The White Knight was recounting the story of one of the trials he had recently attended:

“Ah yes, it was a fine trial. Let’s see if I can remember it… If I recall there were three defendants. Each of them made one statement accusing one of the others, I think, but I’m afraid I can’t remember who accused whom. Can you work out who was guilty?”

“Of course not!” you reply. “You haven’t told me anything yet! Could you at least tell me who lied and who told the truth?”

“Hmmm... Interesting that you should ask that. When I was describing the trial to the Red King he asked me the same question. When I told him the answer, he worked it out. Unfortunately I’ve now quite forgotten what I said.”

“Well, I guess it’s hopeless for me then…” you sigh.

“Interesting that you should say that, too. When I was describing the trial to Humpty Dumpty he asked me the same question, and when I told him I had forgotten what I told the Red King, he too claimed to be at a loss. But then he asked me another question, I can't quite remember what, but when I told him the answer he was able to solve it. I think he either asked me whether two consecutive statements were true, or whether two consecutive statements were false. Unfortunately I can neither remember which one he asked, nor what I answered. I think I’ve given you quite enough information now though, so tell me: who was guilty?”

    Adapted from Raymond Smullyan's Alice in Puzzleland

First off, we know there are three defendants, let's name them D1, D2, and D3.

Each made a statement accusing another.  This leaves several possibilities.  D1 could accuse D2, D2 accuse D3, and D3 accuse D1.  Also, D1 could accuse D2, D2 accuse D1, and D3 accuse D2.  I'm assuming that since it says accuses another, that none accuse themselves, as this would be ridiculous anyway.

Apparantly, not all of them told the truth.  The fact that the Red King was able to work it out from this is irrelevant at this point, as we don't know which ones told the truth yet.

What is important is that Humpty Dumpty figured it out after being told one of two things.  He was either told that either two consecutive statements were true or two consecutive were false, but he didn't say whether he answered the question yes or no.

This leads to six cases.  TTF, FTT, FFT, TFF, TFT, and FTF.

Lets now analyze each case with the first two cases.  Assuming TTF, with the first case of D1 accusing D2, D2 accusing D3, and D3 accusing D1, this gives you D2 and D3 as the perp, because D1 and D2's statements would be true, but there is only one guilty party, so this isn't the case.

Assuming FTT with the same, you get D3 and D1 as the perps, again, only one guilty so this is also not the case.

Assuming FFT with the same, you get D2 and D3 not the perp, and D1 is the perp, this case is plausible, so we'll add it to our set of plausible cases.

Assuming TFF with the same, you get D2 as the perp, and D3 and D1 as not guilty.  This is also plausible for now.

Assuming TFT, you get D2 and D1 as perps, again, there is only one guilty, so this is not the case.

Assuming FTF, you get D3 as the perp, and D1 with D2 not guilty, also a plausible case.

Now looking at the other situation, D1 accuses D2, D2 accuses D1, and D3 accuses D2, you get:

TTF--D2 and D1 perps, not the case.
FTT--D1 and D2 perps, not the case.
FFT--D2 is and isn't the perp, nonsensicle, so not the case.
TFF--same case as before, nonsensicle, so not the case.
TFT--D2 is the perp, so this case is plausible.
FTF--D1 is the perp, so this case is also plausible.

Obviously, in the first case, there would have to be two people lying, so in summary, TFF, FFT, and FTF can work.
In the second case, Only the cases where the two people agree could work, unless they are both lying, in which case you get, TFT and FTF.

Now, in order for the Red King to work it out in the first situation, he would have been told which of the three cases, TFF, FFT, or FTF was right, and also, because he forgot what was said, but told Humpty Dumpty the answer to his question and Humpty Dumpty figured it out as well, if the question answered were is there two consecutive trues, or two consecutive falses, either way, Humpty would have been left with no way to tell which was used, therefore, the only correct question in this case would be to ask is there two consecutive falses, and to be told no, leaving the only choice to be FTF, and therefore, meaning that D3 is the perp, or guilty party.

In the second situation, the Red King would have been told either that TFT, or FTF were the correct situation.  In order for Humpty to figure it out as well, then he would have asked one of the two questions above, but neither question would have eliminated a choice, so there is no way for both to figure out the truth in this situation.

That leaves the guilty party as definitely D3.  The truth teller is D2, the liars are D1 and D3.  The statements given are D1 accused D2, D2 accused D3, and D3 accused D1.  Humpty Dumpty asked if there were two consecutive falses, to which he was told no.  This allowed both the Red King and Humpty Dumpty to figure it out, as well as the reader.

Posted by Joshua on 2004-04-17 03:17:02