Logic Puzzles: Who Owns the Fish?

Having looked at methods for solving four logic puzzles, I’ll close the series with a classic, this one involving five houses, five colors, five countries, five drinks, five cigarettes, and five pets – that’s six attributes to juggle! (Last time, we had only 3 or 5.) Once again, the challenge is to solve it without looking up the answer, so we only give a start to a solution, to suggest a method, which is a little different from the others we’ve seen.

“Einstein’s problem”

Here is the problem as posed to us in 2002:

Who Owns the Fish? (Einstein's Problem)

There are 5 houses sitting next to each other, each with a different color, occupied by 5 guys, each from a different country, and with a favorite drink, cigarette, and pet.  Here are the facts:

The British occupies the red house.
The Swedish owns a dog.
The Danish drinks tea.
The green house is on the left of the white house.
The owner of the green house drinks coffee. 
The person who smokes "Pall Mall" owns a bird.
The owner of the yellow house smokes "Dunhill".
The owner of the middle house drinks milk.
The Norwegian occupies the 1st house.
The person who smokes "Blend" lives next door to the person who owns a cat.
The person who owns a horse live next door to the person who smokes "Dunhill".
The person who smokes "Blue Master" drinks beer.
The German smokes "Prince".
The Norwegian lives next door to the blue house.
The person who smokes "Blend" lives next door to the person who drinks 
water.

The question is: Who owns the fish?

Doctor Ian answered, first giving a link to a site that includes a solution (which I will let you find for yourself if you give up):

This is a famous problem, attributed to Albert Einstein.  If you just want to find out the answer, you can go here:

...

If you'd like to find the answer yourself (it's actually pretty fun once you get into it, so I recommend finding the answer instead of just looking it up), you'll have to set up some systematic way to eliminate the possibilities.

With so many attributes, the full tables we used last time may be too unwieldy. Strategies have to be adjusted to fit the problem.

For example, you might list the houses, along with the possible owners, and information about the owners, e.g., 

  House 1     House 2     House 3     House 4     House 5
  ----------  ----------  ----------  ----------  ----------
  Brit        Brit        Brit        Brit        Brit
  Swede       Swede       Swede       Swede       Swede
  Dane        Dane        Dane        Dane        Dane
  Norwegian   Norwegian   Norwegian   Norwegian   Norwegian
  German      German      German      German      German

  Red         Red         Red         Red         Red
  White       White       White       White       White
  Yellow      Yellow      Yellow      Yellow      Yellow
  Green       Green       Green       Green       Green
  Blue        Blue        Blue        Blue        Blue

  Pall Mall   Pall Mall   Pall Mall   Pall Mall   Pall Mall
  Dunhill     Dunhill     Dunhill     Dunhill     Dunhill
  Blend       Blend       Blend       Blend       Blend
  Blue Master Blue Master Blue Master Blue Master Blue Master
  Prince      Prince      Prince      Prince      Prince

This includes only the houses, countries, colors, and cigarettes, so you will need to add the drinks and pets.

Then you use the clues to eliminate possibilities, until only one possibility is left. For example, the Norwegian occupies the first house.  That means that no one else can be in the house, and the Norwegian can't be in any other houses:

  House 1     House 2     House 3     House 4     House 5
  ----------  ----------  ----------  ----------  ----------
              Brit        Brit        Brit        Brit
              Swede       Swede       Swede       Swede
              Dane        Dane        Dane        Dane
  NORWEGIAN   
              German      German      German      German

  Red         Red         Red         Red         Red
  White       White       White       White       White
  Yellow      Yellow      Yellow      Yellow      Yellow
  Green       Green       Green       Green       Green
  Blue        Blue        Blue        Blue        Blue

  Pall Mall   Pall Mall   Pall Mall   Pall Mall   Pall Mall
  Dunhill     Dunhill     Dunhill     Dunhill     Dunhill
  Blend       Blend       Blend       Blend       Blend
  Blue Master Blue Master Blue Master Blue Master Blue Master
  Prince      Prince      Prince      Prince      Prince

This is equivalent to zeroing out the rest of each row and column last time.

