram's statement is false → matches bit 0
* Meera's statement is true → matches bit 1
* Priya's statement is true → matches bit 1
* Ravi's statement is true → matches bit 0

No other code satisfies all constraints.
Therefore, the correct code is 0110.