Binary Logic - Intermediate Level: binary reasoning INTERMEDIATE

Let's test if all statements can be true:

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

Assumption 2: If Sanjay lies, then Ravi does NOT have silver ring.
Ravi says the same thing - consistent if Ravi tells truth.
Amit says Sanjay lies - consistent if Amit tells truth.

Therefore, all statements CAN be consistent when Sanjay lies, Ravi and Amit 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: Sunil=T, Rahul=T, Anita=L, Kiran=L
Therefore, truth-tellers are Sunil and Rahul.

Key insight: The statement 'Sometimes I tell the truth and sometimes I lie' 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: Neha is the alternator.

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

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.

Identify types by their statements:

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

The key is Harsha'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, Harsha is the alternator.

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

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.

Let's solve step by step:

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

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

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

Identify types by their statements:

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

The key is Kiran'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, Kiran 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 Sunil same type' - true (both truth) - consistent.
p3 (truth) says 'at least one truth-teller' - true - consistent.
Solution: Sanjay=T, Amit=T, Sunil=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 Amit and Sunil are liars. Then Amit liar says 'Sanjay and Sunil same type' - Sanjay liar, Sunil liar -> same type -> true statement, but liar can't make true - contradiction.
- If 3 liars, all are liars. Then Amit liar says 'Sanjay and Sunil 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 by cases:

Case 1: Pooja came first.
Then statement 1 is true, statement 2 is false.
If statement 2 is false, then Manoj is liar.
Statement 3: Vikram says 'I came second' - unknown.
Statement 4: Divya says 'Vikram is lying'.
This leads to multiple possibilities.

Case 2: Manoj came first.
Then statement 1 is false → Pooja is liar.
Statement 2 is true → Manoj is truth-teller.
If Vikram came second, statement 3 is true → Vikram is truth-teller.
Then statement 4 says 'Vikram is lying' - false → Divya is liar.
This gives 2 truth-tellers (Manoj, Vikram) and 2 liars, consistent.

Therefore, the only consistent assignment is Manoj came first.

Let's solve step by step:

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

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

Identify types by their statements:

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

The key is Meera'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, Meera is the alternator.

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:
Rahul (father) = Truth-teller
Manoj (mother) = Liar
Farhan (son) = Truth-teller
Deepa (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.

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

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

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:
Farhan (father) = Truth-teller
Sunil (mother) = Liar
Leena (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.