# Elimination Grids

An **elimination grid** is a tool to solve complex logic problems. It is useful when there is lots of information and works by converting data into a visual grid.

In basic logic puzzles, the sort found on many math and reasoning tests, a grid like the one to the right can be useful to eliminate possibilities, based on provided information.

Some well-known examples of these sorts of puzzles are *Sudoku*, *Einstein's logic puzzle*, and *Cheryl's birthday*. For instance, in Sudoku puzzles, an elimination grid can be used to determine what the right entry is: eliminating numbers 1-9 that have already been used across, down, and in the same box.

#### Contents

## Setting Up the Grid

As the name suggests, when solving these types of problems, it is advisable (though not a necessity) to set up a grid. Typically, these puzzles have a description or a list of facts that describes a group of people or other objects. The puzzle then asks for the solver to uniquely match the people or objects to properties described in the list of facts.

Alice, Bethany, Carly, and Denise each have their own favorite flavor of ice cream (vanilla, chocolate, strawberry, or caramel) and their own favorite topping (sprinkles, nuts, M&Ms, or gummi bears). Right away, a grid can be set up wherein each characteristic (name, flavor, and topping) owns a set of rows that overlap each other characteristic as columns.

We are then given information about which person likes which flavor and topping. For instance, we might be told that Bethany likes chocolate ice cream, but doesn't like nuts. With this in mind, we can check Bethany off as the chocolate lover, while crossing off every other flavor for Bethany and every other name for chocolate. Furthermore, we can mark Bethany and the chocolate-lover off as candidates for liking nuts.

Next, we might be told that Carly doesn't like strawberry or vanilla ice cream, allowing us to eliminate her as candidates for those flavors, revealing that she must be the caramel lover.

With the remaining information, see if you can finish off the puzzle:

- Denise likes gummi bears as her favorite topping.
- The person who likes vanilla ice cream likes sprinkles as her topping.
- Alice doesn't like nuts or M&Ms.

In making a grid we use the columns and rows effectively so that every possibility in the puzzle is identified. Start with one of the easiest clues that gives you a simple fact matching two pieces of information together. For instance, you have the information "Sam has a red ball." Then find the row or column of your grid labeled "Sam," and follow it until you get to a square underneath the column or row labeled "red." Highlight that point to show that Sam and the red ball are connected. Let us try another example now.

Question:There are three people--X, Y, and Z--and they are a doctor, an engineer, and a teacher, not in the same order. They also have A, B, and C as their children, also not in the same order. Consider the following statements:

- X is the father A, but is not a doctor.
- Y can either be a doctor or a teacher.
- Z is an engineer, and is the father of B.
- C is the son of a doctor.
What is the profession of A's father?

Answer:This is how the grid will look:

Then step by step cross out squares that are ruled out from the hints provided. Harder problems have numerous properties, in which you have to fill out a much more complicated matrix.

After fixing their microverse battery, Rick, Morty, and Summer go out to eat ice cream. They each like one distinct flavor: Pink Peppercorn, Chocolate, or Green Tea.

- If Summer likes the Pink Peppercorn flavor and,
- Rick doesn't like the Green Tea flavor,

What flavor does Morty like?

###### No copyright infringement intended.

## Direct Elimination

Aaron, Calvin, David, and Peter each live in one of 4 adjacent townhouses in a row, each of a single color.

Each owns one pet and imbibes one kind of drink.

- Aaron owns the dog.
- The bird lives in the red house.
- Calvin lives in the blue house.
- David does not live in the red house.
- The cat lives where the milk drinker lives.
- Either the fish lives next to the cat
**or**the bird lives next to the coffee drinker. - If the dog lives in the green house, then the cat lives next to the blue house.
- If Peter owns the fish, then
**either**Calvin owns the bird**or**else David owns the cat. - The tea drinker lives two houses away from the coffee drinker.
- The red house resident drinks water if and only if the yellow house resident drinks milk.

Who owns the fish?

Note: Color of residences as shown in photograph have nothing to do with this problem. Also, any pet "owned" is presumed to live in the same place as the owner lives.

Four men--A, B, C, and D--are a sales executive, an accountant, a lawyer, and a doctor, **NOT** in the same order. They are accompanied by their children--P, Q, R, and S (also not in same order)--of which two are boys and two are girls, to a magic show. Each person is with one child.

- D and B came with their sons and none of them is a sales executive or a doctor.
- P is not the doctor's daughter.
- Q is not B's son.
- R's father is not an accountant.
- S is not C's daughter.

Based on this data, answer the questions below:

(1) Who is Q's father and what is he?

