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
Noel	X3			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.

	Cobble	Davis	Morse	Smith	Wiley	candy	icecream	peanuts	popcorn	soda
Andy	X4				X4	X4
Chris
Noel	X3			X3			X3
Randy	X3			X3	X1		X3		X1
Steve							X5	X5
candy	X4	X2
icecream	X3	X5		X3
peanuts		X5
popcorn	X6	O6	X6	X6	X6
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.

	Cobble	Davis	Morse	Smith	Wiley	candy	icecream	peanuts	popcorn	soda
Andy	X4				X4	X4
Chris
Noel	X3			X3			X3
Randy	X3	X7	O7	X3	X1		X3		X1
Steve							X5	X5
candy	X4	X2
icecream	X3	X5		X3
peanuts		X5
popcorn	X6	O6	X6	X6	X6
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

	Cobble	Davis	Morse	Smith	Wiley	candy	icecream	peanuts	popcorn	soda
Andy	X4				X4	X4
Chris
Noel	X3			X3			X3
Randy	X3	X7	O7	X3	X1		X3		X1
Steve							X5	X5
candy	X4	X2
icecream	X3	X5	X8	X3
peanuts		X5
popcorn	X6	O6	X6	X6	X6
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.

	Cobble	Davis	Morse	Smith	Wiley	candy	icecream	peanuts	popcorn	soda
Andy	X4				X4	X4
Chris
Noel	X3			X3			X3
Randy	X3	X7	O7	X3	X1		X3		X1
Steve							X5	X5
candy	X4	X2			X9
icecream	X3	X5	X8	X3	O9
peanuts		X5			X9
popcorn	X6	O6	X6	X6	X6
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

	Cobble	Davis	Morse	Smith	Wiley	candy	icecream	peanuts	popcorn	soda
Andy	X4				X4	X4
Chris
Noel	X3			X3	X10		X3
Randy	X3	X7	O7	X3	X1		X3		X1
Steve					X10		X5	X5
candy	X4	X2			X9
icecream	X3	X5	X8	X3	O9
peanuts		X5			X9
popcorn	X6	O6	X6	X6	X6
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!

	Cobble	Davis	Morse	Smith	Wiley	candy	icecream	peanuts	popcorn	soda
Andy	X4				X4	X4
Chris	X11	X11	X11	X11	O11		O11
Noel	X3			X3	X10		X3
Randy	X3	X7	O7	X3	X1		X3		X1
Steve					X10		X5	X5
candy	X4	X2			X9
icecream	X3	X5	X8	X3	O9
peanuts		X5			X9
popcorn	X6	O6	X6	X6	X6
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.

	Cobble	Davis	Morse	Smith	Wiley	candy	icecream	peanuts	popcorn	soda
Andy	X4				X4	X4	X12
Chris	X11	X11	X11	X11	O11		O11
Noel	X3			X3	X10		X3
Randy	X3	X7	O7	X3	X1		X3		X1
Steve	O12	X12	X12	X12	X10		X5	X5
candy	X4	X2			X9
icecream	X3	X5	X8	X3	O9
peanuts		X5			X9
popcorn	X6	O6	X6	X6	X6
soda		X2			X9

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

	Cobble	Davis	Morse	Smith	Wiley	candy	icecream	peanuts	popcorn	soda
Andy	X4				X4	X4	X12
Chris	X11	X11	X11	X11	O11		O11
Noel	X3			X3	X10		X3
Randy	X3	X7	O7	X3	X1		X3		X1
Steve	O12	X12	X12	X12	X10	X13	X5	X5
candy	X4	X2			X9
icecream	X3	X5	X8	X3	O9
peanuts	X13	X5			X9
popcorn	X6	O6	X6	X6	X6
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

	Cobble	Davis	Morse	Smith	Wiley	candy	icecream	peanuts	popcorn	soda
Andy	X4				X4	X4	X12			X14
Chris	X11	X11	X11	X11	O11		O11			X14
Noel	X3			X3	X10		X3			X14
Randy	X3	X7	O7	X3	X1		X3		X1	X14
Steve	O12	X12	X12	X12	X10	X13	X5	X5	X14	O14
candy	X4	X2			X9
icecream	X3	X5	X8	X3	O9
peanuts	X13	X5			X9
popcorn	X6	O6	X6	X6	X6
soda	O14	X2	X14	X14	X9

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.

	Cobble	Davis	Morse	Smith	Wiley	candy	icecream	peanuts	popcorn	soda
Andy	X4	X15	X15	O15	X4	X4	X12	O15	X15	X14
Chris	X11	X11	X11	X11	O11	X15	O11	X15		X14
Noel	X3		X15	X3	X10	X15	X3	X15		X14
Randy	X3	X7	O7	X3	X1	O15	X3	X15	X1	X14
Steve	O12	X12	X12	X12	X10	X13	X5	X5	X14	O14
candy	X4	X2	O15	X15	X9
icecream	X3	X5	X8	X3	O9
peanuts	X13	X5	X15	O15	X9
popcorn	X6	O6	X6	X6	X6
soda	O14	X2	X14	X14	X9

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