A logician vacationing in the South Seas finds himself on an island inhabited by two proverbial tribes of liars and truth-tellers. Members of one tribe always tell the truth, members of the other always lie. He comes to a fork in a road and has to ask a native bystander which branch he should take to reach a village. He has no way of telling whether the native is a truth-teller or a liar. The logician thinks a moment, then asks one question only. From the reply he knows which road to take. What question does he ask?
(source: My Best Mathematical and Logic Puzzles By Martin Gardner)
Solution:
If we require that the question be answerable by "yes" or "no" there are several solutions, all exploiting the same basic gimmick. For ex, the logician points to one of the roads and says to the native,
1. "If I were to ask you if this road leads to the village, would you say 'yes'?" The native is forced to give the right answer, even if he is a liar! If the road does lead to the village, the liar would say "no" to the direct question, but as the question is put, he lies and says he would respond "yes." Thus the logician can be certain that the road does lead to the village, whether the respondent is a truth-teller or a liar. On the other hand, if the road actually does not go to the village, the liar is forced in the same way to reply "no" to the inquirer's question.
2. "Of the two statements, 'You are a liar' and 'This road leads to the village,' is one and only one of them true?" Here again, a "yes" answer indicates it is the road, a "no" answer that it isn't regardless of whether the native lies or tells the truth.
Twist: Suppose logician knows that 'pish' and 'tush' are the native words for 'yes' and 'no' but has forgotten which is which, though otherwise he can speak the native language, He can still determine which road leads to the village:
He points to one of the roads and asks, 'If I asked you whether the road I am pointing to is the road to the village would you say pish?'
The Fork in the Road
A logician vacationing in the South Seas finds himself on an island inhabited by two proverbial tribes of liars and truth-tellers. Members of one tribe always tell the truth, members of the other always lie. He comes to a fork in a road and has to ask a native bystander which branch he should take to reach a village. He has no way of telling whether the native is a truth-teller or a liar. The logician thinks a moment, then asks one question only. From the reply he knows which road to take. What question does he ask?
(source: My Best Mathematical and Logic Puzzles By Martin Gardner)
Solution:
If we require that the question be answerable by "yes" or "no" there are several solutions, all exploiting the same basic gimmick. For ex, the logician points to one of the roads and says to the native,
1. "If I were to ask you if this road leads to the village, would you say 'yes'?" The native is forced to give the right answer, even if he is a liar! If the road does lead to the village, the liar would say "no" to the direct question, but as the question is put, he lies and says he would respond "yes." Thus the logician can be certain that the road does lead to the village, whether the respondent is a truth-teller or a liar. On the other hand, if the road actually does not go to the village, the liar is forced in the same way to reply "no" to the inquirer's question.
2. "Of the two statements, 'You are a liar' and 'This road leads to the village,' is one and only one of them true?" Here again, a "yes" answer indicates it is the road, a "no" answer that it isn't regardless of whether the native lies or tells the truth.
Twist: Suppose logician knows that 'pish' and 'tush' are the native words for 'yes' and 'no' but has forgotten which is which, though otherwise he can speak the native language, He can still determine which road leads to the village:
He points to one of the roads and asks, 'If I asked you whether the road I am pointing to is the road to the village would you say pish?'
(source: My Best Mathematical and Logic Puzzles By Martin Gardner)
Solution:
If we require that the question be answerable by "yes" or "no" there are several solutions, all exploiting the same basic gimmick. For ex, the logician points to one of the roads and says to the native,
1. "If I were to ask you if this road leads to the village, would you say 'yes'?" The native is forced to give the right answer, even if he is a liar! If the road does lead to the village, the liar would say "no" to the direct question, but as the question is put, he lies and says he would respond "yes." Thus the logician can be certain that the road does lead to the village, whether the respondent is a truth-teller or a liar. On the other hand, if the road actually does not go to the village, the liar is forced in the same way to reply "no" to the inquirer's question.
2. "Of the two statements, 'You are a liar' and 'This road leads to the village,' is one and only one of them true?" Here again, a "yes" answer indicates it is the road, a "no" answer that it isn't regardless of whether the native lies or tells the truth.
Twist: Suppose logician knows that 'pish' and 'tush' are the native words for 'yes' and 'no' but has forgotten which is which, though otherwise he can speak the native language, He can still determine which road leads to the village:
He points to one of the roads and asks, 'If I asked you whether the road I am pointing to is the road to the village would you say pish?'
The Returning Explorer
An old riddle runs as follows. An explorer walks one mile due south, turns and walks one mile due east, turns again and walks one mile due north. He find himself back where he started. He shoots a bear. What color is the bear? The time-honoured answer is: "White," because the explorer must have started at the North Pole. But not long ago someone made the discovery that the North pole is not the only starting point that satisfies the given conditions! Can you think of any other spot on the globe from which one could walk a mil south, a mile east, a mile north and find himself back at his original location?(source: My Best Mathematical and Logic Puzzles By Martin Gardner)
Solution:
Us there any other point on the globe, besides the North Pole, from which you could walk a mile south, a mile east, and a mile north and find yourself back at the starting point? Yes indeed; not just one point but an infinite number of them! You could start from any point on a circle drawn around the south pole at a distance slightly more than 1 + 1/2Pi miles ( 1.16 miles ) from the Pole, the distance is "slightly more" to take into account the curvature of the earth. After walking a mile south, your next walk of one mile east will take you on a complete circle around the Pole, and the walk one mile north from there will then return you to the starting point. Thus your starting point could be any one of the infinite points on the circle with a radius of about 1.16 miles from south pole. You could also start at points close to the Pole, so that the walk east woule carry you just twice around the Pole, or three time, and so on.
Solution:
Us there any other point on the globe, besides the North Pole, from which you could walk a mile south, a mile east, and a mile north and find yourself back at the starting point? Yes indeed; not just one point but an infinite number of them! You could start from any point on a circle drawn around the south pole at a distance slightly more than 1 + 1/2Pi miles ( 1.16 miles ) from the Pole, the distance is "slightly more" to take into account the curvature of the earth. After walking a mile south, your next walk of one mile east will take you on a complete circle around the Pole, and the walk one mile north from there will then return you to the starting point. Thus your starting point could be any one of the infinite points on the circle with a radius of about 1.16 miles from south pole. You could also start at points close to the Pole, so that the walk east woule carry you just twice around the Pole, or three time, and so on.
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