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`, `Alice`, `Arnold`, `Eric`
 - Each person lives in a unique style of house: `victorian`, `ranch`, `craftsman`, `colonial`
 - People have unique favorite music genres: `pop`, `classical`, `rock`, `jazz`
 - They all have a unique favorite flower: `roses`, `lilies`, `carnations`, `daffodils`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`, `cruise`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`

## Clues:
1. The person who loves classical music is Peter.
2. The person in a ranch-style home is not in the fourth house.
3. The person who loves a bouquet of daffodils is the person living in a colonial-style house.
4. The person living in a colonial-style house is the person who loves white.
5. The person who prefers city breaks and the person whose favorite color is red are next to each other.
6. The person who loves the rose bouquet is not in the fourth house.
7. The person who likes going on cruises is in the second house.
8. The person who prefers city breaks is the person who loves rock music.
9. The person who loves pop music is Alice.
10. The person who loves the rose bouquet is the person who prefers city breaks.
11. Eric is the person who enjoys mountain retreats.
12. The person who enjoys mountain retreats is directly left of the person who loves a bouquet of daffodils.
13. The person who loves yellow is somewhere to the left of the person who likes going on cruises.
14. The person who enjoys mountain retreats is the person residing in a Victorian house.
15. The person who loves classical music is the person who likes going on cruises.
16. Peter is somewhere to the right of the person who loves the boquet of lilies.
 What is the value of attribute Color for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: white


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: `Peter`, `Eric`, `Alice`, `Bob`, `Arnold`
 - The people keep unique animals: `cat`, `horse`, `fish`, `dog`, `bird`
 - The mothers' names in different houses are unique: `Penny`, `Holly`, `Janelle`, `Kailyn`, `Aniya`

## Clues:
1. The person whose mother's name is Janelle and The person whose mother's name is Penny are next to each other.
2. There are two houses between the fish enthusiast and The person whose mother's name is Kailyn.
3. There are two houses between Peter and Bob.
4. The bird keeper is in the second house.
5. The person who keeps horses is directly left of the dog owner.
6. The fish enthusiast and Peter are next to each other.
7. The bird keeper is somewhere to the right of Alice.
8. The person whose mother's name is Aniya is somewhere to the left of The person whose mother's name is Penny.
9. The person whose mother's name is Janelle is in the second house.
10. Arnold is directly left of the person who keeps horses.
 What is the value of attribute House for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


Tim is doing a cakewalk with $11$ steps. He takes his first step on step $1$ and takes a total of $139$ steps, walking in a circle (so that after the 11th step he reaches the first step again). Which step of the cakewalk does he end on?

Answer: 7


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`
 - They all have a unique favorite flower: `lilies`, `carnations`, `roses`, `daffodils`
 - Each person has a favorite color: `white`, `green`, `yellow`, `red`
 - Each person has a unique level of education: `bachelor`, `associate`, `master`, `high school`
 - Each person has a unique hobby: `painting`, `photography`, `gardening`, `cooking`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `prince`, `pall mall`

## Clues:
1. The person partial to Pall Mall is the person with an associate's degree.
2. The person who loves white is the Prince smoker.
3. The person with a high school diploma is not in the fourth house.
4. The person with a master's degree is not in the second house.
5. Arnold is not in the fourth house.
6. The person whose favorite color is red is not in the third house.
7. The person whose favorite color is red is somewhere to the left of Eric.
8. The person who loves cooking is somewhere to the right of Peter.
9. The person who loves a bouquet of daffodils is somewhere to the left of the person who paints as a hobby.
10. The person who loves white is the person who loves the boquet of lilies.
11. Alice is the photography enthusiast.
12. Arnold is somewhere to the right of the person partial to Pall Mall.
13. The person who loves white is in the second house.
14. The person whose favorite color is green is directly left of the person who enjoys gardening.
15. The Dunhill smoker is the person with a high school diploma.
16. The person who loves the rose bouquet is somewhere to the right of the person whose favorite color is green.
 What is the value of attribute Cigar for the person whose attribute Hobby is painting? Please reason step by step, and put your final answer within \boxed{}

