Einstein's Puzzle Who Owns the Fish: A Masterclass Foundation

The puzzle has 15 hints and asks—Who owns the fish? Hints on how five nationals in 5 colored houses own pets, smoke cigarettes and drink. Master reasoning.

Einstein's Reasoning Puzzle Who Owns the Fish

Hints:

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

Who owns the fish?

Time for you to find the answer is 45 minutes.

Hints for solving:

• Identify the primary object and its properties mentioned in the hints—a house in a fixed position is the primary object. The rest, including the "occupant" are its properties.
• List all values of each property to keep track.
• Select the hints that create direct assignment of property values to their rightful house positions (not occupants).
• To keep track of the assignments obtained, use a table of 5 columns (with house positions as headers) and of 5 rows, each for one property. Write a property value assigned in the cell against the property row and the house position column.
• Move on to the rest of the hints to identify patterns that would create more assignments.
• Repeat the process till the puzzle is solved.

Don't read more before solving (or failing to solve) the puzzle.

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Solution to Einstein's Puzzle Who Owns the Fish: A Primer on Preparations to Solving Reasoning Puzzles

Understanding the Problem: Identify Roles of Objects, Properties of Objects and their Values

• Identify the primary object: A house—identified by its position. Five houses are in positions 1, 2, 3, 4 and 5. House 1 is the FIRST, and House 3 is at the CENTER.
• Identify the properties of the primary object and its values: Directly related single property is house color:
• Color property values: red, blue, white, green, and yellow. These are the values of the property color of houses.
• Identify the secondary object as a property of a house and its values: An occupant of a house:
• Occupant: Norwegian, Brit, Swede, Dane and German.
• Identify the other properties and values of the properties:
• Pet: birds, cats, horses, dogs and the fish (not mentioned in any hint).
• Drinks: water, beer, milk, coffee, and tea.
• Smokes: Dunhill, Blends, Pallmall, Bluemasters, and Prince.

Reasoning:

• Unless the four pets in the hints get their owners, the owner of the missing fish cannot be identified.
• It makes sense that by assigning all 25 property values to their correct houses, the owner of the fish would automatically be discovered.
• To assign all property values quickly with no random attempts, all the 15 hints are to be used intelligently—not necessarily in the order in which those appear.
• Selection of the right hint to analyze is crucial for quick and assured solution without any extra steps. The way to achieve this forms the Strategies and Techniques for Solving Reasoning puzzles efficiently.

The occupant smokes, keeps pets and drinks—not the house. So, smokes, pets and drinks are first—the properties of an occupant.

As the occupant himself is a property of a house, by the ownership of an occupant, a house effectively owns each property of an owner living in the house as well. This is transitive ownership.

Thus, all properties are the properties of a house.

Example: by the Hint 7: "The owner of the yellow house smokes Dunhill", before knowing who occupies the yellow house, the smokes property value Dunhill gets assigned to the yellow house (usually when its position is known).

This type of reasoning puzzle also belongs to the class of assignment puzzles.

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Keeping Track of the Results of Analyzing the Hints

This is essential for

• Using the assigned values for subsequent analysis and eventual breakthroughs.

The easiest way is to use the 5 column by 5 row assignment table:

• The headers are the house positions 1, 2, 3, 4 and 5 and the row labels are the five properties.
• Row labels: Color, Occupant, Pet, Drinks and Smokes.

The assignment table has no assignments to start with:

Nothing is known at the start.

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Solution to Einstein's Puzzle Who Owns the Fish: Strategies and Techniques for Solving Reasoning Puzzles Efficiently

Objective: Discover assignment of property values—most at the earliest.

Strategy 1: Select the hint that creates an assignment outright—Direct assignment first

Scan all the hints to identify:

• Hint 9: The Norwegian lives in the first house.
• ⇨ Norwegian assigned to House 1 as the occupant—the first breakthrough assignment.

Also,

• Hint 8: The man living in the center house drinks milk.
• ⇨ "milk" as the Drinks is assigned to the person who lives in House 3. Though the occupant of House 3 is not known yet, the value of the property Drinks can be assigned to the House 3.

