Binary Logic - Beginner-Intermediate Level: alternate truth BEGINNER-INTERMEDIATE

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:
Vikram (father) = Truth-teller
Sunil (mother) = Liar
Rohan (son) = Truth-teller
Neha (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.

Rohan tells truth on: Thursday, Monday, Sunday, Wednesday.
Rohan lies on: Tuesday, Friday, Saturday.
On Saturday (a lie day), Rohan says: 'The Earth is flat'.
Since this is a factual false statement, and Rohan lies on this day, the statement is a lie.

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

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

Given: If Meera is a truth-teller

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

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

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

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

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

Let's solve step by step:

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

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

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

Manoj tells truth on: Wednesday, Friday, Sunday, Monday.
Manoj lies on: Tuesday, Thursday, Saturday.
On Sunday (a truth day), Manoj says: 'The Earth orbits the Sun'.
Since this is a factual true statement, and Manoj tells truth on this day, the statement is a truth.

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.

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 Farhan same type' - true (both truth) - consistent.
p3 (truth) says 'at least one truth-teller' - true - consistent.
Solution: Sanjay=T, Deepa=T, Farhan=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 Deepa and Farhan are liars. Then Deepa liar says 'Sanjay and Farhan same type' - Sanjay liar, Farhan liar -> same type -> true statement, but liar can't make true - contradiction.
- If 3 liars, all are liars. Then Deepa liar says 'Sanjay and Farhan 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.

Given: If Divya is a truth-teller

Step 1: Divya tells truth → 'Vikram is liar' is true → Vikram is liar.
Step 2: Vikram (liar) says 'Pooja is truth-teller' → this statement is false → Pooja is liar.
Step 3: Pooja (liar) says 'Manoj and I are different types' → this statement is false → Manoj is SAME type as Pooja → Manoj is liar.
Step 4: Manoj (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, Vikram must be a liar.

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

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: Pooja is the alternator.

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

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

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

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.

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's test if all statements can be true:

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

Assumption 2: If Manoj lies, then Rahul does NOT have artifact.
Rahul says the same thing - consistent if Rahul tells truth.
Neha says Manoj lies - consistent if Neha tells truth.

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

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: Priya=T, Anita=T, Pooja=L, Kiran=L
Therefore, truth-tellers are Priya and Anita.

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

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

Therefore, the only consistent solution is Ravi took the diamond.

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, Meera=T, Harsha=L, Sanjay=L
Therefore, truth-tellers are Divya and Meera.

Given: If Gaurav is a truth-teller

Step 1: Gaurav tells truth → 'Deepa is liar' is true → Deepa is liar.
Step 2: Deepa (liar) says 'Sunil is truth-teller' → this statement is false → Sunil is liar.
Step 3: Sunil (liar) says 'Anita and I are different types' → this statement is false → Anita is SAME type as Sunil → 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, Deepa must be a liar.

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

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

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

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