Home > Logic > Liars and Knights
| Know thy Knaves (Posted on 2007-10-09) |
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Alex, Bert, and Carl are taking a break from being main subjects in these logic puzzles, so Dave and Eddy decided to comment on the rarely seen Fred, Gary, and Hank. Dave and Eddy are both knaves and each one makes four of the eight statements below. The statements are in order, but whether Dave or Eddy made any given statement is not known. Without knowing which statements are Dave's and which are Eddy's, can you determine the types of Fred, Gary, and Hank?
- Fred is a liar.
- Gary is a knave.
- Hank is a knight.
- Fred and Gary are the same type.
- Gary and Hank are different types.
- Fred is a knight.
- Hank is a knave.
- Gary is a liar.
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Submitted by Brian Smith
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Rating: 4.0000 (1 votes)
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Solution:
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Fred and Hank are knights, Gary is a liar.
First, there are four true and four false statements. Also, given that the eight statements are made by two knaves, then there is at least one true and one false statement in any block of three consecutive statements, and there are at least two true and two false statement in any block of five consecutive statements.
At most one of statements 1 and 6 are true, at most one of statements 2 and 8 are true, and at most one of statements 3 and 7 are ture. That means that at least one of statements 4 and 5 are true.
Assume #4 is true and #5 is false. Then Fred, Gary, and Hank are all the same type. That would cause only two of #1,2,3,6,7,8 to be true, making only three true statements. Therefore the assumption that #4 is true and #5 is false is a false assumption.
Assume #4 is true and #5 is true. If Fred and Gary are both liars then Hank is a knave or knight, but that causes five statements to be true (#1,4,5,8 and one of #3 or 7), so Fred and Gary are not both liars. If Fred and Gary are both knaves then Hank is a knight or a liar, but then #6, 7, 8 are all false, so Fred and Gary are not both knaves. If Fred and Gary are both knights then #4, 5, 6 are then all true, so Fred and Gary are not both knights. Therefore the assumption that #4 is true and #5 is true is a false assumption.
#4 must be false and #5 must be true for there to be a solution. Then exactly one of statements #1 and #6 is true, exactly one of statements #2 and #8 is true, and exactly one of statements #3 and #7 is true. Then Fred is either a liar or a knight, Gary is either a knave or a liar, and Hank is a knight or a knave. From the conclusion about #4 and #5, Gary is a different type from both Fred and Hank. That leaves only four possibilities for the types of Fred, Gary, and Hank:
1: Fred is a liar, Gary is a knave, Hank is a knight
2: Fred is a knight, Gary is a knave, Hank is a knight
3: Fred is a knight, Gary is a liar, Hank is a knight
4: Fred is a knight, Gary is a liar, Hank is a knave
The truth of the eight statements for possibility #1 is TTTFTFFF, which is not a solution since there is a block of three true and a block of three false statments. The truth of the eight statements for possibility #2 is FTTFTTFF, which is not a solution since there is a block of five statements containing four true statements. The truth of the eight statements for possibility #4 is FFFFTTTT, which is not a solution since there is a block of three true and a block of three false statments.
Finally, the truth of the eight statements for possibility #3 is FFTFTTFT, which satisfies all the requisites stated in the first paragraph. Dave could have made statements 2-5 and Eddy could have made statements 1, 6, 7, 8.
Therefore the unique answer for the types of Fred, Gary, and Hank is: Fred and Hank are knights and Gary is a liar. |
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