Print

HOW TO SOLVE 3 DIMENSIONAL LOGIC PROBLEMS


Andy, Chris, Noel, Randy and Steve were all recently hired as stand vendors at Yankee stadium. One of their last names is Morse, and each sells only one product.

1.Randy, whose last name is not Wiley sells no popcorn.
2.The Davis boy does not sell soda or candy.
3.The five boys are Noel, Randy, the Smith boy, the Cobble boy, and the ice cream seller.
4.Andy's last name is not Wiley or Cobble, and neither Andy nor the Cobble boy sell candy.
5.Neither Steve nor Davis sell peanuts or ice cream.

What are the full names of the boys, and what does each of them sell?

This is called a 3 dimensional problem because of the 3 categories: first names, last names, and product sold.

To solve this type of problem, you need a 3 dimensional table. The representation is difficult on a 2 dimensional surface like this document, but you can visualize that the first names relate to product, the last names relate to product, and the first and last names relate to each other. Draw the matrix like this:

 Cobble
Davis
Morse
Smith
Wiley
candy
icecream
peanuts
popcorn
soda
Andy
          
Chris
          
Noel
          
Randy
          
Steve
          
candy
          
icecream
          
peanuts          
popcorn
          
soda
          

Now go through the clues placing an 'X' in cells to indicate an elimination, , and an 'O' to indicate a fact. We've gone through the 5 clues and places the X's and O's in the appropriate cell, and annotated then with footnotes corresponding to the clue #s.

 Cobble
Davis
Morse
Smith
Wiley
candy
icecream
peanuts
popcorn
soda
Andy
X4
   X4
X4
    
Chris
          
NoelX3
  X3
  X3
   
Randy
X3
  X3
X1
 X3
 X1
 
Steve
      X5
X5
  
candy
X4
X2
        
icecream
X3
X5
 X3
      
peanuts
 X5        
popcorn
          
soda
 X2
        

6.Now using deductive reasoning, you know that the Davis boy must sell popcorn. Put an 'O' in the Davis/popcorn cell, and fill in the rest of the lastname/popcorn cells with 'X's.

 CobbleDavisMorseSmithWileycandyicecreampeanutspopcornsoda
AndyX4   X4X4    
Chris          
NoelX3  X3  X3   
RandyX3  X3  X3 X1 
Steve      X5X5  
candyX4X2        
icecreamX3X5 X3      
peanuts X5        
popcornX6O6X6X6X6     
soda X2        

 7. And since Randy doesn't sell popcorn, his name can't be Davis. Therefore it must be Morse. Fill in the 'X' in the Randy/Davis cell, implying an 'O' in the Randy/Morse cell.

 CobbleDavisMorseSmithWileycandyicecreampeanutspopcornsoda
AndyX4   X4X4    
Chris          
NoelX3  X3  X3   
RandyX3X7 O7 X3  X3 X1 
Steve      X5X5  
candyX4X2        
icecreamX3X5 X3      
peanuts X5        
popcornX6O6X6X6X6     
soda X2        

8.Since Randy's last name is Morse, now you can take the Randy facts and transfer them to Morse, viz Morse does not sell icecream

 

 CobbleDavisMorseSmithWileycandyicecreampeanutspopcornsoda
AndyX4   X4X4    
Chris          
NoelX3  X3  X3   
RandyX3X7 O7 X3  X3 X1 
Steve      X5X5  
candyX4X2        
icecreamX3X5X8
X3      
peanuts X5        
popcornX6O6X6X6X6     
soda X2        

9.Now it is apparent that Wiley sells icecream. Put an 'O' in Wiley icecream, and 'X's for the other Wiley Products.

