Binary Logic - Beginner Level: binary reasoning BEGINNER

Let's solve by cases:

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

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

Therefore, the only consistent assignment is Priya came first.

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.

Given: If Meera is a truth-teller

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

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

Rohan tells truth on: Friday, Thursday, Tuesday.
Rohan lies on: Monday, Wednesday, Saturday, Sunday.
On Monday (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.

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: Gaurav is the alternator.

Identify types by their statements:

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

The key is Farhan'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, Farhan is the alternator.

Given: If Harsha is a truth-teller

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

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

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

Given: If Sunil is a truth-teller

Step 1: Sunil tells truth → 'Deepa is liar' is true → Deepa is liar.
Step 2: Deepa (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 'Sunil is truth-teller' → this statement is false → Sunil is liar → CONTRADICTION with our assumption!

This shows that Sunil CANNOT be a truth-teller under these statements.
However, the conditional question asks: IF we assume Sunil 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.

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

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

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

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

Therefore, the only consistent assignment is Amit came first.

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

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

Therefore, the only consistent solution is Divya 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: Priya=T, Divya=T, Sanjay=L, Neha=L
Therefore, truth-tellers are Priya and Divya.

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, Sanjay=T, Amit=L, Vikram=L
Therefore, truth-tellers are Deepa and Sanjay.

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

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

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

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.

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: Sunil 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
* Priya's statement is false → matches bit 0
* Neha's statement is true → matches bit 1
* Gaurav's statement is true → matches bit 0

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

Given: If Farhan is a truth-teller

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

This shows that Farhan CANNOT be a truth-teller under these statements.
However, the conditional question asks: IF we assume Farhan 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.

Kiran tells truth on: Tuesday, Thursday, Wednesday, Sunday.
Kiran lies on: Monday, Friday, Saturday.
On Tuesday (a truth day), Kiran says: 'Humans need oxygen to survive'.
Since this is a factual true statement, and Kiran tells truth on this day, the statement is a truth.

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
Farhan (mother) = Liar
Vikram (son) = Truth-teller
Pooja (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.