Binary Logic - Beginner-Intermediate Level: truth-teller liar BEGINNER-INTERMEDIATE

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, Priya=T, Vikram=L, Harsha=L
Therefore, truth-tellers are Rahul and Priya.

Let's test if all statements can be true:

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

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

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

Identify types by their statements:

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

The key is Anita'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, Anita 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: Divya=T, Neha=T, Rohan=L, Rahul=L
Therefore, truth-tellers are Divya and Neha.

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

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

Let's test if all statements can be true:

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

Assumption 2: If Anita lies, then Gaurav does NOT have emerald.
Gaurav says the same thing - consistent if Gaurav tells truth.
Rohan says Anita lies - consistent if Rohan tells truth.

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

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

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

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

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 by cases:

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

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

Therefore, the only consistent assignment is Kiran came first.

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 Amit is liar.
Statement 3: Sanjay says 'I came second' - unknown.
Statement 4: Manoj says 'Sanjay is lying'.
This leads to multiple possibilities.

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

Therefore, the only consistent assignment is Amit came first.

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
Ravi (mother) = Liar
Deepa (son) = Truth-teller
Leena (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.

Let's solve step by step:

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

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

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

Let L = number of liars.

Statement constraints:
1. Gaurav: L ≥ 2
2. Sanjay: L ≤ 3
3. Neha: L = 2
4. Sunil: Harsha is truth-teller
5. Harsha: 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 Harsha 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 (Harsha says 'Gaurav is liar') would be TRUE.
So Harsha would be truth-teller. Then statement 4 (Sunil says 'Harsha is truth-teller') would be TRUE.
So Sunil would be truth-teller. That gives at least 2 truth-tellers (Harsha, Sunil) plus possibly others, contradicting L=1.

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

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.
- Neha 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.
- Farhan and Manoj 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.

Given: If Rohan is a truth-teller

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

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

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

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

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

Let L = number of liars.

Statement constraints:
1. Sunil: L ≥ 2
2. Ravi: L ≤ 3
3. Vikram: L = 2
4. Kiran: Manoj is truth-teller
5. Manoj: Sunil 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 Sunil is liar - if true, then Sunil is liar.
Statement 4 says Manoj is truth-teller - can be true.
This configuration is possible with L=2.

Can L=1? Statement 1 would be false, so Sunil would be liar.
Then statement 5 (Manoj says 'Sunil is liar') would be TRUE.
So Manoj would be truth-teller. Then statement 4 (Kiran says 'Manoj is truth-teller') would be TRUE.
So Kiran would be truth-teller. That gives at least 2 truth-tellers (Manoj, Kiran) plus possibly others, contradicting L=1.

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

Harsha tells truth on: Sunday, Tuesday, Monday, Friday.
Harsha lies on: Wednesday, Thursday, Saturday.
On Monday (a truth day), Harsha says: 'Water freezes at 0 degrees Celsius'.
Since this is a factual true statement, and Harsha tells truth on this day, the statement is a truth.

Let's solve by cases:

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

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

Therefore, the only consistent assignment is Divya came first.

Farhan tells truth on: Sunday, Saturday, Wednesday.
Farhan lies on: Monday, Tuesday, Thursday, Friday.
On Saturday (a truth day), Farhan says: 'Water freezes at 0 degrees Celsius'.
Since this is a factual true statement, and Farhan tells truth on this day, the statement is a truth.