Another clue says that the Norwegian lives next door to the blue house.  This means the second house has to be blue, and no other house can be blue:

  House 1     House 2     House 3     House 4     House 5
  ----------  ----------  ----------  ----------  ----------
              Brit        Brit        Brit        Brit
              Swede       Swede       Swede       Swede
              Dane        Dane        Dane        Dane
  NORWEGIAN   
              German      German      German      German

  Red                     Red         Red         Red
  White                   White       White       White
  Yellow                  Yellow      Yellow      Yellow
  Green                   Green       Green       Green
              BLUE  

  Pall Mall   Pall Mall   Pall Mall   Pall Mall   Pall Mall
  Dunhill     Dunhill     Dunhill     Dunhill     Dunhill
  Blend       Blend       Blend       Blend       Blend
  Blue Master Blue Master Blue Master Blue Master Blue Master
  Prince      Prince      Prince      Prince      Prince

Not all clues are as easy to indicate in this partial table:

Another clue tells us that the Brit lives in the red house.  This means he can't be in the blue house.  And the German smokes Prince, so that can't be smoked by the Norwegian:

  House 1     House 2     House 3     House 4     House 5
  ----------  ----------  ----------  ----------  ----------
                          Brit        Brit        Brit
              Swede       Swede       Swede       Swede
              Dane        Dane        Dane        Dane
  NORWEGIAN
              German      German      German      German

  Red                     Red         Red         Red
  White                   White       White       White
  Yellow                  Yellow      Yellow      Yellow
  Green                   Green       Green       Green
              BLUE  

  Pall Mall   Pall Mall   Pall Mall   Pall Mall   Pall Mall
  Dunhill     Dunhill     Dunhill     Dunhill     Dunhill
  Blend       Blend       Blend       Blend       Blend
  Blue Master Blue Master Blue Master Blue Master Blue Master
              Prince      Prince      Prince      Prince

I haven't included the drinks or the pets, but clues for those would work the same way.

Since not all possible tables have been built, some information can’t be shown yet; but, for example, once we find where the German lives, we’ll know Prince is smoked there and nowhere else. You might want to make one mark next to each clue that has been used up, and a different on for those that have been used partly, but not fully.

Note that you'll probably have to revisit some of the clues several times.  A clue that seems to give you no information at first might prove to be useful later on, after the possibilities have been narrowed down.

For example, the first clue, that the Brit lives in the red house, wasn't useful until we knew that he didn't live in the first house.

So a judicious choice of the order in which to use clues will be a major part of your strategy.

Also, you need to use ALL of the information given to you by each clue.  For example,the clue

  1. The Brit lives in a red house

tells you all of the following:

  (1) If you know that the Brit lives in a
      house, then the house must be red.

  (2) If you know that a house is red, 
      then the Brit must live in it.

  (3) If you know that a house isn't red, 
      then the Brit can't live in it,

  (4) If you know that the Brit doesn't live
      in a house, then the house can't be red.

You might want to break up each clue in this way, and mark off the parts that have been used.

Now we come to the biggest hint here (which you may want to hide until you’ve given it a try on your own):

Just to make sure that the technique I'm suggesting actually works, I went ahead and solved the puzzle.  Here are the clues that I used, in the order that I used them:

   9. The Norwegian lives in the first house.
  14. The Norwegian lives next to the blue house.
   8. The man living in the house right in the center drinks milk.
   1. The Brit lives in a red house.
   4. The green house is on the left of the white house (next to it).
   5. The green house owner drinks coffee.
   4. The green house is on the left of the white house (next to it).
   1. The Brit lives in a red house.
   7. The owner of the yellow house smokes Dunhill.
   3. The Dane drinks tea.
   2. The Swede keeps dogs as pets.
   6. The person who smokes Pall Mall rears birds.
  11. The man who keeps horses lives next to the man who smokes Dunhill.
  15. The man who smokes blend has a neighbor who drinks water.
   1. The Brit lives in a red house.
   3. The Dane drinks tea.
   2. The Swede keeps dogs as pets.
   6. The person who smokes Pall Mall rears birds.
  12. The owner who smokes Blue Master drinks beer.
  13. The German smokes Prince.
   2. The Swede keeps dogs as pets.
  12. The owner who smokes Blue Master drinks beer.
  15. The man who smokes blend has a neighbor who drinks water.
  13. The German smokes Prince.
   3. The Dane drinks tea.
   2. The Swede keeps dogs as pets.
   6. The person who smokes Pall Mall rears birds.
  10. The man who smokes blend lives next to the one who keeps cats.

Clue 1, the Brit in the red house, which was mentioned above, is actually used fourth, and again two more times later on.

Note that I used many of the clues more than once.  Sometimes this is because they yielded extra information the second time around; sometimes it may have been because I didn't notice something the first time I applied a particular clue.  For this reason, just trying to use the list above may not work for you. You're going to have to come up with your own. 

In any case, I can tell you with confidence that the puzzle can be solved by this method (i.e., using a table like the one above), with some patience, some care, and about 30-60 minutes.

Have fun!

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