Binary Logic - Intermediate-Advanced Level: dual logic INTERMEDIATE-ADVANCED

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.
- Anita claims 'Amit is truth-teller'. Without knowing Amit's type, this is ambiguous.
- Amit admits to sometimes lying and sometimes telling truth - this is the hallmark of an alternator.
- Manoj and Rahul make contradictory claims about each other, suggesting one is truth-teller, one liar.

The key is Amit'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, Amit is the alternator.

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.

Let's test if all statements can be true:

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

Assumption 2: If Anita lies, then Leena does NOT have silver ring.
Leena says the same thing - consistent if Leena tells truth.
Harsha says Anita lies - consistent if Harsha tells truth.

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

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

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

Given: If Deepa is a truth-teller

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

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

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

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

Let's solve step by step:

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

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

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

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

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

Farhan tells truth on: Saturday, Wednesday, Tuesday.
Farhan lies on: Monday, Thursday, Friday, Sunday.
On Thursday (a lie day), Farhan says: 'Water freezes at 100 degrees Celsius'.
Since this is a factual false statement, and Farhan lies on this day, the statement is a lie.

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
* Neha's statement is true → matches bit 1
* Pooja's statement is true → matches bit 1
* Farhan's statement is true → matches bit 0

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

Let's solve step by step:

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

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

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

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

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

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.
- Vikram 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.
- Meera and Sunil 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.

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

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

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: Leena=T, Manoj=T, Vikram=L, Sunil=L
Therefore, truth-tellers are Leena and Manoj.

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

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

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, Gaurav=T, Rahul=L, Farhan=L
Therefore, truth-tellers are Neha and Gaurav.

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: Deepa=T, Sunil=T, Priya=L, Leena=L
Therefore, truth-tellers are Deepa and Sunil.