Archit and Ayush are walking around on the set of points  $(x,y)$  for all integers  $-1\leq x,y\leq1$ . Archit starts at  $(1,1)$  and Ayush starts at  $(1,0)$ . Each second, they move to another point in the set chosen uniformly at random among the points with distance  $1$  away from them. If the probability that Archit goes to the point  $(0,0)$  strictly before Ayush does can be expressed as  $\frac{m}{n}$  for relatively prime positive integers  $m,n$ , compute  $m+n$ .

*2020 CCA Math Bonanza Lightning Round #3.2*

Answer: 7


We build a tower of  $2\times 1$  dominoes in the following way. First, we place  $55$  dominoes on the table such that they cover a  $10\times 11$  rectangle; this is the first story of the tower. We then build every new level with  $55$  domioes above the exact same  $10\times 11$  rectangle. The tower is called *stable* if for every non-lattice point of the  $10\times 11$  rectangle, we can find a domino that has an inner point above it. How many stories is the lowest *stable* tower?

Answer: 5


Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard?

SolutionWith probability  $1$  the number of corners covered is  $0$ ,  $1$ , or  $2$  for example by the diameter of a square being  $\sqrt{2}$  so it suffices to compute the probability that the square covers  $2$  corners. This is due to the fact that density implies the mean number of captured corners is  $1$ . For the lattice with offset angle  $\theta \in \left[0,\frac{\pi}{2}\right]$  consider placing a lattice uniformly randomly on to it and in particular say without loss of generality consider the square which covers the horizontal lattice midpoint  $\left(\frac{1}{2},0 \right)$ . The locus of such midpoint locations so that the square captures the  $2$  points  $(0,0),(1,0)$ , is a rectangle. As capturing horizontally adjacent points does not occur when capturing vertically adjacent points one computes twice that probability as  $\frac{4}{\pi} \int_0^{\frac{\pi}{2}} (1-\sin(\theta))(1-\cos(\theta)) d\theta=\boxed{\frac{2(\pi-3)}{\pi}}$
[asy]
draw((0,0)--(80,40));
draw((0,0)--(-40,80));
draw((80,40)--(40,120));
draw((-40,80)--(40,120));
draw((80,40)--(-20,40));
draw((-40,80)--(60,80));
draw((32*sqrt(5),16*sqrt(5))--(-8*sqrt(5),16*sqrt(5)));
draw((40+8*sqrt(5),120-16*sqrt(5))--(40-32*sqrt(5),120-16*sqrt(5)));
draw((12*sqrt(5),16*sqrt(5))--(12*sqrt(5)+2*(40-16*sqrt(5)),16*sqrt(5)+(40-16*sqrt(5))));
draw((12*sqrt(5),16*sqrt(5))--(12*sqrt(5)-(80-16*sqrt(5))/2,16*sqrt(5)+(80-16*sqrt(5))));
draw((40-12*sqrt(5),120-16*sqrt(5))--(40-12*sqrt(5)+(120-16*sqrt(5)-40)/2,120-16*sqrt(5)-(120-16*sqrt(5)-40)));
draw((40-12*sqrt(5),120-16*sqrt(5))--(40-12*sqrt(5)-2*(120-16*sqrt(5)-80),120-16*sqrt(5)-(120-16*sqrt(5)-80)));
[/asy]

Answer: 2-\frac{6}{\pi}


There are  $5$  students on a team for a math competition. The math competition has  $5$  subject tests. Each student on the team must choose  $2$  distinct tests, and each test must be taken by exactly two people. In how many ways can this be done?

Answer: 2040


In the diagram below  $ \angle CAB, \angle CBD$ , and  $\angle CDE$  are all right angles with side lengths  $AC = 3$ ,  $BC = 5$ ,  $BD = 12$ , and  $DE = 84$ . The distance from point  $E$  to the line  $AB$  can be expressed as the ratio of two relatively prime positive integers,  $m$  and  $n$ . Find  $m + n$ .

[asy]
size(300);
defaultpen(linewidth(0.8));
draw(origin--(3,0)--(0,4)--cycle^^(0,4)--(6,8)--(3,0)--(30,-4)--(6,8));
label(" $A$ ",origin,SW);
label(" $B$ ",(0,4),dir(160));
label(" $C$ ",(3,0),S);
label(" $D$ ",(6,8),dir(80));
label(" $E$ ",(30,-4),E);[/asy]

Answer: 5393


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Eric`, `Peter`
 - Everyone has something unique for lunch: `stew`, `spaghetti`, `grilled cheese`, `pizza`, `stir fry`
 - Each person has a favorite color: `green`, `blue`, `yellow`, `red`, `white`
 - The people are of nationalities: `swede`, `brit`, `dane`, `german`, `norwegian`
 - People have unique favorite book genres: `fantasy`, `biography`, `romance`, `mystery`, `science fiction`
 - Each mother is accompanied by their child: `Timothy`, `Samantha`, `Meredith`, `Fred`, `Bella`

## Clues:
1. The Swedish person is the person's child is named Bella.
2. There is one house between the person who loves science fiction books and the person who loves the stew.
3. The person who is a pizza lover is the person's child is named Fred.
4. The person who loves white is not in the fourth house.
5. The person who loves yellow is Peter.
6. The person who loves eating grilled cheese is directly left of the person's child is named Meredith.
7. The German is the person who loves the spaghetti eater.
8. The person who loves romance books is the person who loves the stew.
9. The person's child is named Samantha is directly left of the German.
10. Alice is the Swedish person.
11. Arnold is the person whose favorite color is green.
12. Arnold is in the third house.
13. The person's child is named Meredith is the person who loves biography books.
14. The Norwegian is the person whose favorite color is red.
15. The person who loves romance books is not in the third house.
16. The person who loves the spaghetti eater is Eric.
17. The Dane is somewhere to the left of the person who loves yellow.
18. The person who loves mystery books is directly left of the person whose favorite color is red.
19. The person's child is named Fred is not in the third house.
 What is the value of attribute House for the person whose attribute BookGenre is fantasy? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Eric`
 - Each person has a unique birthday month: `sept`, `feb`, `april`, `jan`
 - The mothers' names in different houses are unique: `Aniya`, `Kailyn`, `Holly`, `Janelle`
 - The people keep unique animals: `horse`, `bird`, `cat`, `fish`
 - Each person has a favorite color: `green`, `white`, `yellow`, `red`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `soccer`

## Clues:
1. The bird keeper is The person whose mother's name is Kailyn.
2. The fish enthusiast is not in the fourth house.
3. The person who loves white is Arnold.
4. The person who keeps horses is Peter.
5. The person who keeps horses is not in the first house.
6. The person whose mother's name is Kailyn is Arnold.
7. The person whose birthday is in January is in the first house.
8. The person whose birthday is in April is somewhere to the left of the person whose birthday is in September.
9. Alice is the person whose birthday is in September.
10. The person whose mother's name is Aniya and the person whose birthday is in September are next to each other.
11. Peter is directly left of the person whose favorite color is red.
12. There is one house between the person whose birthday is in April and the person who loves soccer.
13. The fish enthusiast is the person whose birthday is in September.
14. The person who loves swimming is The person whose mother's name is Holly.
15. The person whose favorite color is green is the person who loves swimming.
16. The person who loves yellow is the person who loves basketball.
 What is the value of attribute House for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Carol`, `Eric`, `Alice`, `Arnold`
 - Each person has a unique favorite drink: `coffee`, `milk`, `water`, `root beer`, `tea`, `boba tea`
 - They all have a unique favorite flower: `tulips`, `lilies`, `daffodils`, `iris`, `roses`, `carnations`
 - Each person has an occupation: `doctor`, `artist`, `nurse`, `engineer`, `teacher`, `lawyer`
 - Each person has a unique hobby: `woodworking`, `photography`, `cooking`, `painting`, `knitting`, `gardening`

## Clues:
1. Alice is somewhere to the right of the person who loves cooking.
2. Peter is not in the second house.
3. The person who loves the rose bouquet is directly left of the person who likes milk.
4. Bob is the person who is a lawyer.
5. The person who loves a carnations arrangement is somewhere to the right of Bob.
6. The person who enjoys knitting is somewhere to the left of the coffee drinker.
7. The person who is a nurse is directly left of the person who is a doctor.
8. Eric is the photography enthusiast.
9. The person who loves the boquet of lilies is the woodworking hobbyist.
10. The person who paints as a hobby is in the first house.
11. The person who loves a carnations arrangement is the root beer lover.
12. The person who is an artist is the person who likes milk.
13. The person who loves a bouquet of daffodils is in the first house.
14. The photography enthusiast is somewhere to the right of the boba tea drinker.
15. The person who loves the rose bouquet is somewhere to the right of the person who is a doctor.
16. The tea drinker is the person who is an engineer.
17. Arnold is directly left of Carol.
18. The coffee drinker is the person who is a nurse.
19. The person who loves the vase of tulips is the person who loves cooking.
20. The person who is a teacher is somewhere to the right of the root beer lover.
21. The person who enjoys gardening is Carol.
 What is the value of attribute Drink for the person whose attribute Occupation is doctor? Please reason step by step, and put your final answer within \boxed{}

Answer: root beer


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Eric`, `Carol`, `Bob`
 - Each person has a unique favorite drink: `water`, `milk`, `root beer`, `tea`, `coffee`, `boba tea`
 - Each person has an occupation: `artist`, `doctor`, `teacher`, `engineer`, `nurse`, `lawyer`