(2) If R and P are cousins, which two men are brothers?

Sandip, Tracy, Jamal, and Sheng are best friends who work together everyday at a banana stand. Since they are so close, their happiness on any given day depends on the happiness of the other three people on the previous day. Suppose they behave as follows:

- Sheng is happy today only if Tracy and Jamal were both happy yesterday.
- Jamal is happy today only if Sandip or Sheng (or both) were happy yesterday.
- Sandip enjoys watching Sheng cry, so Sandip is happy today only if Sheng was sad yesterday.
- Tracy is happy today only if Tracy was happy yesterday, meaning she has an independent streak.

Suppose that on day 1, all four of the friends are sad. After a few days, the friends reach a stable emotional state that repeats itself. What is the emotional state of each person in this repeating state?

Carmen has decided that the inside of her home needs to be refurnished, so she went out to buy some cans of paint (red, blue, green, yellow, indigo) so that she can redecorate five rooms over the course of the next five weeks. Every week she intends to paint a different room such that each room will be painted with a different color. Using the information given below, find the color that will be painted on the hallway.

The kitchen is to be painted red and Carmen will undertake this work earlier than the repainting of the hallway, a job she doesn't intend to do in week 4. The green paint isn't for the hallway.

The blue paint will be used the week before the bathroom is freshened up, but the week after the green paint.

The indigo paint will be used earlier than the color Carmen has chosen for her bedroom.

Carmen will be repainting her bedroom earlier than her living room.

Mr and Mrs Tan have five children: Alfred, Brenda, Charles, Darius, and Eric who are 18, 17, 16, 15 and 14 (not in this order). Each of the five has a distinct favorite color from the list (blue, green, yellow, orange, red), not necessarily in the same order.

Eric is younger than Alfred, with one of them liking yellow.

The 17-year-old likes blue and has an elder brother, Charles.

The orange lover and Brenda are two years apart.

Darius is a year older than the red lover who is in turn a year older than the green lover.

What is Alfred's favourite colour?

## Uniqueness Strategies

In this section we will be solving puzzles by applying the same concepts from K-level thinking. If you are not familiar with that concept, please see the main article first, K-level thinking.

When solving K-level puzzles, it is also possible to create gridlike tables and work our way through to obtain the answer. Because it is difficult to explain this without any question in mind, let us jump into a sample question:

Ed thinks of two positive integers, he then secrely told Fred one number and Greg another number separately. Ed then told both Fred and Greg that the sum of their numbers is 2 or 4. If Fred told Greg that he (Fred) didn't know Greg's number, can you deduce what Fred's number is?

For starters, we need to know what the possible values are for Fred's and Greg's numbers. Let \(F\) and \(G\) denote the possible values for Fred's number and Greg's number, respectively. Because we are told that the sum of their numbers is 2 or 4, it follows that \(F+G = 2\) or \(F+G=4\) must be fulfilled. So \(F\) can take values 1, 2, and 3, and similarly, \(G\) can take values 1, 2, and 3 as well. Let us construct the outline of the grid first:

1 | 2 | 3 | |

Fred | |||

Greg |

Now if Fred's number is 3, then the sum of their numbers must exceed 2, so the only possible number for Greg to have is 1. However, this is impossible because Fred already knows his number and if his number is indeed 3, then he can automatically deduce the fact that Greg's number is 1, which contradicts the fact that Fred says he doesn't know about Greg's number. Hence Fred can't have the value of 3. So, let's mark an X in the table as follows:

1 | 2 | 3 | |

Fred | \(\times\) | ||

Greg |

Similarly if Fred's number is 2, then the sum of their numbers must exceed 2, so the only possible number for Greg to have is 2. However, this is impossible because Fred already knows his number and if his number is indeed 2, then he can automatically deduce the fact that Greg's number is 2, which contradicts the fact that Fred says he doesn't know about Greg's number. Hence Fred can't have the value of 2. So, let's add another X to the table:

1 | 2 | 3 | |

Fred | \(\times \) | \(\times\) | |

Greg |

From here, the only possible solution for Fred's number is \(F=1\). Hence the following table:

1 | 2 | 3 | |

Fred | \(\checkmark \) | \(\times \) | \(\times\) |

Greg |

Now is it possible for Fred to figure out what Greg's number actually is if Greg makes the additional claim that his number is not the same as Fred's? Yes it is! For starters, Greg's number can't be 1 because his number is different from Fred's. Hence the following table:

1 | 2 | 3 | |

Fred | \(\checkmark \) | \(\times \) | \(\times\) |

Greg | \(\times\) |

Because the sum of Fred's and Greg's numbers is either 2 or 4 only, Greg's number must be either \(2-1 = 1\) or \(4-1= 3\) only. Thus Greg's number can't be 2, which gives us 4 as Greg's number, and we have the following table:

1 | 2 | 3 | |

Fred | \(\checkmark \) | \(\times \) | \(\times\) |

Greg | \(\times\) | \(\times \) | \(\checkmark\) |

So we have finally deduced what their numbers are!

