Where is the Arsenic?

Date: 03/12/2002 at 12:37:08
From: Shae
Subject: Russian Roulette

You place six jars (right to left: coffee, arsenic, and sugar on the 
top shelf; snuff, tea, and salt on the bottom shelf) and have somebody 
else fill them up with different ingredients, making sure each jar 
contains something other than what the label says. You are told that 
the salt is located under the jar containing snuff, and that the sugar 
is located to the right of the jar containing coffee. Where is the 
arsenic?

Date: 03/12/2002 at 13:01:12
From: Doctor Ian
Subject: Re: Russian Roulette

Hi Shae,

Let me make sure I understand you correctly. The jars are labeled this 
way:

    Coffee   Arsenic   Sugar
    Snuff    Tea       Salt


But you know that each jar contains something other than what its 
label says. So the possibilities are

    Arsenic  Coffee   Coffee
    Sugar    Sugar    Arsenic
    Snuff    Snuff    Snuff
    Tea      Tea      Tea
    Salt     Salt     Salt
      |        |         |
      v        v         v
    COFFEE   ARSENIC   SUGAR

    SNUFF    TEA       SALT
      ^        ^         ^
      |        |         |
    Coffee   Coffee    Coffee
    Arsenic  Arsenic   Arsenic
    Sugar    Sugar     Sugar
    Tea      Snuff     Snuff
    Salt     Salt      Tea

The salt is under the jar containing snuff. That means that the salt 
is in the bottom row; and it means that the snuff can't be over the 
jar labeled SALT:

    Arsenic  Coffee   Coffee
    Sugar    Sugar    Arsenic
    Snuff    Snuff    
    Tea      Tea      Tea
      |        |         |
      v        v         v
    COFFEE   ARSENIC   SUGAR

    SNUFF    TEA       SALT
      ^        ^         ^
      |        |         |
    Coffee   Coffee    Coffee
    Arsenic  Arsenic   Arsenic
    Sugar    Sugar     Sugar
    Tea      Snuff     Snuff
    Salt     Salt      Tea

The sugar is located to the right of the jar containing coffee. That 
means that the sugar can't be on the left:

    Arsenic  Coffee   Coffee
             Sugar    Arsenic
    Snuff    Snuff    
    Tea      Tea      Tea
      |        |         |
      v        v         v
    COFFEE   ARSENIC   SUGAR

    SNUFF    TEA       SALT
      ^        ^         ^
      |        |         |
    Coffee   Coffee    Coffee
    Arsenic  Arsenic   Arsenic
             Sugar     Sugar
    Tea      Snuff     Snuff
    Salt     Salt      Tea

Nor can it be in the jar labeled ARSENIC:

    Arsenic  Coffee   Coffee
                      Arsenic
    Snuff    Snuff    
    Tea      Tea      Tea
      |        |         |
      v        v         v
    COFFEE   ARSENIC   SUGAR

    SNUFF    TEA       SALT
      ^        ^         ^
      |        |         |
    Coffee   Coffee    Coffee
    Arsenic  Arsenic   Arsenic
             Sugar     Sugar
    Tea      Snuff     Snuff
    Salt     Salt      Tea

If it were in the jar labeled TEA, then the jar labeled SNUFF would 
have to contain coffee. But one of these (TEA or SNUFF) has to contain 
salt. So the sugar has to be in the jar labeled SALT:

    Arsenic  Coffee   Coffee
                      Arsenic
    Snuff    Snuff    
    Tea      Tea      Tea
      |        |         |
      v        v         v
    COFFEE   ARSENIC   SUGAR

    SNUFF    TEA       SALT
      ^        ^         ^
      |        |         |
    Coffee   Coffee  
    Arsenic  Arsenic  
                       Sugar
    Tea      Snuff     
    Salt     Salt       

Since the sugar is in SALT, the coffee must be in TEA, and the salt 
must be in SNUFF:

    Arsenic           Arsenic
    Snuff    Snuff    
    Tea      Tea      Tea
      |        |         |
      v        v         v
    COFFEE   ARSENIC   SUGAR

    SNUFF    TEA       SALT
      ^        ^         ^
      |        |         |
    Salt     Coffee    Sugar
         
Can you take it from here? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/