## Clues:
1. The person who is a lawyer is somewhere to the left of the one who only drinks water.
2. The boba tea drinker is Alice.
3. Arnold is directly left of the coffee drinker.
4. Peter and the person who is an artist are next to each other.
5. Arnold is the person who is a nurse.
6. Carol is not in the first house.
7. The person who is a teacher is somewhere to the right of the person who is a doctor.
8. The root beer lover is somewhere to the left of the tea drinker.
9. The person who is a lawyer is Carol.
10. Bob is the one who only drinks water.
11. Eric is not in the fourth house.
12. The tea drinker is not in the sixth house.
13. The person who is a teacher is the root beer lover.
14. The one who only drinks water is somewhere to the left of Alice.
 What is the value of attribute Occupation for the person whose attribute Drink is root beer? Please reason step by step, and put your final answer within \boxed{}

Answer: teacher


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`
 - The people keep unique animals: `cat`, `horse`, `bird`
 - Each person has a favorite color: `yellow`, `white`, `red`
 - The people are of nationalities: `brit`, `swede`, `dane`
 - They all have a unique favorite flower: `daffodils`, `lilies`, `carnations`

## Clues:
1. The bird keeper and Eric are next to each other.
2. The person whose favorite color is red is the Dane.
3. The Dane is the person who loves a carnations arrangement.
4. The person who loves the boquet of lilies is the bird keeper.
5. The Dane is somewhere to the left of the person who loves a bouquet of daffodils.
6. The British person is somewhere to the left of the person who loves yellow.
7. Arnold is the person who keeps horses.
8. Eric is the Swedish person.
 What is the value of attribute Nationality for the person whose attribute Color is yellow? Please reason step by step, and put your final answer within \boxed{}

Answer: swede


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Eric`
 - People have unique favorite music genres: `rock`, `jazz`, `classical`, `pop`
 - People have unique favorite book genres: `mystery`, `science fiction`, `fantasy`, `romance`
 - Everyone has a unique favorite cigar: `prince`, `blue master`, `pall mall`, `dunhill`
 - Each person has a unique birthday month: `april`, `jan`, `feb`, `sept`
 - Each person has a unique favorite drink: `water`, `tea`, `milk`, `coffee`

## Clues:
1. The person whose birthday is in September is the one who only drinks water.
2. The person who loves rock music and the person who loves jazz music are next to each other.
3. The person who loves mystery books is the person who smokes Blue Master.
4. Alice is the person who loves classical music.
5. The person who loves jazz music is the person who likes milk.
6. The person who loves jazz music is directly left of the person who loves romance books.
7. The person who loves rock music is not in the fourth house.
8. The person whose birthday is in February is Arnold.
9. The Prince smoker and the person who loves classical music are next to each other.
10. The coffee drinker is the person who loves science fiction books.
11. The person who loves mystery books is the person whose birthday is in September.
12. Peter is the person who loves jazz music.
13. The Dunhill smoker is the person whose birthday is in April.
14. The person who loves mystery books is in the second house.
 What is the value of attribute Birthday for the person whose attribute Drink is milk? Please reason step by step, and put your final answer within \boxed{}

Answer: jan


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`
 - People have unique favorite sports: `tennis`, `basketball`, `soccer`
 - Each person has an occupation: `engineer`, `teacher`, `doctor`
 - People own unique car models: `tesla model 3`, `ford f150`, `toyota camry`
 - Everyone has a unique favorite cigar: `blue master`, `prince`, `pall mall`

## Clues:
1. The person who owns a Ford F-150 is not in the first house.
2. The person who owns a Tesla Model 3 is the person who is a teacher.
3. The person who loves basketball is in the first house.
4. The person partial to Pall Mall is the person who is a teacher.
5. Arnold is in the second house.
6. Arnold is somewhere to the right of the Prince smoker.
7. The person who loves tennis is Arnold.
8. The person who is an engineer is Arnold.
9. Eric is the person partial to Pall Mall.
 What is the value of attribute FavoriteSport for the person whose attribute Cigar is blue master? Please reason step by step, and put your final answer within \boxed{}

Answer: tennis


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Peter`, `Arnold`, `Bob`, `Eric`, `Alice`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `oneplus 9`, `huawei p50`, `samsung galaxy s21`, `xiaomi mi 11`
 - People have unique hair colors: `gray`, `auburn`, `red`, `brown`, `black`, `blonde`
 - Each person has a unique favorite drink: `coffee`, `water`, `root beer`, `tea`, `milk`, `boba tea`
 - People have unique favorite music genres: `classical`, `jazz`, `rock`, `country`, `hip hop`, `pop`

## Clues:
1. Alice is the person who uses a Huawei P50.
2. The person who uses a Xiaomi Mi 11 is somewhere to the left of the one who only drinks water.
3. The one who only drinks water is somewhere to the left of Bob.
4. Carol is the person who has gray hair.
5. Eric is the tea drinker.
6. The person who has black hair is not in the sixth house.
7. The person who uses a Samsung Galaxy S21 is directly left of the person who loves classical music.
8. The person who uses a OnePlus 9 is not in the third house.
9. Alice is somewhere to the right of Carol.
10. The person who loves jazz music is Alice.
11. The person who loves jazz music is the boba tea drinker.
12. There is one house between the person who uses a OnePlus 9 and Eric.
13. The person who has brown hair is the person who loves hip-hop music.
14. The person who loves pop music is not in the first house.
15. The root beer lover is in the second house.
16. Eric is the person who uses an iPhone 13.
17. The person who has red hair is the person who uses an iPhone 13.
18. The person who has red hair and the coffee drinker are next to each other.
19. Peter is in the first house.
20. Arnold is somewhere to the right of the person who has auburn hair.
21. The person who has gray hair is not in the second house.
22. The person who loves country music is in the fifth house.
 What is the value of attribute House for the person whose attribute PhoneModel is huawei p50? Please reason step by step, and put your final answer within \boxed{}

Answer: 6


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Peter`, `Eric`, `Carol`, `Arnold`
 - People own unique car models: `toyota camry`, `ford f150`, `bmw 3 series`, `tesla model 3`, `chevrolet silverado`, `honda civic`
 - Each person has a unique hobby: `knitting`, `gardening`, `woodworking`, `painting`, `photography`, `cooking`
 - Each person has a unique favorite drink: `coffee`, `root beer`, `milk`, `water`, `tea`, `boba tea`

## Clues:
1. There is one house between the coffee drinker and the woodworking hobbyist.
2. The person who enjoys knitting is in the first house.
3. There are two houses between the person who loves cooking and the person who likes milk.
4. The person who paints as a hobby is the person who owns a Toyota Camry.
5. Arnold is the tea drinker.
6. There is one house between the person who owns a Chevrolet Silverado and the person who owns a BMW 3 Series.
7. The woodworking hobbyist is somewhere to the right of Carol.
8. The person who paints as a hobby is not in the fourth house.
9. The boba tea drinker is in the third house.
10. Bob is somewhere to the left of the person who likes milk.
11. The person who owns a Chevrolet Silverado is not in the sixth house.
12. The person who owns a Toyota Camry is the root beer lover.
13. The person who owns a Chevrolet Silverado is Alice.
14. Peter is the person who owns a Toyota Camry.
15. The root beer lover is not in the sixth house.
16. The person who owns a Tesla Model 3 is not in the first house.
17. The person who enjoys gardening is Eric.
18. The tea drinker is somewhere to the left of the one who only drinks water.
19. The person who owns a Honda Civic and the boba tea drinker are next to each other.
20. Bob is in the third house.
 What is the value of attribute Hobby for the person whose attribute CarModel is bmw 3 series? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Bob`, `Peter`, `Arnold`
 - People have unique favorite book genres: `fantasy`, `mystery`, `science fiction`, `biography`, `romance`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Kailyn`, `Janelle`
 - People use unique phone models: `oneplus 9`, `iphone 13`, `samsung galaxy s21`, `google pixel 6`, `huawei p50`
 - The people are of nationalities: `dane`, `brit`, `german`, `swede`, `norwegian`
 - Each person lives in a unique style of house: `ranch`, `modern`, `craftsman`, `victorian`, `colonial`

## Clues:
1. The person in a ranch-style home is somewhere to the right of the person residing in a Victorian house.
2. The Dane and the person who uses a Samsung Galaxy S21 are next to each other.
3. The person who uses a Google Pixel 6 is not in the second house.
4. There are two houses between the person who loves fantasy books and the person in a modern-style house.
5. The person whose mother's name is Kailyn is the Swedish person.
6. The Swedish person is Arnold.
7. The British person is somewhere to the left of the person who uses an iPhone 13.
8. The Dane is The person whose mother's name is Penny.
9. The person who uses a Samsung Galaxy S21 is the Norwegian.
10. The Norwegian is The person whose mother's name is Holly.
11. The person who loves fantasy books is Bob.
12. Alice is directly left of the Swedish person.
13. The person who loves romance books is Arnold.
14. The person who loves science fiction books is Peter.
15. The person whose mother's name is Aniya is directly left of the person who loves science fiction books.
16. The person who loves science fiction books is the person who uses a OnePlus 9.
17. The person residing in a Victorian house is the person who loves biography books.
18. The person who loves fantasy books is somewhere to the left of the German.
19. The person who uses an iPhone 13 is the person in a Craftsman-style house.
 What is the value of attribute Mother for the person whose attribute HouseStyle is modern? Please reason step by step, and put your final answer within \boxed{}