Answer: prince


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`, `Alice`, `Eric`, `Peter`, `Bob`, `Carol`
 - They all have a unique favorite flower: `daffodils`, `roses`, `tulips`, `iris`, `carnations`, `lilies`
 - People have unique favorite sports: `basketball`, `swimming`, `soccer`, `volleyball`, `tennis`, `baseball`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `lime`, `blueberry`, `dragonfruit`, `desert`
 - Each person has an occupation: `lawyer`, `doctor`, `teacher`, `artist`, `nurse`, `engineer`

## Clues:
1. The person who loves a bouquet of daffodils is the person who is an artist.
2. The person who loves tennis is not in the second house.
3. The person who is an artist is somewhere to the right of the person who likes Cherry smoothies.
4. The person who loves soccer is somewhere to the right of the person who is a teacher.
5. The person who loves basketball is directly left of the person who loves the vase of tulips.
6. The person who is a nurse is in the sixth house.
7. The person who loves a bouquet of daffodils and the Dragonfruit smoothie lover are next to each other.
8. The person who loves tennis is not in the third house.
9. The person who loves the boquet of iris is not in the second house.
10. The person who is a teacher is not in the second house.
11. The person who loves a carnations arrangement is the Watermelon smoothie lover.
12. There are two houses between the person who loves a bouquet of daffodils and the person who is a nurse.
13. The person who loves volleyball is the Dragonfruit smoothie lover.
14. Peter is the person who is a nurse.
15. There are two houses between the person who is a doctor and the person who is a lawyer.
16. The person who loves baseball is somewhere to the left of the person who drinks Blueberry smoothies.
17. The person who drinks Lime smoothies is somewhere to the left of Carol.
18. The person who loves the vase of tulips is the Desert smoothie lover.
19. The person who drinks Blueberry smoothies is Alice.
20. There is one house between the person who is an artist and Arnold.
21. Eric is directly left of Arnold.
22. There is one house between the person who is a doctor and Alice.
23. The person who loves the boquet of lilies is in the first house.
24. The person who is a doctor is not in the fifth house.
 What is the value of attribute Flower for the person whose attribute Occupation is engineer? Please reason step by step, and put your final answer within \boxed{}

Answer: roses


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`, `Bob`, `Peter`, `Arnold`, `Alice`
 - Each person prefers a unique type of vacation: `cultural`, `beach`, `mountain`, `cruise`, `camping`, `city`
 - The people are of nationalities: `chinese`, `brit`, `norwegian`, `swede`, `dane`, `german`
 - Everyone has a favorite smoothie: `lime`, `desert`, `watermelon`, `blueberry`, `dragonfruit`, `cherry`

## Clues:
1. The person who likes Cherry smoothies is the person who enjoys mountain retreats.
2. Eric is the Dane.
3. The German is Peter.
4. The Desert smoothie lover is in the sixth house.
5. The Swedish person is the Dragonfruit smoothie lover.
6. Arnold is the Dragonfruit smoothie lover.
7. The Watermelon smoothie lover is the British person.
8. The Norwegian is the person who enjoys camping trips.
9. The person who likes going on cruises is the British person.
10. There are two houses between Bob and the person who enjoys camping trips.
11. The person who drinks Blueberry smoothies is directly left of the person who goes on cultural tours.
12. Bob is directly left of the Dragonfruit smoothie lover.
13. The person who likes going on cruises is directly left of Peter.
14. The person who prefers city breaks is not in the sixth house.
15. Carol is in the fifth house.
 What is the value of attribute House for the person whose attribute Smoothie is watermelon? 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: `Eric`, `Peter`, `Alice`, `Bob`, `Arnold`
 - The people are of nationalities: `norwegian`, `brit`, `swede`, `dane`, `german`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `camping`, `beach`, `city`
 - Each person has a unique level of education: `bachelor`, `master`, `associate`, `doctorate`, `high school`
 - Each person has an occupation: `artist`, `doctor`, `engineer`, `teacher`, `lawyer`

