Binary Logic - Advanced Level: alternate truth ADVANCED

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

Let's test if all statements can be true:

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

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

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

The code represents the truth pattern: 1=truth-teller, 0=liar.

Testing each possible code:
- Code 1001 makes all statements consistent:
* Harsha's statement is true → matches bit 1
* Farhan's statement is false → matches bit 0
* Deepa's statement is true → matches bit 0
* Meera's statement is false → matches bit 1

No other code satisfies all constraints.
Therefore, the correct code is 1001.

Let L = number of liars.

Statement constraints:
1. Ravi: L ≥ 2
2. Meera: L ≤ 3
3. Pooja: L = 2
4. Sanjay: Farhan is truth-teller
5. Farhan: Ravi 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 Ravi is liar - if true, then Ravi is liar.
Statement 4 says Farhan is truth-teller - can be true.
This configuration is possible with L=2.

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

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

Let's solve step by step:

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

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

With this constraint, only Step 2 works (two truth-tellers, one liar).
Therefore, Farhan and Rahul are truth-tellers, Gaurav is 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 Manoj is liar.
Statement 3: Sanjay says 'I came second' - unknown.
Statement 4: Pooja says 'Sanjay is lying'.
This leads to multiple possibilities.

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

Therefore, the only consistent assignment is Manoj came first.

Let's test if all statements can be true:

Assumption 1: If Manoj tells truth, then Gaurav has diamond.
But Gaurav claims not to have it - contradiction if Gaurav tells truth.
If Gaurav lies, then Gaurav 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 Gaurav does NOT have diamond.
Gaurav says the same thing - consistent if Gaurav tells truth.
Neha says Manoj lies - consistent if Neha tells truth.

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

Let's solve step by step:

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

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

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: Farhan=T, Ravi=T, Deepa=L, Pooja=L
Therefore, truth-tellers are Farhan and Ravi.

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.

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

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

Let's test if all statements can be true:

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

Assumption 2: If Priya lies, then Gaurav does NOT have ruby.
Gaurav says the same thing - consistent if Gaurav tells truth.
Manoj says Priya lies - consistent if Manoj tells truth.

Therefore, all statements CAN be consistent when Priya lies, Gaurav and Manoj 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: Gaurav=T, Priya=T, Rohan=L, Leena=L
Therefore, truth-tellers are Gaurav and Priya.

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.
- Ravi claims 'Pooja is truth-teller'. Without knowing Pooja's type, this is ambiguous.
- Pooja admits to sometimes lying and sometimes telling truth - this is the hallmark of an alternator.
- Divya and Sanjay make contradictory claims about each other, suggesting one is truth-teller, one liar.

The key is Pooja'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, Pooja is the alternator.

Given: If Sunil is a truth-teller

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

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

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.

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.

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
Anita (son) = Truth-teller
Rohan (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 Gaurav is a liar.
→ Gaurav's statement 'Manoj is a truth-teller' is FALSE → Manoj is a liar.
→ Kiran says 'Gaurav is a liar' - this is TRUE (since Gaurav is liar).
→ If Kiran tells truth, then Kiran is truth-teller.
→ Manoj (liar) says 'Kiran is a liar' - FALSE (since Kiran is truth) → consistent.
This gives: Manoj=L, Kiran=T, Gaurav=L (two liars, one truth-teller).

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

With this constraint, only Step 2 works (two truth-tellers, one liar).
Therefore, Manoj and Gaurav are truth-tellers, Kiran is a liar.

Let L = number of liars.

Statement constraints:
1. Farhan: L ≥ 2
2. Priya: L ≤ 3
3. Amit: L = 2
4. Sunil: Divya is truth-teller
5. Divya: Farhan 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 Farhan is liar - if true, then Farhan is liar.
Statement 4 says Divya is truth-teller - can be true.
This configuration is possible with L=2.

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

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