The Four Doors of Xanth

Date: 02/11/98 at 19:38:10
From: Brent Goodwin
Subject: The four doors of Xanth

Prince Questor is exploring the caves of Xanth. At the end of a 
tunnel, he finds four doors, he finds a scroll. Here is the message 
from the scroll.

Each door conceals one item. The items are a treasure, a rope, a key, 
and a lantern. You must find all four items in a particular order to 
keep the treasure. 

As Questor is reading the scroll, three bats fly in. The first bat 
says, "You must find the key before you find the rope." The second bat 
says, "If you find the lantern before you find the rope, the treasure 
will disappear." The third bat says, "You must find the treasure 
last."

As Questor is puzzling over these remark,s three ogres appear. The 
first one says, "The rope is not behind the 1st or 2nd door." The 
second ogre says, "The treasure is in the room just to the right of 
the lantern." The third ogre says, "The key is behind the fourth 
door."

In what order should Questor open the doors to keep the treasure?

Please help me with this question.

Date: 02/12/98 at 07:02:39
From: Doctor Allan
Subject: Re: The four doors of Xanth

Hi Brent!

This is a fun question, isn't it?

I will tell you how I reason and I hope this can give you an idea of
how to reason generally in cases like this one.

first bat: key before rope
second bat: rope before lantern

   therefore: key before rope before lantern

third bat: everything before treasure

   therefore: key before rope before lantern before treasure.

This gives you the right order:

   1. key
   2. rope
   3. lantern
   4. treasure

Concerning the ogres:

The third one: key behind door 4
The first one: rope is not behind door 1 or door 2 

   therefore the rope is behind door 3 or door 4. 

   But the key is behind door 4, so the rope is behind door 3.

The second one: The treasure is behind the door to the right of the 
                lantern.

   But only doors 1 and 2 are left, so in order to be to the right                  
   of a door you should be behind door 1. 

   So the treasure is behind door 1 and the lantern is behind door 2.

So all in all, you get this order:

   1. door 4: key
   2. door 3: rope
   3. door 2: lantern
   4. door 1: treasure

Hope this helps!

-Doctor Allan,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 06/01/98 at 14:52:11
From: mark norwine
Subject: Word / deduction math problems

I simply love mathmatical brain teasers and number problems. I often 
spend my lunchtime online trying my hand at some of the math problems 
presented by others on your Web site.

Today, I was stumped by The four doors of Xanth. I have a problem 
understanding the concluding logic.

The key is behind "door 4" and the rope is behind "door 3." That's 
easy.

The treasure is "to the right" of the lantern, but in the given of 
this problem, doors 1 through 4 were never oriented. Do they go 
1 = leftmost, 4 = rightmost, or vice versa?

Using the time-honored tradition of "left-to-right," this means door 1 
is leftmost and door 2 is to the right of door 1.

With this in mind, I easily see that the treasure is behind door 2, 
the lantern behind door 1 ["...The treasure is behind the door to the 
right of the lantern..."], and that is exactly the opposite of Dr. 
Allan's conclusion.

Can you shed some light?  
Many thanks.

Date: 09/07/98 at 08:19:11
From: Doctor Allan
Subject: Re: Word / deduction math problems

Hello Mark,

I looked over my original answer, and you're absolutely correct that 
it is pretty confusing. Let me try to correct this:

As you say one easily finds that the key is behind door 4 and the rope 
is behind door 3. No matter the orientation only two doors remain and 
the right-most of these must contain the treasure.

Whether a left-orientation, a right-orientation, or numbers on the 
doors are assumed, once you have concluded where the rope and the key 
is, it is easy to determine where the remaining things are.

I appologise for the bad English in my original answer.

Thanks for pointing it out,
      
- Doctor Allan, The Math Forum
 http://mathforum.org/dr.math/