## Clues:
1. The person who likes going on cruises is the person who is a lawyer.
2. The person who loves beach vacations is directly left of Arnold.
3. The person with a doctorate is somewhere to the left of Bob.
4. The person with an associate's degree is the person who likes going on cruises.
5. Peter is not in the first house.
6. The person who is an artist is Peter.
7. The person who enjoys camping trips is the person with a master's degree.
8. The Dane is somewhere to the right of the person who is a doctor.
9. The person with an associate's degree is directly left of the person who is an engineer.
10. The person who enjoys camping trips is the British person.
11. The Norwegian and the person with a bachelor's degree are next to each other.
12. The person who is an artist is the Swedish person.
13. Bob is not in the fourth house.
14. The person who enjoys camping trips is Eric.
15. Alice is the German.
16. The person who loves beach vacations is somewhere to the left of the person who prefers city breaks.
17. The person who enjoys mountain retreats is in the fifth house.
18. The person who likes going on cruises is somewhere to the right of the person who loves beach vacations.
19. The person with a bachelor's degree is in the third house.
 What is the value of attribute House for the person whose attribute Occupation is engineer? 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`, `Peter`, `Alice`, `Arnold`
 - People have unique hair colors: `brown`, `black`, `blonde`, `red`
 - People have unique favorite sports: `tennis`, `basketball`, `soccer`, `swimming`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `dragonfruit`, `desert`
 - Each person has a unique birthday month: `april`, `sept`, `feb`, `jan`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `iphone 13`, `google pixel 6`

## Clues:
1. The person whose birthday is in January is somewhere to the left of the person who loves basketball.
2. The person whose birthday is in February is directly left of the person who has red hair.
3. Eric is the person who uses a OnePlus 9.
4. The person whose birthday is in April is Arnold.
5. The person who uses a Samsung Galaxy S21 and the person who likes Cherry smoothies are next to each other.
6. The person who has blonde hair and Peter are next to each other.
7. The person who has black hair is somewhere to the right of the person who has brown hair.
8. The person who has black hair is the person whose birthday is in February.
9. The Dragonfruit smoothie lover is in the second house.
10. The person who loves soccer is somewhere to the right of the person who uses a OnePlus 9.
11. There are two houses between the person who loves tennis and the person who loves basketball.
12. The person who has black hair is the person who uses a Samsung Galaxy S21.
13. Alice is the person who likes Cherry smoothies.
14. The Desert smoothie lover is the person who uses a Google Pixel 6.
 What is the value of attribute PhoneModel for the person whose attribute Smoothie is cherry? Please reason step by step, and put your final answer within \boxed{}

Answer: iphone 13


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`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `desert`, `watermelon`
 - People have unique favorite sports: `soccer`, `tennis`, `basketball`, `swimming`
 - People own unique car models: `tesla model 3`, `toyota camry`, `honda civic`, `ford f150`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`

## Clues:
1. The person who owns a Tesla Model 3 is the person who loves the rose bouquet.
2. Peter is the Dragonfruit smoothie lover.
3. The Desert smoothie lover is the person who owns a Toyota Camry.
4. The person who loves tennis is in the first house.
5. The person who owns a Toyota Camry and the person who loves basketball are next to each other.
6. Arnold is the person who loves basketball.
7. The person who owns a Honda Civic is the person who loves a bouquet of daffodils.
8. Eric is the person who loves the rose bouquet.
9. The Watermelon smoothie lover is not in the first house.
10. The person who owns a Honda Civic is somewhere to the right of the Desert smoothie lover.
11. The person who loves basketball is the person who loves the boquet of lilies.
12. The person who loves tennis and the person who loves soccer are next to each other.
 What is the value of attribute CarModel for the person whose attribute FavoriteSport is basketball? Please reason step by step, and put your final answer within \boxed{}

Answer: ford f150


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`, `Eric`, `Alice`, `Arnold`, `Carol`
 - People have unique hair colors: `auburn`, `blonde`, `brown`, `black`, `red`, `gray`
 - People have unique heights: `very tall`, `average`, `very short`, `tall`, `super tall`, `short`

## Clues:
1. The person who has blonde hair is directly left of Bob.
2. Alice is in the fourth house.
3. The person who is short is Arnold.
4. The person who is tall is in the sixth house.
5. The person who has black hair is not in the fourth house.
6. The person who has red hair is Eric.
7. The person who is super tall is somewhere to the right of the person who has an average height.
8. The person who has blonde hair is Carol.
9. There is one house between the person who has gray hair and the person who has red hair.
10. The person who is very short is in the fifth house.
11. Bob is the person who has brown hair.
12. The person who has gray hair is in the third house.
13. The person who has blonde hair is the person who is very tall.
 What is the value of attribute House for the person whose attribute Height is short? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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 Flower for the person whose attribute CarModel is bmw 3 series? Please reason step by step, and put your final answer within \boxed{}