Answer: Holly


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `cherry`
 - People have unique hair colors: `black`, `blonde`, `brown`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `samsung galaxy s21`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `victorian`
 - People have unique heights: `short`, `average`, `very short`

## Clues:
1. The person living in a colonial-style house is the person who is very short.
2. The person who has black hair is Eric.
3. The person who uses a Samsung Galaxy S21 is not in the second house.
4. The person who uses a Google Pixel 6 is somewhere to the right of the person who likes Cherry smoothies.
5. Peter is the person who uses a Google Pixel 6.
6. The person who has brown hair is the person residing in a Victorian house.
7. The person residing in a Victorian house is the person who likes Cherry smoothies.
8. The person who has blonde hair is in the third house.
9. The Watermelon smoothie lover is directly left of the person in a ranch-style home.
10. The person who has an average height is the Desert smoothie lover.
 What is the value of attribute PhoneModel for the person whose attribute HouseStyle is victorian? Please reason step by step, and put your final answer within \boxed{}

Answer: samsung galaxy s21


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `colonial`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`
 - People have unique favorite book genres: `romance`, `mystery`, `science fiction`
 - People own unique car models: `ford f150`, `toyota camry`, `tesla model 3`
 - Each person has a unique level of education: `associate`, `high school`, `bachelor`

## Clues:
1. The person who loves science fiction books is in the second house.
2. The person with a high school diploma is the person residing in a Victorian house.
3. The person who loves a bouquet of daffodils is in the second house.
4. The person with an associate's degree is the person in a ranch-style home.
5. The person who owns a Tesla Model 3 is not in the third house.
6. The person who loves the boquet of lilies is the person residing in a Victorian house.
7. Arnold is the person in a ranch-style home.
8. The person who owns a Tesla Model 3 and Eric are next to each other.
9. The person who owns a Ford F-150 is the person who loves romance books.
10. The person who owns a Tesla Model 3 is the person residing in a Victorian house.
 What is the value of attribute HouseStyle for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: victorian


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Bob`, `Peter`, `Alice`, `Carol`
 - People have unique heights: `very tall`, `tall`, `average`, `short`, `super tall`, `very short`
 - Each person has a unique birthday month: `sept`, `may`, `jan`, `april`, `feb`, `mar`
 - People have unique hair colors: `blonde`, `red`, `black`, `auburn`, `gray`, `brown`
 - People have unique favorite book genres: `science fiction`, `romance`, `biography`, `mystery`, `fantasy`, `historical fiction`
 - Each person has a unique type of pet: `bird`, `hamster`, `fish`, `dog`, `cat`, `rabbit`

## Clues:
1. The person with an aquarium of fish is the person who loves mystery books.
2. The person who has black hair is the person who owns a dog.
3. The person who has gray hair is the person who loves science fiction books.
4. The person who loves historical fiction books is somewhere to the left of the person who has black hair.
5. The person who loves historical fiction books is directly left of the person who owns a rabbit.
6. The person whose birthday is in February is the person who owns a dog.
7. Bob is the person who keeps a pet bird.
8. The person who keeps a pet bird is the person who loves romance books.
9. Alice is directly left of the person who loves fantasy books.
10. The person who owns a rabbit is directly left of the person who is very short.
11. The person who is short is somewhere to the left of the person who is super tall.
12. The person who has an average height is in the sixth house.
13. The person who has red hair is not in the first house.
14. The person who is very tall is somewhere to the right of the person who is super tall.
15. There are two houses between the person who loves romance books and Eric.
16. The person with an aquarium of fish is in the sixth house.
17. The person who has blonde hair is the person who loves historical fiction books.
18. The person who is tall is somewhere to the right of the person whose birthday is in February.
19. The person who has auburn hair is Carol.
20. The person with an aquarium of fish is the person whose birthday is in March.
21. The person who loves science fiction books is directly left of the person with an aquarium of fish.
22. The person whose birthday is in September is not in the first house.
23. The person who loves biography books is directly left of the person who has a cat.
24. The person whose birthday is in January is the person who loves science fiction books.
25. Arnold is the person whose birthday is in May.
 What is the value of attribute House for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Eric`, `Peter`, `Arnold`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `huawei p50`, `google pixel 6`, `oneplus 9`
 - People have unique favorite sports: `soccer`, `swimming`, `baseball`, `basketball`, `tennis`
 - Everyone has a favorite smoothie: `cherry`, `dragonfruit`, `lime`, `desert`, `watermelon`
 - Each person prefers a unique type of vacation: `city`, `cruise`, `beach`, `mountain`, `camping`
 - The people keep unique animals: `horse`, `fish`, `cat`, `dog`, `bird`

## Clues:
1. Peter is somewhere to the right of the person who uses an iPhone 13.
2. The person who loves tennis is directly left of the person who uses a Google Pixel 6.
3. There is one house between the Watermelon smoothie lover and the person who loves beach vacations.
4. The person who loves baseball is in the fourth house.
5. The dog owner is Peter.
6. There is one house between the person who prefers city breaks and the person who keeps horses.
7. The fish enthusiast is directly left of the cat lover.
8. Arnold is the Desert smoothie lover.
9. The person who likes Cherry smoothies is the person who enjoys mountain retreats.
10. The person who loves swimming is not in the first house.
11. Bob is the person who prefers city breaks.
12. The person who uses a Huawei P50 is the fish enthusiast.
13. The person who loves tennis is the person who enjoys mountain retreats.
14. The person who loves baseball is somewhere to the left of the person who loves soccer.
15. The person who uses a OnePlus 9 is Peter.
16. There is one house between the person who prefers city breaks and Alice.
17. The person who enjoys camping trips is the Dragonfruit smoothie lover.
18. The person who loves baseball is the person who enjoys camping trips.
19. The person who loves beach vacations is directly left of the person who enjoys camping trips.
20. Bob is the fish enthusiast.
 What is the value of attribute PhoneModel for the person whose attribute Animal is dog? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Peter`, `Eric`
 - Each person lives in a unique style of house: `victorian`, `craftsman`, `ranch`, `colonial`
 - The people keep unique animals: `fish`, `cat`, `horse`, `bird`
 - Each person has a unique hobby: `cooking`, `painting`, `gardening`, `photography`

## Clues:
1. The person who loves cooking is in the second house.
2. The fish enthusiast is not in the first house.
3. Alice is in the second house.
4. The cat lover is the person residing in a Victorian house.
5. The person who enjoys gardening is the person residing in a Victorian house.
6. The person in a ranch-style home is not in the first house.
7. The person who keeps horses is somewhere to the right of the person residing in a Victorian house.
8. Peter is somewhere to the left of the person who paints as a hobby.
9. The person in a Craftsman-style house is Eric.
10. The person residing in a Victorian house is in the third house.
11. Peter is directly left of the person living in a colonial-style house.
 What is the value of attribute Hobby for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Eric`, `Arnold`, `Carol`, `Alice`, `Peter`
 - Each person has a unique hobby: `cooking`, `painting`, `knitting`, `photography`, `woodworking`, `gardening`
 - People have unique hair colors: `brown`, `auburn`, `black`, `red`, `gray`, `blonde`
 - People have unique favorite sports: `swimming`, `volleyball`, `basketball`, `baseball`, `tennis`, `soccer`
 - The mothers' names in different houses are unique: `Sarah`, `Holly`, `Penny`, `Janelle`, `Kailyn`, `Aniya`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `pall mall`, `prince`, `yellow monster`, `blue master`

## Clues:
1. The person who has gray hair is directly left of The person whose mother's name is Janelle.
2. The person who loves soccer is in the sixth house.
3. Arnold is the person who has blonde hair.
4. The photography enthusiast is the person who smokes Yellow Monster.
5. There is one house between the Dunhill smoker and the person who loves cooking.
6. The person who paints as a hobby is the person who has blonde hair.
7. The Dunhill smoker is directly left of Alice.
8. The person who paints as a hobby is directly left of the person who loves volleyball.
9. The person who loves soccer is The person whose mother's name is Aniya.
10. The person partial to Pall Mall is somewhere to the left of Carol.
11. The person who enjoys knitting and the person who loves basketball are next to each other.
12. The person partial to Pall Mall is directly left of The person whose mother's name is Holly.
13. The person who loves soccer is the Prince smoker.
14. The person who loves swimming is The person whose mother's name is Penny.
15. The person whose mother's name is Kailyn is the person who loves volleyball.
16. The person who loves basketball is The person whose mother's name is Janelle.
17. The person who enjoys knitting is the person who loves volleyball.
18. The person who enjoys knitting is not in the third house.
19. The person who has auburn hair is somewhere to the left of the photography enthusiast.
20. The person whose mother's name is Sarah is not in the fifth house.
21. Bob is directly left of the person who has red hair.
22. The Dunhill smoker is the person who enjoys gardening.
23. The person who loves baseball is the person who has auburn hair.
24. Eric is the person who smokes many unique blends.
25. The person who has brown hair is somewhere to the right of The person whose mother's name is Penny.
 What is the value of attribute Hobby for the person whose attribute Mother is Penny? Please reason step by step, and put your final answer within \boxed{}