Strategy 2: Select the hint that refers to an already assigned property value—Link Search Technique

• Hint 14. The Norwegian lives next to the blue house.
• ⇨ With the Norwegian living in the first house, the color of the second house is now known as "blue"—a bonus assignment.
• Always look for this type of additional assignments by linked reference.

Strategy 3: Advanced technique: Identify a horizontally related pair of values of the same property—a Horizontal Bond

This promising reasoning puzzle solving technique often produces an unexpected breakthrough.

The positions of the two colors green and white are related to each other in a fixed way in Hint 4. This horizontal interlinking of values of the same property is the Horizontal Bond.

• Hint 4: The green house is on the immediate left of the white house.
• Focus on "green". Being on the left, it cannot occupy position 1. With position 2 already assigned to "blue", the "green-white" horizontal bond can be in:
• Positions 3 and 4, or
• Positions 4 and 5.

The assignment table till now captures this uncertainty of two possibilities of the "green-white" pair of colors. Probable location of "green" is position 3 or 4. This reduces the uncertainty of assigning "green" (and hence "white") to a large extent to two positions only.

Advantages of horizontal interlinking—a Horizontal Bond:

• It restricts the positions of the horizontal bond to TWO only (more than two won't be useful).
• One of the two possibilities will be eliminated later by another suitably chosen hint. This will create two assignments at once.

Search for a mention of "green"—use link search technique:

• Hint 5: The owner of the green house drinks coffee.

This creates the opportunity for using the second promising pattern—the Vertical Bond.

Strategy 4: Identify a promising relationship between two values of properties for the same position—the Vertical Bond—assignment by Conflict

The hint 5 creates such a relationship between color "green" and Drinks "coffee" of the same owner (in same position).

• Hint 5: The owner of the green house drinks coffee.

This pair of values "green-coffee" for the same house could have taken up position 3 or 4 (as "green" can only be in these two positions). But, position 3 is taken up by "milk" (conflicting with "coffee") and is eliminated for "green-coffee".

• Result: The owner of the green house cannot be in House 3. He can only be in House 4 and owner of the white house in House 5—a double assignment by Conflict created by a promising Vertical Bond of "green-coffee".
• A bonus assignment:
• ⇨ Drinks "coffee" for House 4.

This is a frequently used advanced technique for solving difficult reasoning puzzles.

A golden rule in solving reasoning puzzles:

After each assignment, use every opportunity for more simple assignments.

With Color row heavily assigned, grab the opportunities for further assignments using conflict in color first:

Strategy: With only 2 out of 5 places vacant, the Color row with most assignments at present, is the most promising row for getting a breakthrough.

• Hint 1: The Brit lives in the red house.
• ⇨ The only position for this vertical bond is House 3—"blue" blocks position 2 for "red" and position 1 already occupied by the Norwegian—a double conflict.
• ⇨ "Brit-red" for House 3.
• Hint 7: The owner of the yellow house smokes Dunhill.
• ⇨ The only color left "yellow" is automatically assigned to House 1 by default and bonus assignment "Dunhill" to House 1.
• ⇨ "yellow-Dunhill" for House 1.
• Hint 11: The man who keeps the horses lives next to the man who smokes Dunhill.
• ⇨ The pet of House 2 is assigned "horses" by link search technique finding mention of already assigned "Dunhill" for House 1.
• ⇨ The pet "horses" for House 2.

The assignment table:

Strategy 5: Breakthrough by a Cell Block Created by a Pair of Vertical Bonds

The pair of promising vertical bonds identified are:

• Hint 3: The Dane drinks tea.
• ⇨ The "Dane-tea" vertical bond can occupy the positions 2 or 5.
• Hint 12: The man who smokes Bluemasters drinks beer.
• ⇨ The "Bluemasters-beer" vertical bond can also occupy the same pair of positions 2 or 5.

When the pair of vertical bonds are merged, a new powerful pattern "Cell Block" is formed in positions 2 and 5 for Drinks property with pair of values limited to only "tea/beer".

How the Cell Block works:

• The Drinks property in positions 2 and 5 can have only two values—tea or beer.
• Though these two positions cannot yet be assigned a Drinks value with certainty, these cannot also have any Drinks value other than "tea" or "beer".
• Effectively, the "tea/beer" pair of values blocks the two positions 2 and 5 for Drinks property.