Answer: tulips


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`, `Arnold`, `Alice`
 - People have unique favorite music genres: `pop`, `classical`, `jazz`, `rock`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `photography`
 - Each person has a unique level of education: `master`, `associate`, `high school`, `bachelor`
 - Each person has a unique type of pet: `fish`, `bird`, `dog`, `cat`

## Clues:
1. The person who loves classical music is not in the fourth house.
2. Arnold is the person with a master's degree.
3. Alice is the person who owns a dog.
4. The person who enjoys gardening is not in the third house.
5. The person with a bachelor's degree is Alice.
6. Alice is the photography enthusiast.
7. Arnold is directly left of the person who loves pop music.
8. The person with an associate's degree is directly left of the photography enthusiast.
9. The person who loves rock music is the person who loves cooking.
10. The person with an associate's degree is Eric.
11. The person who has a cat is the person who loves jazz music.
12. Arnold is somewhere to the right of the person with an aquarium of fish.
 What is the value of attribute Hobby for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: painting


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 House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: soup


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`, `Arnold`, `Bob`, `Carol`, `Alice`, `Eric`
 - People have unique heights: `very tall`, `very short`, `super tall`, `average`, `tall`, `short`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `blueberry`, `desert`, `dragonfruit`, `lime`
 - Everyone has a unique favorite cigar: `prince`, `blue master`, `yellow monster`, `dunhill`, `blends`, `pall mall`
 - Each person prefers a unique type of vacation: `camping`, `city`, `beach`, `cultural`, `mountain`, `cruise`

## Clues:
1. The person who is very short is directly left of the person who drinks Lime smoothies.
2. The person who smokes Yellow Monster is directly left of the person who drinks Lime smoothies.
3. The person who is short and the person who goes on cultural tours are next to each other.
4. The person who smokes many unique blends is the person who loves beach vacations.
5. There are two houses between the person who smokes Blue Master and the person who drinks Blueberry smoothies.
6. The Prince smoker is not in the fourth house.
7. The person who is tall is not in the first house.
8. The person who smokes Yellow Monster is the person who enjoys mountain retreats.
9. The person who is very tall is Alice.
10. The person who likes Cherry smoothies is the person who smokes Blue Master.
11. Bob is in the first house.
12. The person who smokes many unique blends is Alice.
13. The person who goes on cultural tours is the person partial to Pall Mall.
14. Eric is the person who drinks Blueberry smoothies.
15. The person who prefers city breaks is not in the sixth house.
16. The Prince smoker is the person who prefers city breaks.
17. The person who enjoys camping trips is somewhere to the left of Carol.
18. The person who has an average height is the person who prefers city breaks.
19. The Desert smoothie lover is in the fifth house.
20. The Watermelon smoothie lover is Arnold.
21. The person who is very tall is the person who drinks Lime smoothies.
22. Carol is the Dunhill smoker.
 What is the value of attribute House for the person whose attribute Cigar is dunhill? 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: `Peter`, `Arnold`, `Eric`, `Alice`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `lilies`
 - People have unique heights: `very short`, `short`, `tall`, `average`
 - The mothers' names in different houses are unique: `Janelle`, `Kailyn`, `Holly`, `Aniya`
 - Each person has an occupation: `engineer`, `doctor`, `teacher`, `artist`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `soccer`

## Clues:
1. The person who loves swimming is the person who loves the rose bouquet.
2. The person who loves the rose bouquet is Eric.
3. Arnold is the person who is tall.
4. The person who loves a bouquet of daffodils is somewhere to the right of the person who is an engineer.
5. The person who loves soccer is the person who is short.
6. The person who is a teacher is in the first house.
7. The person whose mother's name is Janelle is the person who loves a carnations arrangement.
8. The person who loves basketball is the person who has an average height.
9. Arnold is not in the third house.
10. The person whose mother's name is Holly is somewhere to the right of the person who has an average height.
11. Peter is the person who is a doctor.
12. The person whose mother's name is Aniya is Alice.
13. Arnold is the person who loves the boquet of lilies.
 What is the value of attribute Mother for the person whose attribute Flower is lilies? 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: `Peter`, `Eric`, `Arnold`, `Bob`, `Alice`, `Carol`
 - The people keep unique animals: `rabbit`, `dog`, `fish`, `bird`, `cat`, `horse`
 - Each person has a unique hobby: `cooking`, `knitting`, `painting`, `woodworking`, `photography`, `gardening`
 - People have unique favorite sports: `basketball`, `soccer`, `swimming`, `tennis`, `baseball`, `volleyball`
 - Each person prefers a unique type of vacation: `cruise`, `cultural`, `city`, `camping`, `mountain`, `beach`
 - Each person has a unique favorite drink: `water`, `root beer`, `boba tea`, `tea`, `milk`, `coffee`