Now, doing the same kind of reasoning as above, can you solve the following example?

Alice thinks of two positive integers, secretly tells Bob one, and secretly tells Charlie the other. Then the following conversation occurs:

Alice: "The product of my numbers was 8. Or maybe it was 16."

Bob (to Charlie): "I don't know your number."

Charlie (to Bob): "I don't know your number."

Bob (to Charlie): "I don't know your number."

Charlie (to Bob): "I don't know your number."

Bob (to Charlie): "I know your number!"

What was Charlie's number?

Difficulty increases with the number of rows and columns. It could also be unclear how one should choose to present the data and follow the argument. An excellent example is Josh Silverman's solution to the viral problem Cheryl's Birthday.

Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates:

May 15 | May 16 | May 19 |

June 17 | June 18 | |

July 14 | July 16 | |

August 14 | August 15 | August 17 |

Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively.

Albert: I don't know when Cheryl's birthday is, but I know that Bernard does not know too.

Bernard: At first I don't know when Cheryl's birthday is, but I know now.

Albert: Then I also know when Cheryl's birthday is.

So, when is Cheryl's birthday?

###### Image credit: Wikipedia Tatiana Sapateiro

## Example Problems

Four friends have been identified as suspects in a car theft. One of them really stole the car, and the following are the statements they made to the investigating authorities:

Thomas said, “Simon did it.”

Kim said, “I did not do it.”

Simon said, “Harper did it.”

Harper said, “Simon lied when he said that I did it.”

If the authorities also know that exactly one of the four suspects is telling the truth, who did it?

Set up a truth table for all the possibilities as follows:

Then the first four columns show the suspects and the scenarios of who possibly lied and who possibly told the truth. The second four columns show what can be understood by assuming that the respective suspect is the only one who told the truth.

Thomas is telling the truth: This results in both Simon and Kim being guilty, which can't be.

Kim is telling the truth: Since Simon and Harper contradict each other, one of them must be telling the truth. But since we are assuming only Kim is telling the truth in this scenario, this cannot be the case.

Simon is telling the truth: This results in both Harper and Kim being guilty, which can't be.

Harper is telling the truth: This means that Kim is the criminal, which is the correct answer. \(_\square\)

Let's try a harder problem.

Saeed and Grover are two mathematicians, and \(f_1\) and \(f_2\) are two integers such that each is greater than \(1\) and their sum is not greater than \(100.\) Saeed is told the total \(f_1+f_2\), and Grover is told the product \(f_1\times f_2\).

Both Saeed and Grover have the above information and they are both perfect in their logic. Then the following conversation occurs:

Grover says, "I do not know \(f_1\) and \(f_2\)."

Saeed says, "I know that you don't know."

Grover says, "Now I also know \(f_1\) and \(f_2\)!"

Saeed says, "So do I."

What are the values \(f_1\) and \(f_2\)?

After Grover’s first statement, we know that there is not a unique decomposition of the product \(f_1 \times f_2\) in \(f_{1}, f_{2} > 1\) with \(f_{1} + f_{2} ≤ 100\). From this we conclude:

- \(f_{1}\times f_{2}\) cannot be the product of two distinct primes, for then Grover could deduce the numbers.
- \(f_{1}\times f_{2}\) cannot be the cube of a prime, such as \(3^{3} = 27\), for then \(3×9 =27\) would be a unique factorization; or the fourth power of a prime.
- \(f_{1}\times f_{2}\) does not contain a prime factor greater than \(50\).
From Saeed's first statement, we learn that all splittings of the sum in \(f_{1}\) and \(f_{2}\) satisfy the above conditions. This implies that the sum is odd (for Goldbach’s conjecture that all even numbers except \(2\) and \(4\) are the sum of two primes, holds below \(100\)). Finally the sum is not of the form \(f_{1}+f_{2}=q+2,\) where \(q\) is a prime number and the sum is less than \(50\) (since \(53\) is the smallest prime greater than \(50\)).