⇨ Critical Breakthrough:

• The fifth Drinks value "water" is automatically assigned to the only vacant House 1 by default.

The assignment table:

A Cell Block is a powerful pattern that provides a critical breakthrough with certainty.

Assignment of "water" to position 1 leads to the solution with no further difficulty.

Ultimate Solution: The owner of the pet "fish" discovered

The chain of hints leading to the solution:

• Hint 15: The man who smokes Blends has a neighbor who drinks water.
• ⇨ Smokes value "Blends" assigned to position 2.
• ⇨ Bonus assignments: "Bluemasters-beer" to position 5 and "Dane-tea" to position 2.
• Hint 13: The German smokes Prince.
• ⇨ The vertical bond "German-Prince" can only be assigned to House 4.
• ⇨ Bonus assignments: The fifth value "Pallmall" assigned to position 3 by default. The same way, "Swede" assigned by default to position 5.
• Hint 2: The Swede keeps dogs as pets.
• ⇨ Position 5 pets is assigned "dogs".
• Hint 6: The person who smokes Pallmall rears birds.
• ⇨ Pet "birds" assigned to position 3.

With "birds" taking up position 3, the only neighbor to the one smoking "Blends" remains in House 1 keeping "cats" by,

• Hint 10: The man who smokes Blends lives next to the one who keeps cats.
• ⇨ Pet "cats" assigned to House 1.

The pet still unknown is the "fish" kept by the occupant "German" living in House 4.

Answer: The German owns the fish.

Assignments for solution shown:

Before summing up, I will show you a second reasoning chain of hints to solve this popular reasoning puzzle.

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The Second way to Solve Einstein's Puzzle Who Owns the Fish Efficiently

Assumed: The strategies and techniques for efficient solution of reasoning puzzles already known.

Stage 1: Direct assignments first

Three direct assignments:

• Hint 9: The Norwegian lives in the first house.
• Hint 14: The Norwegian lives next to the blue house.
• Hint 8: The man living in the center house drinks milk.

I will avoid using the powerful horizontal bond relationship "green-white" at this stage and take a different path of reasoning.

Stage 1 assignment table:

Stage 2: Assignment by logical reasoning using vertical bond and horizontal bond

Color for position 1:

• Cannot be green or white because of Hint 4: "The green house is on the immediate left of the white house"—partial use of the horizontal bond.
• Cannot be red because of the conflict in Hint 1: "The Brit lives in the red house".
• Result: The fifth color "yellow" assigned to House 1 by default.

"yellow" appears in Hint 7—link reference:

• Hint 7: The owner of the yellow house smokes Dunhill.
• Result: Norwegian in the first house smokes Dunhill.

"Dunhill" appears in Hint 11—link reference:

• Hint 11: The man who keeps horses lives next to the man who smokes Dunhill.
• Result: The owner of the second house keeps "horses" as pets.

In both assignments link search used.

Stage 2 assignment table:

Stage 3: Assignments by using the Horizontal Bond fully and Breakthrough by the Cell Block

Assignments by using the Horizontal Bond of "green-white" pair of colors to its full potential.

• Hint 4: "The green house is on the left of the white house". The horizontal bond of "green-white" color pair can take positions 3-4 or 4-5.
• Hint 5: "The green house's owner drinks coffee". The vertical bond "green-coffee" conflicts with Drinks "milk" in position 3.
• Results: "green" to House 4, "white" to House 5, "coffee" to House 4, by default the fifth color "red" to House 3.
• Hint 1: "The Brit lives in the red house". The Brit lives in House 3 by link reference of "red".

The Cell Block created by merging a pair of vertical bonds in the same pair of positions:

• Hint 3: "The Dane drinks tea": Creates a vertical bond in positions 2 and 5.
• Hint 12: "The owner who smokes Bluemasters drinks beer": Creates a second vertical bond in the same positions 2 and 5.
• Merging the two vertical bonds creates the breakthrough pattern of Cell Block for Drinks values "tea/beer" in cells 2 and 5.
• Breakthrough assignment: The fifth value of Drinks "water" assigned to House 1 by default in the only vacant cell.

The last assignment of "water" is the critical breakthrough after which rest of the assignments are routine.