## Clues:
1. The person who enjoys camping trips is the person who loves cooking.
2. The person who loves baseball is the photography enthusiast.
3. The person who loves volleyball is somewhere to the left of the person who prefers city breaks.
4. Carol is the person who loves tennis.
5. The dog owner is the person who loves swimming.
6. The photography enthusiast is the tea drinker.
7. The person who loves baseball is directly left of the person who goes on cultural tours.
8. Peter is the person who enjoys camping trips.
9. The person who enjoys knitting is the person who keeps horses.
10. The coffee drinker is directly left of the person who loves soccer.
11. The person who enjoys mountain retreats is the person who loves soccer.
12. The boba tea drinker is not in the fourth house.
13. The cat lover is somewhere to the left of the dog owner.
14. The person who loves beach vacations is Bob.
15. The coffee drinker is somewhere to the right of Alice.
16. The tea drinker is the fish enthusiast.
17. The person who likes milk is Carol.
18. There is one house between the rabbit owner and the woodworking hobbyist.
19. The person who loves cooking is not in the third house.
20. The person who loves tennis is the rabbit owner.
21. The dog owner is directly left of the person who likes going on cruises.
22. The woodworking hobbyist is the dog owner.
23. There are two houses between the person who paints as a hobby and the person who keeps horses.
24. Arnold is the root beer lover.
 What is the value of attribute Hobby for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: woodworking


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`, `Bob`, `Eric`, `Alice`
 - People have unique hair colors: `gray`, `brown`, `black`, `red`, `blonde`
 - Each person prefers a unique type of vacation: `city`, `mountain`, `camping`, `cruise`, `beach`
 - They all have a unique favorite flower: `roses`, `tulips`, `carnations`, `lilies`, `daffodils`
 - Everyone has something unique for lunch: `stew`, `stir fry`, `spaghetti`, `pizza`, `grilled cheese`
 - Each person has a unique hobby: `painting`, `photography`, `gardening`, `knitting`, `cooking`

## Clues:
1. The person who loves the stew is directly left of the person who loves the vase of tulips.
2. The person who enjoys camping trips is the person who loves eating grilled cheese.
3. The person who has black hair is somewhere to the left of Bob.
4. The photography enthusiast is the person who has gray hair.
5. The person who is a pizza lover is the person who likes going on cruises.
6. Alice is the person who prefers city breaks.
7. The person who loves beach vacations is directly left of the person who paints as a hobby.
8. The person who loves the spaghetti eater is Arnold.
9. Bob is in the fifth house.
10. The person who loves the boquet of lilies is in the third house.
11. The person who enjoys knitting is in the fourth house.
12. The person who has brown hair is directly left of the person who paints as a hobby.
13. There are two houses between the person who likes going on cruises and the person who enjoys knitting.
14. The person who has red hair is the person who loves the spaghetti eater.
15. The person who loves a carnations arrangement is not in the second house.
16. Eric is somewhere to the left of the person who enjoys gardening.
17. The photography enthusiast is the person who loves the stew.
18. Eric is the person who loves a bouquet of daffodils.
 What is the value of attribute House for the person whose attribute HairColor is red? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


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`, `Carol`, `Eric`, `Alice`, `Peter`, `Arnold`
 - Each person prefers a unique type of vacation: `camping`, `cruise`, `mountain`, `cultural`, `city`, `beach`
 - Each person has a unique type of pet: `fish`, `hamster`, `bird`, `rabbit`, `dog`, `cat`
 - The people are of nationalities: `brit`, `swede`, `german`, `norwegian`, `dane`, `chinese`
 - People have unique favorite book genres: `biography`, `romance`, `mystery`, `fantasy`, `science fiction`, `historical fiction`
 - People own unique car models: `tesla model 3`, `bmw 3 series`, `honda civic`, `chevrolet silverado`, `toyota camry`, `ford f150`

