MARK WAUGH and Steve Waugh, twin brothers, are sitting outside the office of the Australian Cricket Board. There have been allegations that one of them has fixed some matches. The board has summoned them to record their confessions. However, it will not be an easy job. It is well known in the Australian team that one of the twins always lies and the other always tells the truth. No one knows who tells the truth and who is a liar.

Now, the truth-teller is also totally accurate in all his beliefs. He believes all true propositions to be true and all false propositions to be false. The lying brother is totally inaccurate in his beliefs. He believes all true propositions to be false and all false propositions to be true. Also, each brother will give the same answer to the same question.

If you ask them whether one plus one is equal to two, the truth-teller knows that it is so and will truthfully answer yes. The liar will believe that one plus one is not equal to two (since he is inaccurate) and will then lie and say that it does. His answer will also be yes. In such a situation, one wonders how will the board find out who is the liar.

The situation reminds a board member of a similar situation. He says, "Doctors in a mental hospital were thinking of releasing patient. They decided to test him under a lie detector. One of the questions they asked him was, ‘Are you Napoleon?’ He replied, ‘No.’ The machine showed that he was lying!"

Meanwhile, two board members are having an argument over the following question: Suppose we meet one of the two brothers alone, is it possible by asking him any number of yes-no questions to find out who is he — liar or the truth- teller? One of these two says, "It will not be possible, because whatever answers we receive to our questions, the other brother will have the same answers."

However, the second member says that it is possible to find out who is who. The person who is recording the minutes of the meeting is somewhat of a mathematician. He knows that the second member is right. However, he is unable to decide how many questions are necessary and what is wrong with the first member’s argument?

The second member says, "One question is enough to know which brother you are addressing. Just ask him if he is the truth-teller. If he is, he will know that he is (since he is accurate) and will truthfully answer yes. If he is the liar, he will believe that he is the truth- teller (since he is inaccurate), but then he will lie and say no. In this way, the truth- teller will answer yes and the liar no to this question."

"Hey! don’t the two brothers give the same answer to the same question?" says the first member. "They do, but if I ask one person if he is the truth- teller and then ask the second the same thing, I am actually asking two different questions since the identical word you has a different reference in each case," says the second member.

Another member says he know a better question to ask the two, which is: "If I were to ask your brother whether you always tell the truth, what would he say?" A reply of no means you are talking to the truth-teller and a reply of yes means you are talking to the liar. Another possible question is: "If I were to ask you whether you always tell the truth, what would you say?" A reply of yes means you are talking to the truth-teller and a no means you are talking to the liar.

The culprit is found out and is sentenced to a century of imprisonment. However, since the board members love riddles, they throw this guy into a room with two doors — one leading to jail and the other to freedom. There are two guards at each door. One of them is a perfect liar, the other always tells the truth. The condemned brother is allowed to ask one guard one yes-no question before deciding which door to take. Which question should he ask to find the door to freedom?

— Aditya Rishi