Stage 3 assignments:

Stage 4: A Series of Assignments Leads to the Owner of the Fish

After the breakthrough of "water" in position 1, the rest are routine.

• Hint 15: "The man who smokes Blends has a neighbor who drinks water."
• Results: With reference of "water" drinker as the neighbor, "Blends" gets position 2, "Bluemasters-beer" position 5, "Dane-tea" position 2.
• Hint 13: "The German smokes Prince."
• Results: "German-Prince" vertical bond gets position 4 (Smokes in position 5 taken up by Bluemasters).
• Hint 2: "The Swede keeps dogs as pets."
• Results: "Swede-dogs" vertical bond gets position 5.
• Hint 6: "The person who smokes Pall Mall rears birds."
• Results: "Pallmall-birds" vertical bond gets position 3.
• This eliminates the second neighbor of Blends smoking person leaving House 1 for the pet "cats".
• Hint 10: "The man who smokes Blends lives next to the one who keeps cats."
• Result: Pet "cats" is assigned to House 1.
• Four out of five pets are now known.

The single unknown pet owned by the German living in House 4 must be the fish, not mentioned in any hint.

Answer: The German owns the fish.

Ultimate assignments shown:

The second solution chain of hints explored the solution paths through the 15 hints thoroughly. This confirms:

To find the owner of the fish, all 24 other property values are to be assigned using all 15 hints.

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Consolidated Foundational Concepts, Strategies and Techniques for Solving Reasoning Puzzles with Ease

• Direct assignment first:
• Assign directly selecting the right hints first, not necessarily in the order these appear in the puzzle.
• Example: Hint 9 was the first to produce a direct assignment.
• Search and use hints that mention a value already assigned—Link Search Technique:
• Example: Hint 14 mentions the house color of the Norwegian already living in House 1 as ""blue". This is one of the most frequently used technique.
• A golden rule: grab all opportunities for additional assignments after every breakthrough.
• This is based on the concept: the more the assignments, the less the overall uncertainty and the easier the subsequent assignments.
• Horizontal Bond: This is a fixed relationship between two values in a row of the same property. This is a more promising pattern that invariably produces a breakthrough.
• Example: Hint 4 fixes the green house on the immediate left of the white house restricting positions for "green" to just two—position 3 and 4.
• Hint 5 eliminates position 3 for "green" by the conflict in Drinks "coffee" with "milk" in position 3.
• This created a triple assignment of "green-white-coffee".
• Use of a group of hints together: Often, two or more closely related hints together provide important breakthroughs.
• Example: Hint 4 and Hint 5 together provided the breakthrough with the help of the horizontal bond.
• Vertical Bond: Two values of separate properties of the same owner or position produces a relationship of vertical bonding between the two.
• The puzzle has nine vertical bonds: Hint 1 (Brit-red), Hint 2 (Swede-dogs), Hint 3 (Dane-tea), Hint 5 (green-coffee), Hint 6 (Pallmall-birds), Hint 7 (yellow-Dunhill), Hint 9 (Norwegian-House 1), Hint 12 (Bluemasters-beer), and Hint13 (German-Prince).
• The obvious advantage of a vertical bond is, once one value is assigned to a position, the second gets automatic assignment. In addition, a pair of vertical bonds may create a more powerful pattern of a Cell Block.
• Cell Block: When a pair of positions can take any of a pair of property values only, a cell block is created on the pair of positions.
• No value other than the blocking pair of property values can be assigned to the pair of positions.
• This creates a virtual assignment for two positions leaving other values for other positions.
• Example: "Dane-tea" and "Bluemasters-beer" both with probable positions 2 and 5 merge to form a cell block of "tea/beer" for positions 2 and 5.
• This cell block provides the all important critical breakthrough of default "water" assigned to leftover position 1.

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Comments

This is the most balanced reasoning puzzle treading on the fine line between being hard and not too hard to solve. It has a good mix of basic concepts and the introduction to more advanced concepts and techniques for solving reasoning puzzles.

To get a feel of how advanced the techniques can be, try solving a hard Sudoku puzzle, a special form of reasoning puzzle with 81 value cells.

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Enjoy puzzle solving while learning problem solving techniques.

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