 CobbleDavisMorseSmithWileycandyicecreampeanutspopcornsoda
AndyX4   X4X4    
Chris          
NoelX3  X3  X3   
RandyX3X7 O7 X3  X3 X1 
Steve      X5X5  
candyX4X2  X9     
icecreamX3X5X8
X3O9     
peanuts X5  X9     
popcornX6O6X6X6X6     
soda X2  X9     

10.Using the icecream information, you now know that Noel, Randy and Steve cannot be named Wiley, because these three boys do not sell icecream.. Enter the 'X's. We already know from #5 that Randy's last name is not Wiley

 CobbleDavisMorseSmithWileycandyicecreampeanutspopcornsoda
AndyX4   X4X4    
Chris          
NoelX3  X3X10 X3   
RandyX3X7 O7 X3  X3 X1 
Steve    X10 X5X5  
candyX4X2  X9     
icecreamX3X5X8
X3O9     
peanuts X5  X9     
popcornX6O6X6X6X6     
soda X2  X9     

11.Now we conclude that Chris's last name is Wiley, that Chris sells Ice cream (because Wiley does), and not Cobble, Davis, or Morse or Smith. They're falling like dominoes now!

 CobbleDavisMorseSmithWileycandyicecreampeanutspopcornsoda
AndyX4   X4X4    
ChrisX11X11X11X11O11  O11   
NoelX3  X3X10 X3   
RandyX3X7 O7 X3  X3 X1 
Steve    X10 X5X5  
candyX4X2  X9     
icecreamX3X5X8
X3O9     
peanuts X5  X9     
popcornX6O6X6X6X6     
soda X2  X9     

12. We also know that Andy doesn't sell ice cream (because Chris Wiley does), that Steve's name is Cobble, and no one else's is.

 CobbleDavisMorseSmithWileycandyicecreampeanutspopcornsoda
AndyX4   X4X4X12   
ChrisX11X11X11X11O11 O11   
NoelX3  X3X10 X3   
RandyX3X7 O7 X3  X3 X1 
SteveO12X12X12X12X10 X5X5  
candyX4X2  X9     
icecreamX3X5X8
X3O9     
peanuts X5  X9     
popcornX6O6X6X6X6     
soda X2  X9     

13.Now transfer to Steve what you know about Cobble, and to Cobble what you know about Steve

 CobbleDavisMorseSmithWileycandyicecreampeanutspopcornsoda
AndyX4   X4X4X12   
ChrisX11X11X11X11O11 O11   
NoelX3  X3X10 X3   
RandyX3X7 O7 X3  X3 X1 
SteveO12X12X12X12X10X13X5X5  
candyX4X2  X9     
icecreamX3X5X8
X3O9     
peanuts X5  X9     
popcornX6O6X6X6X6     
soda X2  X9     

14. So Cobble sells soda, therefore Steve sells soda. Therefore Andy, Chris Noel, Randy, Morse and Smith do not sell soda. Fill in the 'O's and 'X's

 CobbleDavisMorseSmithWileycandyicecreampeanutspopcornsoda
AndyX4   X4X4X12  X14
ChrisX11X11X11X11O11 O11  X14
NoelX3  X3X10 X3  X14
RandyX3X7 O7 X3  X3 X1X14
SteveO12X12X12X12X10X13X5X5X14O14
candyX4X2  X9     
icecreamX3X5X8
X3O9     
peanuts X5  X9     
popcornX6O6X6X6X6     
sodaO14X2X14X14X9     

15. By elimination, Smith must be Andy's last name. Therefore Smith sells no candy. Therefore he must sell peanuts. That means that Morse sells candy, and so does Randy.

 CobbleDavisMorseSmithWileycandyicecreampeanutspopcornsoda
AndyX4 O15X4X4X12O15X15X14
ChrisX11X11X11X11O11X15O11X15 X14
NoelX3  X3X10X15X3X15 X14
RandyX3X7 O7 X3 O15X3X15X1X14
SteveO12X12X12X12X10X13X5X5X14O14
candyX4X2O15 X9     
icecreamX3X5X8
X3O9     
peanuts X5X15O15X9     
popcornX6O6X6X6X6     
sodaO14X2X14X14X9     

By elimination, the answer is
Noel Davis popcorn
Andy Smith peanuts
Chris Wiley Ice cream
Randy Morse candy
Steve Cobble soda

Copyright © 2001 by Paladin Consultants,LLC
Chatham, NJ
All Rights Reserved
This page revised   August 3, 2004