Answer: photography


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Bob`, `Arnold`, `Carol`, `Alice`
 - Each person has a unique birthday month: `may`, `sept`, `jan`, `mar`, `feb`, `april`
 - People have unique hair colors: `brown`, `blonde`, `red`, `auburn`, `black`, `gray`
 - Each mother is accompanied by their child: `Alice`, `Samantha`, `Fred`, `Meredith`, `Timothy`, `Bella`
 - Everyone has a favorite smoothie: `blueberry`, `cherry`, `watermelon`, `lime`, `dragonfruit`, `desert`
 - The mothers' names in different houses are unique: `Sarah`, `Aniya`, `Janelle`, `Holly`, `Kailyn`, `Penny`

## Clues:
1. The person whose birthday is in September and the person's child is named Alice are next to each other.
2. The person whose birthday is in April is the person who is the mother of Timothy.
3. The person who likes Cherry smoothies is the person who has gray hair.
4. Alice is somewhere to the right of Arnold.
5. The person's child is named Alice is somewhere to the right of The person whose mother's name is Sarah.
6. Peter is the person who drinks Blueberry smoothies.
7. The person whose mother's name is Janelle is the person whose birthday is in May.
8. The person whose birthday is in January is somewhere to the left of The person whose mother's name is Sarah.
9. The Watermelon smoothie lover is the person who has brown hair.
10. The person who has red hair is somewhere to the right of the person who has gray hair.
11. The person who drinks Lime smoothies is in the fifth house.
12. The person who has black hair and the person's child is named Meredith are next to each other.
13. The person whose birthday is in March is somewhere to the right of the person who has red hair.
14. The person whose mother's name is Sarah is somewhere to the left of the person who has auburn hair.
15. The person who has black hair is directly left of The person whose mother's name is Penny.
16. The person's child is named Samantha is The person whose mother's name is Holly.
17. The person who has blonde hair and Bob are next to each other.
18. The person who has gray hair is the person whose birthday is in January.
19. The person who has black hair is in the first house.
20. The person whose mother's name is Sarah is the person who has blonde hair.
21. The person whose mother's name is Kailyn is Arnold.
22. The person whose birthday is in January is The person whose mother's name is Kailyn.
23. The person whose birthday is in September is Carol.
24. The person who drinks Lime smoothies is somewhere to the right of the Desert smoothie lover.
25. The person's child is named Fred is not in the third house.
 What is the value of attribute Mother for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: Kailyn


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Carol`, `Peter`, `Bob`
 - Each person lives in a unique style of house: `mediterranean`, `modern`, `craftsman`, `ranch`, `colonial`, `victorian`
 - People have unique favorite music genres: `country`, `hip hop`, `pop`, `jazz`, `classical`, `rock`
 - Each person has a unique hobby: `cooking`, `painting`, `photography`, `woodworking`, `gardening`, `knitting`

## Clues:
1. The person who loves rock music is in the fifth house.
2. The person who loves classical music and the woodworking hobbyist are next to each other.
3. The person in a Mediterranean-style villa is the person who loves hip-hop music.
4. There are two houses between Arnold and the person residing in a Victorian house.
5. The person who loves jazz music is directly left of Eric.
6. The person who loves hip-hop music is somewhere to the left of the person who enjoys knitting.
7. Carol is the person who loves hip-hop music.
8. The person in a Craftsman-style house is Arnold.
9. The person in a ranch-style home is Eric.
10. The woodworking hobbyist is the person residing in a Victorian house.
11. The person who loves country music is in the first house.
12. There is one house between the person who paints as a hobby and the person living in a colonial-style house.
13. Alice is the photography enthusiast.
14. The person who enjoys gardening is Eric.
15. Bob is in the third house.
 What is the value of attribute House for the person whose attribute MusicGenre is classical? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `beach`, `city`
 - Each person has a unique birthday month: `jan`, `feb`, `sept`, `april`
 - The people are of nationalities: `swede`, `dane`, `brit`, `norwegian`

## Clues:
1. The person who enjoys mountain retreats is Alice.
2. The Norwegian is somewhere to the left of the person whose birthday is in September.
3. The person who prefers city breaks is the British person.
4. Alice and Eric are next to each other.
5. The person who likes going on cruises is not in the fourth house.
6. Alice is somewhere to the left of the person whose birthday is in January.
7. The person who enjoys mountain retreats is the Norwegian.
8. Alice is directly left of the Swedish person.
9. Eric is in the third house.
10. The person whose birthday is in April is Peter.
11. The Dane is the person whose birthday is in September.
 What is the value of attribute House for the person whose attribute Nationality is norwegian? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Eric`, `Arnold`, `Alice`
 - Each person has a favorite color: `blue`, `red`, `green`, `yellow`, `white`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `pall mall`, `blends`, `blue master`
 - People have unique hair colors: `blonde`, `brown`, `red`, `gray`, `black`
 - Each person has a unique favorite drink: `tea`, `milk`, `coffee`, `water`, `root beer`
 - The people are of nationalities: `dane`, `swede`, `german`, `brit`, `norwegian`

## Clues:
1. The Norwegian is directly left of the Prince smoker.
2. The person who loves yellow is somewhere to the right of the tea drinker.
3. The person who has blonde hair is the Norwegian.
4. The root beer lover is somewhere to the left of the Dane.
5. There is one house between the person whose favorite color is red and the person who has black hair.
6. The German is in the third house.
7. The person who loves blue is not in the fourth house.
8. The person whose favorite color is red and the Dunhill smoker are next to each other.
9. The root beer lover is directly left of the German.
10. The Norwegian is Eric.
11. The British person is the person who likes milk.
12. Peter is the person who smokes many unique blends.
13. The coffee drinker is directly left of the Swedish person.
14. The person who smokes Blue Master is somewhere to the left of the person who smokes many unique blends.
15. The person who has gray hair is directly left of the person who has red hair.
16. The person who loves blue is somewhere to the right of the one who only drinks water.
17. Arnold is the person who has brown hair.
18. The person who has gray hair is not in the third house.
19. The coffee drinker is the person whose favorite color is green.
20. The Dunhill smoker is Alice.
 What is the value of attribute HairColor for the person whose attribute Cigar is prince? Please reason step by step, and put your final answer within \boxed{}

Answer: gray


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Arnold`, `Alice`, `Eric`, `Carol`
 - The people keep unique animals: `rabbit`, `bird`, `dog`, `horse`, `cat`, `fish`
 - Each person has an occupation: `doctor`, `teacher`, `lawyer`, `artist`, `engineer`, `nurse`

## Clues:
1. Arnold is not in the sixth house.
2. The person who is an engineer is directly left of Eric.
3. Carol and the rabbit owner are next to each other.
4. The person who is an artist is Peter.
5. The person who keeps horses is in the second house.
6. Eric is directly left of the person who is a lawyer.
7. The person who is an artist is somewhere to the left of the person who is a doctor.
8. The cat lover is not in the sixth house.
9. Peter is in the fifth house.
10. The dog owner is the person who is an artist.
11. The person who keeps horses is directly left of the rabbit owner.
12. The bird keeper is Alice.
13. The person who is a teacher and Alice are next to each other.
14. Alice is not in the first house.
 What is the value of attribute House for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Alice`, `Arnold`
 - People have unique hair colors: `brown`, `blonde`, `red`, `black`
 - People have unique heights: `short`, `average`, `tall`, `very short`
 - Each person has a favorite color: `green`, `yellow`, `white`, `red`

## Clues:
1. Eric is the person who has an average height.
2. The person whose favorite color is red is the person who is tall.
3. The person whose favorite color is green is somewhere to the left of Peter.
4. Arnold is in the fourth house.
5. The person who has an average height is not in the first house.
6. The person who is very short is in the fourth house.
7. The person who loves white is the person who has black hair.
8. The person who is tall is directly left of the person who has brown hair.
9. The person who has blonde hair is the person who has an average height.
 What is the value of attribute Color for the person whose attribute Height is tall? Please reason step by step, and put your final answer within \boxed{}

Answer: red


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Carol`, `Peter`, `Bob`, `Arnold`, `Eric`
 - They all have a unique favorite flower: `carnations`, `iris`, `tulips`, `daffodils`, `lilies`, `roses`
 - Each person prefers a unique type of vacation: `camping`, `mountain`, `city`, `beach`, `cultural`, `cruise`
 - Each person lives in a unique style of house: `colonial`, `modern`, `mediterranean`, `craftsman`, `ranch`, `victorian`
 - People have unique heights: `short`, `very short`, `average`, `tall`, `super tall`, `very tall`
 - People have unique hair colors: `red`, `black`, `gray`, `brown`, `auburn`, `blonde`

