Knights always tell the truth. Liars always lie. Knaves alternate between telling the truth and lying. Pages tell the truth if the last statement was a lie and lie if the last statement was a truth. Squires tell the truth if the last statement was a truth and lie if the last statement was a lie. You meet five people, A, B, C, D, and E. One of them is a knight, one of them is a liar, one is a knave, one is a page, and one is a squire. They make the following statements.
A:I am a liar.
B:I am a liar.
C:I am a squire.
D:A is a page.
A:E is not a knight.
What are A, B, C, D, and E?
Definitions
Knights always tell the truth.
Liars always lie.
Knaves alternate between telling the truth and lying.
Pages tell the truth if the last statement was a lie and lie if the last statement was a truth.
Squires tell the truth if the last statement was a truth and lie if the last statement was a lie.
Observations:
A:I am a liar.
B:I am a liar.
C:I am a squire.
D:A is a page.
A:E is not a knight.
It's obvious that A and B are neither liars nor knights and C is not a knight.
Since A was lying, B was either a knave in his lying phase or was a squire imitating A's telling a lie.
Since B was lying, C would be lying if he really were a squire, so he was not a squire, and was thus lying. So he wasn't a page either. He was not a knight, and so the only things he could be were Liar or Knave in his lying phase.
The possibilities for A would be
Knave in lying phase first and telling truth last.
Page
Squire
If A was a knave, then E is indeed not a knight and D must be the knight, but that would make A a page--a contradiction, so A is not a knave. This leaves Page and Squire. If he was a page, then D was telling the truth and A would be lying this second time, so E is a knight. If he was a squire, D would have been telling a lie, but again, E would be a knight.
So E is the knight in the group. We also know that either C or D was the liar.
If A were a knave he'd tell the truth in the last statement, but he didn't--he's not a knave.
Kt L Kv S P
A n n n - -
B n n - - n
C n - - n n
D n - - - -
E Y n n n n
If A is a page, D is telling the truth and is not himself a page or liar. If further he were a squire he'd be violating the squire's rules by telling the truth after a falsehood, so he'd be the knave, and C would be the liar and B the squire. Does that work?
Kt L Kv S P
A n n n n Y Page
B n n n Y n Squire
C n Y n n n Liar
D n n Y n n Knave
E Y n n n n Knight
Notes on statements:
| A:I am a liar. Actually Page, lies at the end because D told truth before that.
| B:I am a liar. Actually Squire, lies just because A lied just before that.
| C:I am a squire. Is a liar; no further explanation needed.
| D:A is a page. As a knave he must be in his truth-telling phase.
| A:E is not a knight. Again, D told the truth, so A, as a Page, must lie.
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This is a consistent explanation. If we assume there's only one solution, we have found it.
Someone else might want to follow the path of assuming A is the Squire (the only other possibility at the point at which we made our assumption he was a Page), and assure that that leads to a contradiction.
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Posted by Charlie
on 2020-10-08 13:47:48 |
solution without proof of uniqueness
