Re: More Knight-Knave Metapuzzles

On Thu, 13 Jul 2006 12:30:23 -0400 (EDT), "Arthur J. O'Dwyer"
<ajonospam@xxxxxxxxxxxxxx> wrote:

Here are a couple of logic puzzles I wrote last night. They take place
on the Island of Knights and Knaves, where all natives are either knights,
who never lie, or knaves, who never tell the truth. I've spiced up the
mythos by postulating the convention that all natives, when they travel,
wear pointy hats of the appropriate color. But sometimes these hats get
mixed up...


PROBLEM 1. A hats puzzle.

Alan, Bill, Charles, and Daniel --- all four natives of the Island of
Knights and Knaves --- come for a dinner party hosted by Horace, also a
native. As is the custom on the Island, each of the guests arrives wearing
a pointy hat: knights wear white hats, and knaves wear black hats.

As the party is breaking up, the guests go to the cloakroom to retrieve
their hats, but in the darkness, each guest puts on another guest's hat.
The guests don't realize the mistake until they're all outside, at which
point the following conversation ensues:

ALAN. This isn't my hat!
BILL. Well, you're certainly not wearing mine.
CHARLES. At least the hat I'm wearing is the right color for me!
DANIEL. Bill is wearing my hat.
HORACE. Gentlemen, gentlemen, it doesn't matter; all the hats
are the same color anyway!
CHARLES. Horace is lying, or I am not a knight!
ALAN. Horace is lying, or I am not a knave!
DANIEL. Horace is lying, and I am wearing a white hat!
BILL. Horace is lying, and I am wearing a black hat!

Who's wearing whose hat?

Number the statements A1, B1, C1, D1, H1, C2, A2, D2, B2. Denote the
first parts of compound statements with the suffix P1, the second
parts with P2.

A1 is true, so Alan is a knight, so A2 is true (A2P2 is true, so A2P1
tells us nothing about Horace).

Consider C2. If Charles is a knave, then C2P2 is true, so C2 is true;
contradiction; so Charles is a knight, C2P2 is false, so C2P1 must be
true; Horace is a knave, H1 is false, so Bill and/or Daniel is a
knave. Also, C1 is true, so Charles is wearing some other knight's
hat.

If Bill is a knave and Daniel is a knight, then:
B1 is false, so Alan is wearing Bill's hat.
D1 is true, so Bill is wearing Daniel's hat.
D2 is true, so Daniel is wearing some other knight's hat.
B2 is false, so Bill is wearing some knight's hat (already known
from D1).
This leads to a valid solution:
Alan is wearing Bill's hat
Bill is wearing Daniel's hat
Charles is wearing Alan's hat
Daniel is wearing Charles's hat

If Bill is a knight and Daniel is a knave, then:
B1 is true, so Alan is not wearing Bill's hat.
D1 is false, so Bill is not wearing Daniel's hat.
D2 is false, so Daniel is wearing some other knave's hat.
B2 is true, so Bill is wearing a knave's hat.
This leads to a contradiction: there's no valid choice for which
hat Bill is wearing.

If Bill and Daniel are both knaves, then:
B1 is false, so Alan is wearing Bill's hat.
D1 is false, so Bill is not wearing Daniel's hat.
D2 is false, so Daniel is wearing some other knave's hat.
B2 is false, so Bill is wearing some knight's hat.
This leads to a contradiction: there's no valid choice for which
hat Daniel is wearing.

An excellent puzzle! I'll try #2 next.
.