Therefore the list of plausible sums are:

\[f_{1}+f_{2}\in S=\{11, 17, 23, 27, 29, 35, 37, 41, 47, 51, 53\}.\]

Grovers statement "Now I also know \(f_1\) and \(f_2\)!" implies he now knows that sum is one of the values listed above. If this enables him to deduce \(f_{1}\) and \(f_{2}\), then, of the eligible factorizations of \(f_{1}\) times \(f_{2}\), there must be precisely one for which the sum of the factors is in the list. Now, we make the following observation:

- Are the numbers \({ f }_{ 1 }\times { f }_{ 2 }={ 2 }^{ k }\cdot q\) with \(q\) prime, \(k>1\) and \(2^{k}+q\in f_{1}+f_{2}\) eligible (because all decompositions of \(f_{1}\times f_{2}\) have an odd sum)?
Saeed's final statement "So do I" implies that the sum \(f_{1}+f_{2}\) must admit exactly one eligible product. We can observe that \(11, 23, 27, 29, 35, 37, 41, 47, 51, 53\) all admit at least two eligible products. \(17\) is the only number that admits only one eligible product, where \(k=2\) and \(q=13\). Thus the values of \(f_{1}\) and \(f_{2}\) are \(4\) and \(13\). \(_\square\)

## Sample problems

Pop Quiz! Submit your answers, in order, to Questions 1-4.

**Q1.** Which is the first question where c) is the correct answer?

a) Q3

b) Q4

c) Q1

d) Q2

**Q2.** Which is the first question where a) is the correct answer?

a) Q4

b) Q2

c) Q3

d) Q1

**Q3.** Which is the first question where d) is the correct answer?

a) Q1

b) Q2

c) Q4

d) Q3

**Q4.** Which is the first question where b) is the correct answer?

a) Q2

b) Q4

c) Q3

d) Q1

\[\]
**Problems from Brilliant:**

- https://brilliant.org/problems/four-townhouses-logic-puzzle-2/
- https://brilliant.org/problems/relationship-problems/
- https://brilliant.org/problems/its-always-happy-at-the-banana-stand/
- https://brilliant.org/problems/who-is-sitting-where/
- https://brilliant.org/problems/an-algebra-problem-by-lakkoju-dilip/
- https://brilliant.org/problems/birthday-girl/
- https://brilliant.org/problems/house-number/
- https://brilliant.org/problems/question-1-or-2-or-3-or-wait-what/
- https://brilliant.org/problems/guess-the-bday/
- https://brilliant.org/problems/logical-maths-is-not-the-only-maths/
- https://brilliant.org/problems/the-amazing-race/
- https://brilliant.org/problems/men-children-and-professions-confusing/
- https://brilliant.org/problems/partitioned-paintings/
- https://brilliant.org/problems/the-maze-runners/
- https://brilliant.org/problems/my-kids-love-colors/
- https://brilliant.org/problems/celebrating-anniversaries/
- https://brilliant.org/problems/correct-correspondence/

\[\]
**Reference from other sites:**

EASY

21. http://www.braingle.com/brainteasers/49299/snack-time-at-rachels.html

22. http://www.braingle.com/brainteasers/48968/multicolour-doors-easy.html

23. http://www.braingle.com/brainteasers/49107/getting-a-pet.html

24. http://www.braingle.com/brainteasers/44337/computer-mix-up.html

25. http://www.braingle.com/brainteasers/45084/let-there-be-cake.html

26. http://www.braingle.com/brainteasers/45005/video-games.html

27. http://www.braingle.com/brainteasers/49973/gone-fishing.html

28. http://www.braingle.com/brainteasers/21187/colors.html

29. http://www.braingle.com/brainteasers/49098/restaurant.html

30. http://www.braingle.com/brainteasers/23198/people-and-pepsi.html

31. http://www.braingle.com/brainteasers/42926/the-dance-off.html

HARD

32. http://www.braingle.com/brainteasers/24178/a-day-of-shopping.html

33. http://www.braingle.com/brainteasers/23446/a-band-of-musicians.html

34. http://www.braingle.com/brainteasers/27316/book-reports-who-read-what.html

35. http://www.braingle.com/brainteasers/49444/jazz-band-solos.html

36. http://www.braingle.com/brainteasers/35800/diet-time.html

37. http://www.braingle.com/brainteasers/23378/well-used-truck.html

38. http://www.braingle.com/brainteasers/46013/clowns-in-the-subway.html

39. http://www.braingle.com/brainteasers/44985/five-cousins.html

40. http://www.braingle.com/brainteasers/24243/olympic-swim-team.html

**Cite as:**Elimination Grids.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/elimination-grids/