## Clues:
1. The person who loves the boquet of lilies is not in the fifth house.
2. The person who is very short is the person who has auburn hair.
3. Arnold is the person who enjoys camping trips.
4. The person who has auburn hair is Eric.
5. The person who goes on cultural tours is somewhere to the left of the person who is super tall.
6. There are two houses between the person who has gray hair and the person who is short.
7. The person who is tall is somewhere to the left of the person who has brown hair.
8. The person in a Craftsman-style house is Bob.
9. The person who is very short is somewhere to the left of the person who has blonde hair.
10. The person who is very tall is Bob.
11. There are two houses between the person who loves the rose bouquet and Peter.
12. The person in a modern-style house is not in the fourth house.
13. The person who prefers city breaks is the person who is short.
14. The person living in a colonial-style house is somewhere to the left of the person who loves the vase of tulips.
15. The person who loves a carnations arrangement is the person who has brown hair.
16. The person who has red hair is the person who loves the boquet of lilies.
17. The person who enjoys mountain retreats is the person residing in a Victorian house.
18. The person who loves a bouquet of daffodils is the person in a modern-style house.
19. Arnold is the person who has an average height.
20. The person in a Mediterranean-style villa is not in the first house.
21. The person who is tall is not in the fifth house.
22. The person who loves the vase of tulips is not in the fourth house.
23. Carol is directly left of Bob.
24. The person who is very tall is the person who goes on cultural tours.
25. The person in a Mediterranean-style villa is the person who has red hair.
26. The person who is super tall is the person who has black hair.
27. Eric is the person who loves the rose bouquet.
28. The person who enjoys mountain retreats is directly left of the person who likes going on cruises.
 What is the value of attribute Height for the person whose attribute Vacation is cruise? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Peter`, `Arnold`, `Eric`
 - People have unique hair colors: `black`, `red`, `brown`, `gray`, `blonde`
 - Each person has a unique level of education: `bachelor`, `associate`, `high school`, `master`, `doctorate`
 - Each mother is accompanied by their child: `Bella`, `Timothy`, `Samantha`, `Meredith`, `Fred`
 - The people keep unique animals: `horse`, `cat`, `fish`, `bird`, `dog`

## Clues:
1. The person who has red hair is directly left of the bird keeper.
2. Peter is the person with a doctorate.
3. The fish enthusiast is somewhere to the right of the person who has brown hair.
4. The cat lover is the person with a bachelor's degree.
5. The person's child is named Samantha is somewhere to the right of the person who has black hair.
6. The fish enthusiast is the person's child is named Bella.
7. Arnold and the person with an associate's degree are next to each other.
8. The person who keeps horses is the person who is the mother of Timothy.
9. The person who has brown hair is somewhere to the right of Bob.
10. The person who has blonde hair is the bird keeper.
11. The person with a master's degree is the fish enthusiast.
12. The person with a high school diploma is the person's child is named Fred.
13. The person with a high school diploma is Eric.
14. The person with a bachelor's degree is the person who has red hair.
15. The person who has gray hair is the person with an associate's degree.
 What is the value of attribute Animal for the person whose attribute HairColor is black? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


Let $n = 2^{19} \times 3^{12}$. Let $M$ denote the number of positive divisors of $n^2$ which are less than $n$ but would not divide $n$. What is the number formed by taking the last two digits of $M$ (in the same order)?

Answer: 28


Let $ABC$ be a triangle. Let the $A$-altitude meet $BC$ at $D$, the $B$-altitude meet $AC$ at $E$, and let $T \neq A$ be the point on the circumcircle of $\triangle ABC$ such that $AT \parallel BC$. Given that $D, E, T$ are collinear, if $BD = 3$ and $AD = 4$, then the area of $\triangle ABC$ can be written as $a + \sqrt{b}$, where $a$ and $b$ are positive integers. What is $a + b$?

Answer: 112


Triangle $ABC$ is inscribed in a unit circle $\omega$. Let $H$ be its orthocenter and $D$ be the foot of the perpendicular from $A$ to $BC$. Let $\triangle XYZ$ be the triangle formed by drawing the tangents to $\omega$ at $A$, $B$, $C$. If $\overline{AH} = \overline{HD}$ and the side lengths of $\triangle XYZ$ form an arithmetic sequence, the area of $\triangle ABC$ can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$, $q$. What is $p + q$?

Answer: 11


A trapezoid has side lengths $10, 10, 10$, and $22$. Each side of the trapezoid is the diameter of a semicircle. The two semicircles on the parallel sides of the trapezoid face outside the trapezoid, while the other two semicircles face inside, as shown in the figure.

The region bounded by these four semicircles has an area expressed as $m + n\pi$, where $m$ and $n$ are positive integers. Find $m + n$.

Answer: 176


A child lines up $2020^2$ pieces of bricks in a row and removes bricks whose positions are square numbers (i.e., the 1st, 4th, 9th, 16th, ... bricks). Then, he lines up the remaining bricks again and removes those that are in a 'square position'. This process is repeated until the number of bricks remaining drops below $250$. How many bricks remain in the end?

Answer: 240


Let $a_1, a_2, a_3, \ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j < k$, $a_j \neq a_k$. (So $1 \in S$; $2 \in S$ if and only if $a_1 \neq a_2$; $3 \in S$ if and only if $a_1 \neq a_3$ and $a_2 \neq a_3$; and so on.) In simplest form, let $\frac{p}{q}$ be the expected number of positive integers $m$ such that $m$ and $m+1$ are in $S$. Compute $pq$.

Answer: 4030


Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability $\frac{9}{10}$. Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability $\frac{3}{4}$. Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 73


Triangle $ABC$ is isosceles, and $\angle ABC = x^{\circ}$. If the sum of the possible measures of $\angle BAC$ is $240^{\circ}$, find $x$.

Answer: 20


The edges and diagonals of convex pentagon $ABCDE$ are all colored either red or blue. How many ways are there to color the segments such that there is exactly one monochromatic triangle with vertices among $A$, $B$, $C$, $D$, $E$; that is, triangles whose edges are all the same color?

Answer: 260


Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$. Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$. If $\angle MPN = 40^\circ$, find the degree measure of $\angle BPC$.

Answer: 160


Let $\triangle ABC$ have median $CM$ ($M \in AB$) and circumcenter $O$. The circumcircle of $\triangle AMO$ bisects $CM$. Determine the least possible perimeter of $\triangle ABC$ if it has integer side lengths.

Answer: 24


Let $ABCD$ be a convex quadrilateral with $\angle DAB = \angle BDC = 90^\circ$. Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$, respectively, with $P$ lying in between $B$ and $Q$. If $AD = 999$ and $PQ = 200$, then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$?

Answer: 799


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Peter`, `Eric`, `Bob`, `Alice`, `Arnold`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `city`, `camping`, `cultural`, `beach`
 - Each person has a unique hobby: `knitting`, `photography`, `cooking`, `painting`, `gardening`, `woodworking`
 - Everyone has something unique for lunch: `stew`, `pizza`, `spaghetti`, `grilled cheese`, `stir fry`, `soup`
 - People own unique car models: `chevrolet silverado`, `honda civic`, `toyota camry`, `ford f150`, `tesla model 3`, `bmw 3 series`
 - They all have a unique favorite flower: `tulips`, `daffodils`, `carnations`, `lilies`, `iris`, `roses`

## Clues:
1. Arnold is the person who enjoys camping trips.
2. The photography enthusiast is the person who loves the soup.
3. Carol is the person who loves eating grilled cheese.
4. The person who owns a Honda Civic is the person who loves a carnations arrangement.
5. The person who loves beach vacations is not in the first house.
6. The person who enjoys gardening is the person who owns a Toyota Camry.
7. There is one house between the person who loves the boquet of iris and the person who loves the vase of tulips.
8. The person who prefers city breaks is the person who loves the stew.
9. The person who loves a bouquet of daffodils is Carol.
10. The person who prefers city breaks is Eric.
11. Carol is the person who loves cooking.
12. Eric is directly left of the photography enthusiast.
13. The person who loves the rose bouquet is Bob.
14. There is one house between Arnold and the person who likes going on cruises.
15. The person who owns a Tesla Model 3 and the person who enjoys mountain retreats are next to each other.
16. The person who loves the rose bouquet is the person who owns a Toyota Camry.
17. The person who owns a BMW 3 Series is somewhere to the right of the person who loves cooking.
18. The woodworking hobbyist is not in the third house.
19. The person who owns a Ford F-150 is the person who enjoys mountain retreats.
20. The person who is a pizza lover is not in the second house.
21. The person who loves the vase of tulips is not in the fourth house.
22. The person who loves the spaghetti eater is somewhere to the left of the person who enjoys knitting.
23. Peter is somewhere to the right of the photography enthusiast.
24. The person who owns a Honda Civic is the person who is a pizza lover.
25. The person who loves cooking is in the fifth house.
26. The person who likes going on cruises is the person who enjoys knitting.
 What is the value of attribute Food for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: spaghetti


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`, `Bob`
 - Each person has a unique type of pet: `cat`, `dog`, `bird`, `fish`, `hamster`
 - Everyone has a unique favorite cigar: `pall mall`, `blends`, `prince`, `dunhill`, `blue master`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `dragonfruit`, `cherry`, `lime`
 - People use unique phone models: `google pixel 6`, `oneplus 9`, `samsung galaxy s21`, `iphone 13`, `huawei p50`
 - People have unique favorite sports: `baseball`, `soccer`, `basketball`, `swimming`, `tennis`

## Clues:
1. The person who loves swimming is directly left of the Prince smoker.
2. The person who smokes Blue Master is somewhere to the right of Eric.
3. The person who uses a OnePlus 9 is Arnold.
4. Arnold is somewhere to the right of the person with a pet hamster.
5. The person who owns a dog is Peter.
6. Eric is the Desert smoothie lover.
7. The person who loves basketball is not in the fifth house.
8. The person who has a cat is the person who uses a Samsung Galaxy S21.
9. The person who loves baseball is directly left of Peter.
10. There is one house between the Desert smoothie lover and the Dunhill smoker.
11. The person who loves baseball and the person who uses a Google Pixel 6 are next to each other.
12. The person who keeps a pet bird is in the second house.
13. The person who uses a Huawei P50 is in the fourth house.
14. The person who drinks Lime smoothies is in the fourth house.
15. The person partial to Pall Mall is in the fifth house.
16. The Dragonfruit smoothie lover is in the second house.
17. The person who loves tennis is the person with a pet hamster.
18. Bob is somewhere to the right of the person who owns a dog.
19. The Watermelon smoothie lover is Arnold.
20. The Prince smoker is somewhere to the left of the Dunhill smoker.
 What is the value of attribute Pet for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `beach`, `city`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `victorian`, `colonial`
 - The mothers' names in different houses are unique: `Kailyn`, `Holly`, `Aniya`, `Janelle`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`
 - Each person has a favorite color: `green`, `white`, `yellow`, `red`

