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Know thy Knaves (Posted on 2007-10-09) Difficulty: 4 of 5
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?
  1. Fred is a liar.
  2. Gary is a knave.
  3. Hank is a knight.
  4. Fred and Gary are the same type.
  5. Gary and Hank are different types.
  6. Fred is a knight.
  7. Hank is a knave.
  8. Gary is a liar.

See The Solution Submitted by Brian Smith    
Rating: 4.0000 (1 votes)

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Solution Full solution (explanation and answer) | Comment 2 of 7 |

Ok, so there need to be four Ts and four Fs.

We cannot have three Ts (or three Fs) in a row at any time. This is because a knave cannot say two Ts (or two Fs) in a row, and even if we had Dave and Eddy alter during those three statements that would mean the first statement and the third were made by one person, "in a row" for them.

So this means anytime there are two Ts (or two Fs) in a row, they have to be preceded and followed by an F (T), or nothing if they are the beginning or the end of the conversation. That probably didn't make sense. What I mean is if we have TT somewhere, it has to be part of a group that goes FTTF, or if the TT comes at the beginning or end of the conversation, then we can have TTF or FTT respectively. And for FF, it must be TFFT (or FFT and TFF). I will call this the No Triple Rule (or NTR).

Also we cannot have too many Ts (or Fs) close together. While TTFTT (or FFTFF) satisfies the NTR, we still have a problem. We know the first two statements have to be made by different people. Let's say Dave speaks first so we have DE. The next statement could be made by either, so we either have DED or DEE. If we have DED, the next statement must be made by Dave to keep Eddy from speaking two Ts (or two Fs) in a row. But then no matter who speaks the last statement, he will be speaking two Ts (or two Fs) in a row. If we have DEE, the next statement must be made by Eddy to keep Dave from speaking two Ts (or two Fs) in a row. But then no matter who speaks the last statement, he will be speaking two Ts (or two Fs) in a row.

Therefore we cannot have TTFTT (or FFTFF). I will call this the No Single on Double Sandwich Rule (or NSDSR). Think of a ham on rye sandwich.

So let's look at statements 2 and 3 and see how everything else falls into place.

Statements 2 and 3 can either go TT, FF, TF or FT.

--------------------------------------------

If 2 and 3 are TT, then
1: Based on the NTR, this must be false. So Fred is not a liar.
2: Based on our assumption, this is true. So Gary is indeed a knave.
3: Based on our assumption, this is true. So Hank is indeed a knight.
4: Based on the NTR, this must be false. So Fred is not a knave like Gary. So Fred is a knight.
5: This is true.
6: This is true.
7: This is false.
8: This is false.

So the T/F order is FTTFTTFF. Uh oh, we have violated the NSDSR. So it cannot be the case that 2 and 3 are both true.

--------------------------------------------

If 2 and 3 are FF, then
1: Based on the NTR, this must be true. So Fred is indeed a liar.
2: Based on our assumption, this is false. So Gary is not a knave.
3: Based on our assumption, this is false. So Hank is not a knight.
4: Based on the NTR, this must be true. So Gary is also a liar like Fred.
6: This is false.
5: Based on the NSDSR, this must be true. So Hank is not a liar like Gary. So Hank is a knave.
7: This is true.
8: This is true.

So the T/F order is TFFTTFTT. Uh oh, we have two problems. We have five Ts and three Fs, and we have violated the NSDSR. So it cannot be the case that 2 and 3 are both false.

--------------------------------------------

If 2 and 3 are TF, then
2: Based on our assumption, this is true. So Gary is indeed a knave.
3: Based on our assumption, this is false. So Hank is not a knight.
8: This is false.

Hmmm, this doesn't let us draw a lot of conclusions, even with the NTR and NSDSR. Let's make a guess that statement 4 is false. Then we can add that…

4: Based on our guess, this is false. So Fred is not a knave like Gary.
5: Based on the NTR, this must be true. So Hank is not a knave like Gary. So Hank is a liar.
7: This is false.
6: Based on the NSDSR, this must be true. So Fred is a knight. This is consistent with statement 4 being false.
1: This is false.

So the T/F order is FTFFTTFF. Uh oh, we have three Ts and five Fs. So it cannot be the case that 2/3/4 are TFF.

So if 2 and 3 are TF, then 4 cannot be false, so…
4: This must be true, so Fred is a knave like Gary.
6: This is false.
7: Based on the NTR, this must be true. So Hank in indeed a knave.
5: This is false.
1: This is false.

So the T/F order is FTFTFFTF. Uh oh, we have three Ts and five Fs. So it cannot be the case that 2 and 3 are TF.

--------------------------------------------

Therefore it must be the case that 2 and 3 are FT. So…
2: This is false. So Gary is not a knave.
3: This is true. So Hank is indeed a knight.
7: This is false.

Again, we didn't get too far. So let's make a guess that statement 4 is true. Then we can add that…

5: Based on the NTR, this must be false. So Gary is also a knight like Hank.
4: Based on our guess, this is true, so Fred is a knight like Gary.
1: This is false.
6: This is true.
8: This is false.

So the T/F order is FFTTFTFF. Uh oh, we have three Ts and five Fs. So it cannot be the case that 4 is true.

Therefore it must be the case that 2/3/4 are FTF. So the order so far is ?FTF??F?. Since there are exactly 4 false statements, this means that there is only one false statement left. Let's look at 1 and 6. They can't both be false since there is only one false statement left. So Fred is not a knave. They can't both be true, so one of them must be the remaining false statement. That means that 5 and 8 must be true. So Gary is a liar. Since 4 is false, Fred is not a liar like Gary. Therefore Fred is a knight.

So the answer is Fred is a knight, Gary is a liar, and Hank is a knight. And the statements are FFTFTTFT.

The only things we know about who said each statement is that one of each said statements 1 and 2, one of each said statements 5 and 6, and one of them said both 3 and 4 while the other said both 7 and 8. So I think that means there are 8 possibilities of who said what:

DEDDDEEE
DEDDEDEE
DEEEDEDD
DEEEEDDD
EDDDDEEE
EDDDEDEE
EDEEDEDD
EDEEDEDD

So technically I had to look through 6 cases. Well, I don't think that's too bad considering all the T/F combos I could have looked through during a brute force method.

Edited on October 9, 2007, 6:16 pm
  Posted by nikki on 2007-10-09 14:53:49

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