## Clues:
1. The person who loves fantasy books is Peter.
2. The Chinese is somewhere to the right of the Dane.
3. The person who owns a Tesla Model 3 is the person with a pet hamster.
4. Carol is the person who loves beach vacations.
5. The person who has a cat is in the second house.
6. There is one house between the person who loves mystery books and the person who prefers city breaks.
7. The person who owns a Honda Civic is in the first house.
8. The person who loves science fiction books is directly left of the person who prefers city breaks.
9. The British person is the person who owns a Honda Civic.
10. There is one house between the person who loves fantasy books and the person who owns a Toyota Camry.
11. The person who keeps a pet bird is the person who owns a Toyota Camry.
12. The Dane is not in the third house.
13. Eric is not in the sixth house.
14. The person who loves romance books is the person who enjoys camping trips.
15. Arnold is the person who likes going on cruises.
16. The person who owns a dog is in the fifth house.
17. The person who enjoys camping trips is directly left of the person who owns a rabbit.
18. The person who owns a Ford F-150 is the person who owns a rabbit.
19. The person who loves fantasy books is directly left of Eric.
20. The German is the person who owns a BMW 3 Series.
21. The person who goes on cultural tours is not in the first house.
22. The person who loves mystery books is not in the sixth house.
23. The person who loves historical fiction books is somewhere to the left of the person who enjoys camping trips.
24. The Dane is Bob.
25. There is one house between the Dane and the Chinese.
26. The Swedish person is somewhere to the right of the person who likes going on cruises.
27. Eric is not in the third house.
 What is the value of attribute BookGenre for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


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`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `google pixel 6`
 - People have unique heights: `very short`, `average`, `short`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`
 - People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`

## Clues:
1. Peter is somewhere to the right of Eric.
2. The person living in a colonial-style house is in the second house.
3. The person who owns a Tesla Model 3 is the person who is very short.
4. The person who is short is directly left of the person who uses a Samsung Galaxy S21.
5. The person who uses an iPhone 13 is directly left of the person who uses a Google Pixel 6.
6. The person living in a colonial-style house is somewhere to the right of the person in a ranch-style home.
7. Arnold is in the second house.
8. The person who owns a Ford F-150 is somewhere to the right of the person who owns a Toyota Camry.
9. The person who has an average height is in the first house.
 What is the value of attribute PhoneModel for the person whose attribute CarModel is tesla model 3? Please reason step by step, and put your final answer within \boxed{}

Answer: samsung galaxy s21


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`, `Alice`, `Carol`, `Eric`, `Arnold`
 - People have unique favorite sports: `swimming`, `tennis`, `baseball`, `soccer`, `volleyball`, `basketball`
 - People have unique heights: `super tall`, `average`, `very short`, `very tall`, `short`, `tall`
 - People own unique car models: `chevrolet silverado`, `tesla model 3`, `toyota camry`, `ford f150`, `bmw 3 series`, `honda civic`

## Clues:
1. The person who is very short is the person who loves swimming.
2. The person who loves baseball is the person who is short.
3. Alice is the person who owns a Honda Civic.
4. The person who loves baseball is somewhere to the right of Bob.
5. The person who loves soccer is somewhere to the right of Carol.
6. There are two houses between the person who loves volleyball and the person who has an average height.
7. Arnold is in the third house.
8. The person who loves tennis is not in the fourth house.
9. The person who owns a Chevrolet Silverado and the person who owns a Toyota Camry are next to each other.
10. Alice is directly left of the person who is super tall.
11. The person who owns a Ford F-150 is the person who loves swimming.
12. Eric is in the fourth house.
13. The person who is short is the person who owns a Chevrolet Silverado.
14. The person who loves baseball is in the fifth house.
15. Eric is the person who owns a BMW 3 Series.
16. The person who loves swimming is somewhere to the left of the person who is very tall.
17. The person who has an average height is the person who owns a BMW 3 Series.
 What is the value of attribute House for the person whose attribute Name is Bob? 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: `Arnold`, `Alice`, `Eric`, `Peter`, `Bob`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Samantha`, `Meredith`, `Timothy`
 - Each person has a unique hobby: `painting`, `photography`, `cooking`, `knitting`, `gardening`