## Clues:
1. Alice is directly left of the person whose favorite color is red.
2. The person who loves a bouquet of daffodils is The person whose mother's name is Janelle.
3. Eric is not in the first house.
4. The person who loves the boquet of lilies is directly left of the person living in a colonial-style house.
5. The person residing in a Victorian house is in the second house.
6. The person who loves the rose bouquet is the person who loves yellow.
7. The person whose mother's name is Aniya is the person who loves yellow.
8. The person who enjoys mountain retreats is in the first house.
9. The person who likes going on cruises is the person whose favorite color is red.
10. The person who loves yellow is in the second house.
11. The person who loves white is the person who prefers city breaks.
12. The person whose mother's name is Holly is not in the fourth house.
13. The person whose favorite color is red is directly left of Peter.
14. The person who loves a bouquet of daffodils is the person in a ranch-style home.
 What is the value of attribute HouseStyle for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: ranch


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`
 - The mothers' names in different houses are unique: `Kailyn`, `Aniya`, `Holly`, `Janelle`
 - Each person lives in a unique style of house: `craftsman`, `victorian`, `ranch`, `colonial`
 - The people keep unique animals: `cat`, `fish`, `bird`, `horse`
 - Each person has a unique level of education: `bachelor`, `high school`, `master`, `associate`

## Clues:
1. The person with a bachelor's degree is directly left of The person whose mother's name is Aniya.
2. The person who keeps horses is not in the second house.
3. Eric is not in the second house.
4. The person with a high school diploma is the person in a Craftsman-style house.
5. Peter is somewhere to the left of the person with a master's degree.
6. Alice is in the third house.
7. The person in a Craftsman-style house is somewhere to the left of The person whose mother's name is Janelle.
8. The person living in a colonial-style house is Peter.
9. The fish enthusiast is the person with a master's degree.
10. The person with an associate's degree is in the third house.
11. The person whose mother's name is Kailyn is the bird keeper.
12. The person whose mother's name is Kailyn is the person residing in a Victorian house.
 What is the value of attribute House for the person whose attribute Education is associate? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Bob`, `Eric`, `Peter`
 - They all have a unique favorite flower: `roses`, `tulips`, `carnations`, `lilies`, `daffodils`
 - Each mother is accompanied by their child: `Fred`, `Meredith`, `Timothy`, `Bella`, `Samantha`

## Clues:
1. The person's child is named Meredith is the person who loves a carnations arrangement.
2. The person's child is named Bella is directly left of the person who loves the vase of tulips.
3. The person's child is named Bella is the person who loves the rose bouquet.
4. The person who loves the boquet of lilies is Eric.
5. There are two houses between Bob and Peter.
6. Alice is the person's child is named Meredith.
7. Peter is the person who loves the rose bouquet.
8. The person's child is named Fred is in the third house.
9. The person's child is named Samantha is in the fifth house.
10. The person's child is named Meredith is in the second house.
 What is the value of attribute Children for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: Meredith


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Bob`, `Eric`, `Carol`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `spaghetti`, `pizza`, `soup`, `stew`
 - People use unique phone models: `xiaomi mi 11`, `iphone 13`, `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `huawei p50`
 - People have unique favorite music genres: `pop`, `country`, `classical`, `jazz`, `rock`, `hip hop`
 - Each person has a unique favorite drink: `water`, `boba tea`, `milk`, `root beer`, `tea`, `coffee`
 - Each person has an occupation: `nurse`, `lawyer`, `artist`, `doctor`, `teacher`, `engineer`

## Clues:
1. The tea drinker is somewhere to the right of the person who is a teacher.
2. Eric is the person who uses a Google Pixel 6.
3. The person who loves the spaghetti eater and the person who loves hip-hop music are next to each other.
4. The boba tea drinker is somewhere to the left of the person who uses a Google Pixel 6.
5. The person who uses a Huawei P50 is somewhere to the left of the person who uses a OnePlus 9.
6. The person who is a pizza lover is the person who loves rock music.
7. The person who is a pizza lover is somewhere to the right of the person who loves the soup.
8. The person who likes milk is somewhere to the left of the person who loves hip-hop music.
9. The person who uses a OnePlus 9 is Arnold.
10. The person who is a lawyer is the person who loves stir fry.
11. The person who is an artist is Alice.
12. Bob is the person who loves classical music.
13. The person who loves rock music is directly left of the person who loves classical music.
14. The person who is a doctor is the person who uses a Huawei P50.
15. Bob is the root beer lover.
16. There are two houses between the person who is an engineer and the person who uses a OnePlus 9.
17. Eric is the person who loves jazz music.
18. The person who uses a Xiaomi Mi 11 is the person who is an artist.
19. Carol is directly left of the person who uses a Huawei P50.
20. The person who is a lawyer is the coffee drinker.
21. The person who loves eating grilled cheese is the person who loves pop music.
22. The person who loves hip-hop music is the person who uses a Samsung Galaxy S21.
23. The coffee drinker is somewhere to the left of the person who likes milk.
 What is the value of attribute MusicGenre for the person whose attribute Occupation is teacher? Please reason step by step, and put your final answer within \boxed{}

Answer: hip hop


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - They all have a unique favorite flower: `carnations`, `tulips`, `lilies`, `roses`, `daffodils`, `iris`
 - People have unique favorite book genres: `science fiction`, `fantasy`, `romance`, `mystery`, `biography`, `historical fiction`
 - People use unique phone models: `huawei p50`, `oneplus 9`, `iphone 13`, `xiaomi mi 11`, `samsung galaxy s21`, `google pixel 6`
 - People have unique hair colors: `black`, `gray`, `red`, `auburn`, `blonde`, `brown`
 - Each person has a favorite color: `yellow`, `green`, `red`, `white`, `blue`, `purple`

## Clues:
1. There is one house between the person who has gray hair and the person who uses an iPhone 13.
2. The person who loves the rose bouquet is the person who has blonde hair.
3. There is one house between the person who has gray hair and the person who loves yellow.
4. Carol is somewhere to the right of the person who loves a bouquet of daffodils.
5. The person who loves white is somewhere to the left of Peter.
6. The person who loves white is somewhere to the left of the person who loves science fiction books.
7. Arnold is directly left of the person who loves historical fiction books.
8. The person who uses a Huawei P50 is the person who loves a bouquet of daffodils.
9. The person who loves purple is somewhere to the left of the person who loves fantasy books.
10. Eric is the person who loves science fiction books.
11. The person who loves a bouquet of daffodils is the person who has auburn hair.
12. Alice is the person whose favorite color is green.
13. The person who loves fantasy books is in the second house.
14. The person who uses a Samsung Galaxy S21 is Carol.
15. The person who uses a Xiaomi Mi 11 is not in the fourth house.
16. Eric is the person who has auburn hair.
17. The person who loves the rose bouquet is somewhere to the left of the person who has red hair.
18. The person who loves historical fiction books is the person who loves the rose bouquet.
19. The person who uses a Xiaomi Mi 11 is somewhere to the right of the person who loves white.
20. The person who has black hair is in the fourth house.
21. The person whose favorite color is red is the person who loves mystery books.
22. The person who loves historical fiction books and the person who loves the vase of tulips are next to each other.
23. The person who has brown hair is the person who uses a OnePlus 9.
24. The person who loves romance books is somewhere to the left of the person who loves fantasy books.
25. The person who loves the boquet of lilies is in the second house.
26. The person who loves the boquet of iris is somewhere to the right of the person whose favorite color is green.
 What is the value of attribute PhoneModel for the person whose attribute BookGenre is science fiction? Please reason step by step, and put your final answer within \boxed{}

Answer: huawei p50


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Eric`
 - Each person has a unique birthday month: `sept`, `feb`, `april`, `jan`
 - The mothers' names in different houses are unique: `Aniya`, `Kailyn`, `Holly`, `Janelle`
 - The people keep unique animals: `horse`, `bird`, `cat`, `fish`
 - Each person has a favorite color: `green`, `white`, `yellow`, `red`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `soccer`

## Clues:
1. The bird keeper is The person whose mother's name is Kailyn.
2. The fish enthusiast is not in the fourth house.
3. The person who loves white is Arnold.
4. The person who keeps horses is Peter.
5. The person who keeps horses is not in the first house.
6. The person whose mother's name is Kailyn is Arnold.
7. The person whose birthday is in January is in the first house.
8. The person whose birthday is in April is somewhere to the left of the person whose birthday is in September.
9. Alice is the person whose birthday is in September.
10. The person whose mother's name is Aniya and the person whose birthday is in September are next to each other.
11. Peter is directly left of the person whose favorite color is red.
12. There is one house between the person whose birthday is in April and the person who loves soccer.
13. The fish enthusiast is the person whose birthday is in September.
14. The person who loves swimming is The person whose mother's name is Holly.
15. The person whose favorite color is green is the person who loves swimming.
16. The person who loves yellow is the person who loves basketball.
 What is the value of attribute FavoriteSport for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: tennis


