A distant island has four types of people.
The first type are the Truthtellers, who always tell the truth.
The second type are the Liars, who always lie.
The third type are the Alternators, who strictly alternate between telling the truth and a lie, or, vice-versa.
The fourth type are the Doubloons.
A Doubloon is a type similar to an Alternator, except that a Doubloon's truth pattern is precisely two true statements
followed by two false statements or, two false statements followed by two true statements.
Aaron, Beau and Chuck are inhabitants of this island.
It is known that the three individuals belong to three different types.
They make these statements:
AARON
1. I am a Truthteller.
2. Beau is a Doubloon.
3. Chuck is the Alternator.
BEAU
1. I am the Alternator.
2. Chuck is the Doubloon.
3. Aaron is the Liar.
CHUCK
1. I am a Liar.
2. Aaron is the Doubloon.
3. Beau is the Truthteller.
Determine the types corresponding to each of these three individuals.
Solution to the Problem:
Aaron is the Doubloon
Beau is the Liar
Chuck is the Alternator
Begin with Aaron.
There are four possibilites -- he can be a member of any one of the four types.
Case 1:
If Aaron is a Truthteller:
Then Beau must be a Doubloon.
But all three of Beau's statements would be false so Beau is not the Doubloon and Aaron cannot be the Truthteller. You do not need to check Chuck.
Here is the truth table for this scenario:
| Name | Type | S1 | S2 | S3 |
|---|---|---|---|---|
| Aaron | Truthteller | T | T | T |
| Beau | Doubloon | F | F | F |
| Chuck | Alternator |
Case 2:
If Aaron is the Liar:
Then Beau must be an Alternator or a Truthteller.
But Beau cannot be a Truthteller since he says he is the Alternator.
If he is the Alternator, then his second statement must be false (since his first and third statements would be true).
However, that means Chuck is not the Doubloon, so he must be the Truthteller.
But that can't be since he says that he is a Liar.
Here is the truth table for this last scenario:
| Name | Type | S1 | S2 | S3 |
|---|---|---|---|---|
| Aaron | Liar | F | F | F |
| Beau | Alternator | T | F | T |
| Chuck | Truthteller | F |
Case 3:
If Aaron is the Alternator:
Then his first statement is false, so his pattern must be F, T, F.
Then Beau must be the Doubloon and Chuck is not the Alternator.
But Beau's first and third statements would be false, so he cannot be the Doubloon.
You do not need to check Chuck's statements.
Here is the truth table for this last scenario:
| Name | Type | S1 | S2 | S3 |
|---|---|---|---|---|
| Aaron | Alternator | F | T | F |
| Beau | Doubloon | F | ? | F |
| Chuck |
Case 4:
If Aaron is the Doubloon (actually Aaron MUST be the Doubloon since we have proved that he can't be the other three):
So his first statement must be false (since he says he is a Truthteller).
Then his second statement must be false and his third statement must be true.
That means that Beau is not the Doubloon and Chuck is the Alternator.
Since Beau cannot be the Truthteller (he says he is the Alternator), that leaves only one choice for him -- he is the Liar.
Here is the truth table for this last scenario:
| Name | Type | S1 | S2 | S3 |
|---|---|---|---|---|
| Aaron | Doubloon | F | F | T |
| Beau | Liar | F | F | F |
| Chuck | Alternator | F | T | F |
Correctly solved by:
| 1. Colin (Yowie) Bowey | Beechworth, Victoria, Australia |
| 2. Ivy Joseph | Pune, Maharashtra, India |
| 3. Benny Varghese | Cochin, Kerala, India |
| 4. Eloise A. Michael | Greenfield, Massachussetts |
| 5. Davit Banana | Istanbul, Turkey |
| 6. Dr. Hari Kishan |
D.N. College, Meerut, Uttar Pradesh, India |
| 7. Jubilee Ortiz |
Central High School, Grand Junction, Colorado |
| 8. Elizabeth Johnson | Colorado |
| 9. Ritwik Chaudhuri | Santiniketan, West Bengal, India |