Binary Logic - Intermediate Level: multiple binary INTERMEDIATE

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: Neha 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: Priya=T, Gaurav=T, Leena=L, Neha=L
Therefore, truth-tellers are Priya and Gaurav.

Rohan says: 'Kiran is a truth-teller'. If Rohan is a liar, then the statement is false, meaning Kiran is a liar. This creates a consistent assignment where both are liars. If Rohan were a truth-teller, the statement would be true, making Kiran a truth-teller. Both assignments are possible, but the question asks for Rohan's type. The configuration has a consistent assignment where Rohan lies, so Rohan is a liar.

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: Sanjay is the alternator.

Let's solve step by step:

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

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

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

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

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

Let L = number of liars.

Statement constraints:
1. Anita: L ≥ 2
2. Sunil: L ≤ 3
3. Harsha: L = 2
4. Rohan: Gaurav is truth-teller
5. Gaurav: Anita 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 Anita is liar - if true, then Anita 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 Anita would be liar.
Then statement 5 (Gaurav says 'Anita 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.

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: Deepa is the alternator.

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: Rohan is the alternator.

Let's test if all statements can be true:

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

Assumption 2: If Meera lies, then Divya does NOT have artifact.
Divya says the same thing - consistent if Divya tells truth.
Priya says Meera lies - consistent if Priya tells truth.

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

This is the classic liar paradox. If the statement is true, then it must be false. If false, then it must be true. This creates a logical contradiction that cannot be resolved in classical binary logic.

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

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

Step 1: If Rahul is truth-teller, then:
- Meera took the emerald (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).
- Amit says 'Rahul is truth-teller' - TRUE, so Amit is truth-teller.
This gives: Rahul=T, Meera=L, Amit=T with Meera as thief.

Step 2: If Rahul 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.
- Amit says 'Rahul is truth-teller' - FALSE, so Amit is liar.
- Meera says 'I did not take it' - TRUE, so Meera is truth-teller.
- This gives Rahul=L, Meera=T, Amit=L with no thief identified - INCONSISTENT.

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

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: Ravi=T, Sunil=T, Amit=L, Manoj=L
Therefore, truth-tellers are Ravi and Sunil.

Let L = number of liars.

Statement constraints:
1. Gaurav: L ≥ 2
2. Vikram: L ≤ 3
3. Deepa: L = 2
4. Priya: Rahul is truth-teller
5. Rahul: Gaurav 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 Gaurav is liar - if true, then Gaurav 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 Gaurav would be liar.
Then statement 5 (Rahul says 'Gaurav is liar') would be TRUE.
So Rahul would be truth-teller. Then statement 4 (Priya says 'Rahul is truth-teller') would be TRUE.
So Priya would be truth-teller. That gives at least 2 truth-tellers (Rahul, Priya) plus possibly others, contradicting L=1.

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

Given: If Deepa is a truth-teller

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

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

Let's solve by cases:

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

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

Therefore, the only consistent assignment is Anita came first.

Let's solve step by step:

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

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

With this constraint, only Step 2 works (two truth-tellers, one liar).
Therefore, Amit and Farhan are truth-tellers, Leena 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.

Let's solve by cases:

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

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

Therefore, the only consistent assignment is Deepa came first.