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Bob`, `Eric`
 - People have unique favorite sports: `tennis`, `swimming`, `soccer`, `basketball`, `baseball`
 - Each person has a unique level of education: `doctorate`, `associate`, `master`, `high school`, `bachelor`
 - They all have a unique favorite flower: `daffodils`, `tulips`, `lilies`, `carnations`, `roses`

## Clues:
1. The person who loves swimming is the person who loves a carnations arrangement.
2. The person with an associate's degree is the person who loves basketball.
3. The person who loves a carnations arrangement is directly left of the person with a master's degree.
4. The person with a master's degree is Eric.
5. Alice is the person with a bachelor's degree.
6. Peter is the person who loves the rose bouquet.
7. The person who loves swimming is somewhere to the right of the person who loves soccer.
8. The person who loves the boquet of lilies is directly left of the person who loves the vase of tulips.
9. The person with a high school diploma is in the third house.
10. The person who loves a carnations arrangement is the person with a doctorate.
11. Arnold is not in the second house.
12. The person who loves baseball is in the first house.
 What is the value of attribute Education for the person whose attribute Flower is roses? Please reason step by step, and put your final answer within \boxed{}

Answer: high school


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`
 - Each person has a unique type of pet: `cat`, `fish`, `dog`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Everyone has something unique for lunch: `spaghetti`, `pizza`, `grilled cheese`

## Clues:
1. The person who owns a Ford F-150 is not in the first house.
2. The person who owns a Ford F-150 is not in the third house.
3. The person who loves science fiction books is somewhere to the right of the person who is a pizza lover.
4. The person with an aquarium of fish is the person who is a pizza lover.
5. The person who is a pizza lover is somewhere to the left of Peter.
6. Peter is the person who owns a Tesla Model 3.
7. The person who loves science fiction books is directly left of the person who loves mystery books.
8. Arnold is the person who loves the spaghetti eater.
9. The person who has a cat is the person who loves the spaghetti eater.
 What is the value of attribute CarModel for the person whose attribute BookGenre is romance? Please reason step by step, and put your final answer within \boxed{}

Answer: toyota camry


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Bob`, `Carol`, `Alice`
 - The people keep unique animals: `horse`, `bird`, `fish`, `cat`, `rabbit`, `dog`
 - Each person lives in a unique style of house: `modern`, `colonial`, `victorian`, `ranch`, `craftsman`, `mediterranean`
 - Each person has an occupation: `engineer`, `lawyer`, `artist`, `nurse`, `doctor`, `teacher`

## Clues:
1. The person who is a teacher is the person in a Craftsman-style house.
2. Bob is in the second house.
3. There are two houses between the person residing in a Victorian house and Eric.
4. The cat lover is directly left of the person who keeps horses.
5. The person who is an engineer is somewhere to the left of the person in a modern-style house.
6. The person in a Craftsman-style house is directly left of the person who is an artist.
7. The person in a Mediterranean-style villa is somewhere to the right of the person who is a doctor.
8. The dog owner is not in the first house.
9. The person in a Mediterranean-style villa is somewhere to the left of the cat lover.
10. The person in a ranch-style home is in the sixth house.
11. The person in a Mediterranean-style villa is Alice.
12. The fish enthusiast is the person in a ranch-style home.
13. The bird keeper is the person in a modern-style house.
14. The person living in a colonial-style house is Peter.
15. The person who is a lawyer is Arnold.
 What is the value of attribute House for the person whose attribute Animal is horse? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Arnold`, `Bob`, `Peter`
 - They all have a unique favorite flower: `lilies`, `roses`, `carnations`, `tulips`, `daffodils`
 - Each person has a unique birthday month: `sept`, `april`, `feb`, `mar`, `jan`
 - Everyone has something unique for lunch: `pizza`, `grilled cheese`, `spaghetti`, `stir fry`, `stew`
 - Each person lives in a unique style of house: `colonial`, `craftsman`, `victorian`, `ranch`, `modern`

## Clues:
1. The person whose birthday is in September is the person who loves the spaghetti eater.
2. The person whose birthday is in February is Alice.
3. The person in a Craftsman-style house is Eric.
4. The person who loves the stew is the person in a modern-style house.
5. The person who loves the vase of tulips is somewhere to the left of the person who is a pizza lover.
6. The person who loves the stew is in the fourth house.
7. The person whose birthday is in March is in the third house.
8. The person whose birthday is in September is in the first house.
9. The person who loves the boquet of lilies is the person whose birthday is in February.
10. The person residing in a Victorian house is directly left of the person who is a pizza lover.
11. The person whose birthday is in January is the person who loves a carnations arrangement.
12. The person who loves eating grilled cheese is directly left of the person who loves the stew.
13. The person living in a colonial-style house is the person whose birthday is in March.
14. The person who is a pizza lover is somewhere to the left of Arnold.
15. There is one house between the person who loves the rose bouquet and the person whose birthday is in April.
16. The person in a modern-style house is Peter.
 What is the value of attribute Name for the person whose attribute Flower is carnations? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Bob`, `Eric`, `Carol`, `Alice`, `Arnold`
 - The people keep unique animals: `rabbit`, `dog`, `horse`, `fish`, `bird`, `cat`
 - People have unique favorite sports: `basketball`, `swimming`, `tennis`, `soccer`, `baseball`, `volleyball`
 - People have unique hair colors: `blonde`, `gray`, `auburn`, `red`, `black`, `brown`
 - People have unique heights: `average`, `very short`, `very tall`, `tall`, `super tall`, `short`
 - Everyone has a unique favorite cigar: `pall mall`, `yellow monster`, `blue master`, `prince`, `dunhill`, `blends`

## Clues:
1. The person who is very tall and the dog owner are next to each other.
2. There are two houses between the person who loves tennis and the person who has an average height.
3. The cat lover is the person who loves baseball.
4. The person who is very short is not in the fifth house.
5. There is one house between the fish enthusiast and the person who has red hair.
6. The cat lover is somewhere to the left of the Dunhill smoker.
7. The rabbit owner is somewhere to the left of the person who is super tall.
8. The Dunhill smoker is Alice.
9. The person who has red hair is the Dunhill smoker.
10. The person who is super tall is not in the fifth house.
11. The Prince smoker is somewhere to the left of the person who smokes Yellow Monster.
12. The person who loves volleyball is somewhere to the right of the person who smokes Yellow Monster.
13. The person who smokes Blue Master is directly left of the person who has black hair.
14. There is one house between the person who loves basketball and the person who smokes Blue Master.
15. The person who smokes Yellow Monster is the person who is tall.
16. Bob and the fish enthusiast are next to each other.
17. The Dunhill smoker is the person who loves soccer.
18. The person who loves tennis is not in the third house.
19. Arnold is the person who smokes many unique blends.
20. The Prince smoker is Peter.
21. There is one house between the person who has blonde hair and the person who is tall.
22. The person who loves swimming is directly left of Alice.
23. The person who is super tall is Eric.
24. The person who is short is the rabbit owner.
25. The person who keeps horses is not in the fifth house.
26. The cat lover and the person who has brown hair are next to each other.
27. The person who is short and the person who has auburn hair are next to each other.
 What is the value of attribute FavoriteSport for the person whose attribute Animal is rabbit? Please reason step by step, and put your final answer within \boxed{}

Answer: tennis


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Eric`, `Peter`, `Alice`, `Bob`, `Arnold`
 - Everyone has something unique for lunch: `spaghetti`, `soup`, `grilled cheese`, `stir fry`, `stew`, `pizza`
 - Each person has a favorite color: `green`, `purple`, `blue`, `red`, `white`, `yellow`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Kailyn`, `Sarah`, `Janelle`
 - Each person lives in a unique style of house: `colonial`, `craftsman`, `modern`, `ranch`, `victorian`, `mediterranean`

## Clues:
1. The person whose mother's name is Janelle is the person living in a colonial-style house.
2. Peter is the person whose favorite color is green.
3. The person who loves blue is directly left of the person who loves the spaghetti eater.
4. The person whose mother's name is Penny is directly left of the person whose favorite color is red.
5. The person who loves blue is The person whose mother's name is Penny.
6. The person whose mother's name is Aniya is directly left of the person whose favorite color is green.
7. The person who loves white is somewhere to the left of The person whose mother's name is Janelle.
8. Eric is the person who loves eating grilled cheese.
9. The person whose mother's name is Holly is directly left of Alice.
10. The person whose mother's name is Holly and the person who loves the stew are next to each other.
11. Alice is the person in a ranch-style home.
12. Carol is not in the sixth house.
13. The person who loves stir fry is the person who loves purple.
14. The person whose favorite color is red and the person living in a colonial-style house are next to each other.
15. The person who loves the soup is Peter.
16. The person who is a pizza lover is somewhere to the left of The person whose mother's name is Sarah.
17. The person residing in a Victorian house is the person who loves eating grilled cheese.
18. The person who loves the spaghetti eater is not in the fifth house.
19. There are two houses between the person who loves eating grilled cheese and The person whose mother's name is Aniya.
20. Arnold is the person whose favorite color is red.
21. The person in a modern-style house is Arnold.
22. The person who loves yellow is the person in a Mediterranean-style villa.
 What is the value of attribute Food for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: spaghetti


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Eric`, `Peter`, `Bob`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `dunhill`, `pall mall`, `blue master`

