hypothetical_count_change Advanced Worksheet: Focus on exam-oriented approach hypothetical_count_change ADVANCED

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 Neha same type' - true (both truth) - consistent.
p3 (truth) says 'at least one truth-teller' - true - consistent.
Solution: Sanjay=T, Leena=T, Neha=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 Leena and Neha are liars. Then Leena liar says 'Sanjay and Neha same type' - Sanjay liar, Neha liar -> same type -> true statement, but liar can't make true - contradiction.
- If 3 liars, all are liars. Then Leena liar says 'Sanjay and Neha 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.

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

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

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

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

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

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

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 Leena same type' - true (both truth) - consistent.
p3 (truth) says 'at least one truth-teller' - true - consistent.
Solution: Priya=T, Farhan=T, Leena=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 Farhan and Leena are liars. Then Farhan liar says 'Priya and Leena same type' - Priya liar, Leena liar -> same type -> true statement, but liar can't make true - contradiction.
- If 3 liars, all are liars. Then Farhan liar says 'Priya and Leena 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.

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

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

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

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

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 Gaurav same type' - true (both truth) - consistent.
p3 (truth) says 'at least one truth-teller' - true - consistent.
Solution: Farhan=T, Ravi=T, Gaurav=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 Ravi and Gaurav are liars. Then Ravi liar says 'Farhan and Gaurav same type' - Farhan liar, Gaurav liar -> same type -> true statement, but liar can't make true - contradiction.
- If 3 liars, all are liars. Then Ravi liar says 'Farhan and Gaurav 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.

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 Rahul same type' - true (both truth) - consistent.
p3 (truth) says 'at least one truth-teller' - true - consistent.
Solution: Manoj=T, Anita=T, Rahul=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 Anita and Rahul are liars. Then Anita liar says 'Manoj and Rahul same type' - Manoj liar, Rahul liar -> same type -> true statement, but liar can't make true - contradiction.
- If 3 liars, all are liars. Then Anita liar says 'Manoj and Rahul 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.

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

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

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 Farhan same type' - true (both truth) - consistent.
p3 (truth) says 'at least one truth-teller' - true - consistent.
Solution: Sanjay=T, Rohan=T, Farhan=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 Rohan and Farhan are liars. Then Rohan liar says 'Sanjay and Farhan same type' - Sanjay liar, Farhan liar -> same type -> true statement, but liar can't make true - contradiction.
- If 3 liars, all are liars. Then Rohan liar says 'Sanjay and Farhan 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.

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

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

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

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

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

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

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

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

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

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

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

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

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, Anita=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 Anita and Sunil are liars. Then Anita 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 Anita 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.

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

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

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

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