February Problem to Ponder
 Neighborhood Sleuth 

 A private detective is working a case and has managed to verify the following information: 

1. There are five houses in the neighborhood.  

2. In each house resides a person of a different nationality—an Australian, an Italian, a Ukrainian, a Norwegian, and a Thailander—each of whom has one type of pet and consumes one type of candy and one type of drink. 
3. The Australian lives in the red house. 
4. The Italian owns a dog. 
5. Coffee is drunk in the green house. 
6. The Ukrainian drinks tea. 
7. The green house is immediately to the right of the ivory house. 
8. The person who eats Baby Ruths owns snails. 
9. M&Ms are eaten in the yellow house. 
10. Milk is drunk in the middle house. 
11. The Norwegian lives in the first house. 
12. The person who eats Reese’s Pieces lives next door to the woman with the fox. 
13. M&Ms are eaten in the house next to the house with a horse. 
14. The person who eats Tootsie Rolls drinks orange juice. 
15. The Thailander eats Snickers.   
16. The Norwegian lives next door to the blue house. 

To solve the case, the detective must uncover the answers to the following questions: Who drinks water? Who owns the zebra?  

In March we worked through part of the problem, using implications and connections among the clues to determine that not only does the Norwegian drink water, but we know other things, as shown in this diagram.

We still need to determine the answer to the question, Who owns the zebra?

At this point, we have some branching possibilities to consider.

The information that we know about the houses so far, together with clues 3 and 5, give us the following possibilities for the third, fourth, and fifth houses:

(i) We know that the third house is either red or ivory, and if it is red, then according to clue 3, the Australian lives there.

(ii) We know that the fourth house is either ivory or green, and if it is green, then according to clue 5, coffee is drunk there.

(iii) We know that the fifth house is either red or green. If it is red, the Australian lives there, and if it is green, coffee is drunk there (from clues 3 and 5 again).

Clue 6 tells us that the Ukrainian drinks tea, so we know that the Ukrainian cannot live in the fifth house, since according to (iii) either coffee or an Australian is in the fifth house. We also know that the Ukrainian can’t live in the first or third house, because we’ve already established that water and milk, respectively, are consumed in those houses. Clue 6 together with other information implies that the Ukrainian who drinks tea lives either in the second or in the fourth house.

Case 1: Let’s suppose that the Ukrainian who drinks tea is in the fourth house. Then from (ii) and (iii) we know that the fifth house must be green, and whoever lives there drinks coffee.  This forces the third house to be red and to be home to the Australian, while placing the Ivory house in the fourth position, since it’s the only one left. The only beverage that we haven’t assigned to a house is orange juice, and the only slot left is the second house—and we also know that with the orange juice come Tootsie Rolls.

Clue 4 tells us that the Italian owns a dog, so we know that the Italian can’t be in the second house, since we already deduced that whoever lives there has a horse. We also know that the Norwegian lives in the first house, and the Australian lives in the third house, and we have supposed that the Ukrainian lives in the fourth house. So the Italian must then live in the fifth house.

Under the supposition that the Ukrainian who drinks tea lives in the fourth house, we have determined that the profile of the houses must look as follows: 

The only place for the Thailander, then, is the second house. But clue 15 tells us that the Thailander eats Snickers! This leads to a contradiction, since we already have placed Tootsie Rolls in the second house!!  —> * <— !!!

Let’s try again…

Case 1: The assumption that the Ukrainian who drinks tea lives in the fourth house does not work because it leads to a contradiction with the given clues.

That means that the Ukrainian who drinks tea must be in the second house.

Then orange juice, together with Tootsie Rolls, must be consumed in either the fourth or the fifth house, since we already know the beverages consumed at the other three houses.

Case 2: What if we supposed that the orange juice and the Tootsie Rolls were consumed in the fifth house? Then from (ii) and (iii) above, we would place the coffee and the green house in the fourth spot, , and from (i) and (ii) above, we would place the ivory house in the third spot, , and that would leave us with the Australian in the red house in the fifth spot. Thus, the housing situation would be as shown below:

Clues 4, 8, and 15 give us three pairs of events: (a) Italian and dog; (b) snails and Baby Ruths, and (c) Thailander and Snickers. However, each of those three pairs has to be in either the third or fourth house, because the first, second, and fifth houses already have a person, pet, or snack. That means that at least two of the three pairs have to be in the same house—either the third house or the fourth house (pigeonhole principle). But pairs (a) and (b) can’t both be in the same house, or else there would be two pets; pairs (a) and (c) can’t both be in the same house, or else there would be two people; and pairs (b) and (c) can’t both be in the same house, or else there would be two snacks.

Let’s try this…

Case 2: The assumption that orange juice is in the fifth house leads us to another contradiction. Thus, orange juice and Tootsie Rolls must be consumed in the fourth house, not the fifth.

If orange juice is consumed in the fourth house, then that means that coffee (and the green house) are in the fifth position, and that in turn means that the Australian and the red house are in the third position, and that the house in the fourth position is ivory. The picture thus becomes that shown below:

The only place for the Thailander with Snickers would be the fifth house, which would put the Italian in the fourth house with the dog, while Baby Ruths and Snails would be in the third house:

We see then that the only place left for the Reese’s Pieces is in the second house with the Ukrainian. Clue 12 then implies that it’s the Norwegian (woman) who has a fox, next door to the eater of Reese’s Pieces—since we already know that the Australian has pet snails on the other side of the Ukrainian.

AND THEREFORE (fanfare)—The Thailander has the zebra!!!