## Clues:
1. Alice is not in the fifth house.
2. The person who smokes Blue Master is not in the fifth house.
3. There is one house between Eric and the Dunhill smoker.
4. Bob is somewhere to the left of the person partial to Pall Mall.
5. Arnold is the Dunhill smoker.
6. Bob is somewhere to the right of the person who smokes many unique blends.
7. Alice is not in the first house.
8. There are two houses between Eric and the person partial to Pall Mall.
 What is the value of attribute Name for the person whose attribute Cigar is prince? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Bob`, `Arnold`
 - Each person has a unique type of pet: `hamster`, `fish`, `cat`, `dog`, `bird`
 - Each person has an occupation: `doctor`, `engineer`, `artist`, `lawyer`, `teacher`
 - Each person has a favorite color: `green`, `blue`, `yellow`, `red`, `white`
 - The people keep unique animals: `bird`, `cat`, `horse`, `fish`, `dog`

## Clues:
1. The person who owns a dog is Arnold.
2. The bird keeper is in the fourth house.
3. The person who keeps a pet bird is directly left of the dog owner.
4. The person who loves white is somewhere to the left of the person who is a lawyer.
5. The person who loves yellow is directly left of the person whose favorite color is green.
6. The cat lover is Bob.
7. The cat lover is somewhere to the left of Eric.
8. The person who keeps horses is in the fifth house.
9. The person who is a lawyer is directly left of the person who is a teacher.
10. The person who is a doctor is in the first house.
11. Alice is the person who loves yellow.
12. The person who loves blue is directly left of the person with an aquarium of fish.
13. The person who loves yellow is in the first house.
14. The person with a pet hamster is the person who is an artist.
15. Eric is the dog owner.
 What is the value of attribute Occupation for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: teacher


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Eric`, `Bob`
 - Each person has a unique favorite drink: `root beer`, `tea`, `water`, `coffee`, `milk`
 - People have unique favorite music genres: `pop`, `hip hop`, `classical`, `jazz`, `rock`
 - The people are of nationalities: `dane`, `norwegian`, `swede`, `german`, `brit`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `blue master`, `pall mall`, `blends`

## Clues:
1. The Dunhill smoker is not in the second house.
2. The person who loves rock music is directly left of the person who loves classical music.
3. Peter is the person who loves jazz music.
4. Arnold is directly left of the Dane.
5. The root beer lover is the German.
6. The tea drinker is the Swedish person.
7. The Swedish person is the person who loves classical music.
8. The Dunhill smoker is not in the fourth house.
9. The person who loves jazz music is the Prince smoker.
10. Eric and the person who loves hip-hop music are next to each other.
11. There are two houses between the person partial to Pall Mall and the Swedish person.
12. Arnold is somewhere to the right of Eric.
13. The coffee drinker is somewhere to the right of the Dunhill smoker.
14. The Norwegian is somewhere to the right of the person who smokes Blue Master.
15. There is one house between the one who only drinks water and the person who smokes many unique blends.
16. Alice is the tea drinker.
 What is the value of attribute House for the person whose attribute Nationality is brit? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Alice`
 - The people keep unique animals: `fish`, `cat`, `horse`, `bird`
 - Everyone has a unique favorite cigar: `pall mall`, `blue master`, `dunhill`, `prince`
 - They all have a unique favorite flower: `lilies`, `roses`, `daffodils`, `carnations`
 - People have unique favorite music genres: `classical`, `jazz`, `rock`, `pop`

## Clues:
1. The Dunhill smoker is Eric.
2. Eric is the cat lover.
3. The person who loves the boquet of lilies is the person who loves pop music.
4. The person partial to Pall Mall is the fish enthusiast.
5. The person who smokes Blue Master is the person who keeps horses.
6. Arnold is not in the third house.
7. The person who loves rock music is somewhere to the left of the person who loves a carnations arrangement.
8. The person who loves classical music is somewhere to the left of the fish enthusiast.
9. The person who loves rock music is Peter.
10. The person who loves jazz music is the person who smokes Blue Master.
11. The person who loves a bouquet of daffodils is in the fourth house.
 What is the value of attribute Flower for the person whose attribute Cigar is pall mall? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Carol`, `Peter`, `Eric`, `Bob`, `Alice`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `modern`, `craftsman`, `mediterranean`, `victorian`
 - Everyone has something unique for lunch: `pizza`, `stew`, `spaghetti`, `grilled cheese`, `stir fry`, `soup`
 - Each person prefers a unique type of vacation: `cultural`, `cruise`, `mountain`, `camping`, `city`, `beach`
 - People have unique heights: `average`, `very tall`, `very short`, `short`, `tall`, `super tall`
 - Everyone has a unique favorite cigar: `yellow monster`, `prince`, `dunhill`, `pall mall`, `blue master`, `blends`

## Clues:
1. Alice is in the fifth house.
2. The person who loves stir fry is the person living in a colonial-style house.
3. Alice is the person who loves the spaghetti eater.
4. Arnold is the person who loves the stew.
5. There is one house between the person who has an average height and Peter.
6. The person in a Craftsman-style house is not in the third house.
7. The person who has an average height is the person who loves stir fry.
8. The person who loves beach vacations is the person in a ranch-style home.
9. Eric is in the fourth house.
10. There is one house between the person living in a colonial-style house and the person who enjoys camping trips.
11. The person who enjoys mountain retreats is the person who smokes Yellow Monster.
12. The person who enjoys mountain retreats is the person who is very tall.
13. The person who enjoys mountain retreats and the Dunhill smoker are next to each other.
14. The person who loves the spaghetti eater is the person residing in a Victorian house.
15. The person who is tall is the person who loves beach vacations.
16. The person who is tall is somewhere to the left of the person residing in a Victorian house.
17. The person who loves stir fry is directly left of Bob.
18. The person in a modern-style house is somewhere to the left of Alice.
19. The person in a Craftsman-style house is somewhere to the left of the person who is short.
20. The person who loves stir fry is somewhere to the left of the Prince smoker.
21. There are two houses between the person who loves eating grilled cheese and the person who is super tall.
22. The person in a ranch-style home is the person who smokes Blue Master.
23. The person who smokes many unique blends is directly left of the person who smokes Blue Master.
24. The person who goes on cultural tours is the person who is a pizza lover.
25. The person who is a pizza lover is somewhere to the left of the person who likes going on cruises.
 What is the value of attribute Food for the person whose attribute HouseStyle is modern? Please reason step by step, and put your final answer within \boxed{}

Answer: grilled cheese


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Peter`, `Bob`, `Arnold`
 - Each person lives in a unique style of house: `colonial`, `victorian`, `craftsman`, `ranch`, `modern`

## Clues:
1. The person residing in a Victorian house is in the fourth house.
2. The person in a ranch-style home is not in the fifth house.
3. The person in a modern-style house is Eric.
4. The person in a modern-style house is not in the third house.
5. Arnold is the person in a Craftsman-style house.
6. Peter is somewhere to the left of Alice.
7. The person in a Craftsman-style house is not in the second house.
8. The person residing in a Victorian house is somewhere to the left of Bob.
9. Arnold is somewhere to the left of Peter.
 What is the value of attribute House for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Carol`, `Arnold`, `Bob`, `Alice`
 - Each person has a favorite color: `blue`, `red`, `green`, `yellow`, `purple`, `white`
 - Each person prefers a unique type of vacation: `city`, `beach`, `cruise`, `cultural`, `mountain`, `camping`
 - People have unique hair colors: `blonde`, `black`, `gray`, `brown`, `red`, `auburn`
 - Each person has an occupation: `artist`, `lawyer`, `engineer`, `teacher`, `doctor`, `nurse`
 - They all have a unique favorite flower: `roses`, `iris`, `daffodils`, `carnations`, `tulips`, `lilies`

## Clues:
1. The person who loves a bouquet of daffodils is the person who is an artist.
2. The person who enjoys camping trips is the person whose favorite color is green.
3. The person who has brown hair is in the first house.
4. The person who loves blue is somewhere to the right of the person who enjoys mountain retreats.
5. The person who loves the vase of tulips is the person who enjoys mountain retreats.
6. Eric is in the second house.
7. The person who likes going on cruises and Arnold are next to each other.
8. The person who loves yellow and the person who enjoys camping trips are next to each other.
9. The person who loves yellow is somewhere to the left of Peter.
10. The person who is a lawyer and Arnold are next to each other.
11. The person who has red hair is Arnold.
12. Alice is the person who is a doctor.
13. The person who loves purple is directly left of the person who loves the vase of tulips.
14. The person who has brown hair is somewhere to the left of the person who is an engineer.
15. The person who has gray hair is somewhere to the left of the person whose favorite color is red.
16. The person who is an artist is somewhere to the right of Arnold.
17. Alice is the person who loves the boquet of lilies.
18. The person whose favorite color is red is directly left of the person who prefers city breaks.
19. The person who prefers city breaks is somewhere to the right of the person who has blonde hair.
20. There is one house between Alice and Bob.
21. The person who loves the rose bouquet is directly left of the person who is a nurse.
22. The person whose favorite color is green is in the fourth house.
23. The person who loves beach vacations is somewhere to the left of the person who has red hair.
24. The person who has black hair is Alice.
25. The person who loves blue is the person who loves a carnations arrangement.
 What is the value of attribute Occupation for the person whose attribute Color is yellow? Please reason step by step, and put your final answer within \boxed{}

Answer: nurse