## Clues:
1. The person's child is named Fred is the person who enjoys gardening.
2. The person's child is named Meredith is directly left of the person who enjoys gardening.
3. Bob is not in the second house.
4. Eric is directly left of the person who paints as a hobby.
5. The person who loves cooking is not in the fourth house.
6. Bob is directly left of Alice.
7. The person who is the mother of Timothy is directly left of the person who loves cooking.
8. The person who enjoys knitting is the person's child is named Bella.
9. The person's child is named Bella is somewhere to the left of the photography enthusiast.
10. The person who paints as a hobby is the person's child is named Samantha.
11. Peter is not in the third house.
 What is the value of attribute Name for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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: `Arnold`, `Eric`, `Peter`
 - Each person has a unique hobby: `cooking`, `photography`, `gardening`
 - The people are of nationalities: `dane`, `brit`, `swede`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`
 - Everyone has a favorite smoothie: `cherry`, `desert`, `watermelon`

## Clues:
1. The Desert smoothie lover is the person who enjoys gardening.
2. The photography enthusiast is the Swedish person.
3. Peter is somewhere to the left of the Desert smoothie lover.
4. The person who is a pizza lover is in the third house.
5. The person who enjoys gardening is the Dane.
6. There is one house between the person who loves eating grilled cheese and Arnold.
7. Arnold is the Swedish person.
8. The Watermelon smoothie lover is somewhere to the left of the person who likes Cherry smoothies.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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`
 - Each person has a unique birthday month: `jan`, `sept`, `april`
 - The people are of nationalities: `brit`, `swede`, `dane`
 - Each person has a favorite color: `yellow`, `white`, `red`

## Clues:
1. The Dane is the person whose favorite color is red.
2. The person who loves yellow is the person whose birthday is in September.
3. The person whose birthday is in September is directly left of the person whose birthday is in April.
4. The person whose birthday is in September is somewhere to the left of Arnold.
5. The Dane is in the second house.
6. The person whose birthday is in January is Peter.
7. The British person is the person who loves white.
 What is the value of attribute Birthday for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


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 has a unique favorite drink: `milk`, `tea`, `water`
 - People have unique favorite sports: `soccer`, `basketball`, `tennis`
 - The people keep unique animals: `bird`, `horse`, `cat`
 - People have unique favorite book genres: `mystery`, `science fiction`, `romance`

## Clues:
1. The person who loves mystery books is the person who loves tennis.
2. The person who loves tennis is the tea drinker.
3. There is one house between the person who keeps horses and the bird keeper.
4. Peter is the person who likes milk.
5. Eric is the person who keeps horses.
6. The person who loves soccer is in the third house.
7. The person who loves science fiction books is the person who loves basketball.
8. The tea drinker is Arnold.
9. The person who likes milk is the person who loves soccer.
 What is the value of attribute Drink for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: milk


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`, `Eric`, `Bob`, `Alice`, `Peter`
 - Each person has a unique birthday month: `feb`, `mar`, `sept`, `jan`, `may`, `april`
 - Everyone has something unique for lunch: `stew`, `soup`, `grilled cheese`, `stir fry`, `spaghetti`, `pizza`
 - People have unique heights: `very short`, `average`, `super tall`, `short`, `very tall`, `tall`
 - People own unique car models: `chevrolet silverado`, `ford f150`, `bmw 3 series`, `tesla model 3`, `toyota camry`, `honda civic`

## Clues:
1. The person who owns a Honda Civic is the person who is short.
2. The person who owns a Ford F-150 is in the fifth house.
3. The person who loves stir fry is somewhere to the left of Eric.
4. The person whose birthday is in May is somewhere to the left of Carol.
5. The person who is very short is somewhere to the left of the person whose birthday is in April.
6. The person who owns a BMW 3 Series is not in the third house.
7. There are two houses between the person who loves stir fry and the person who is a pizza lover.
8. The person who loves the soup is directly left of Eric.
9. The person who loves the spaghetti eater and the person whose birthday is in May are next to each other.
10. Alice is directly left of the person who owns a BMW 3 Series.
11. The person who owns a Tesla Model 3 is somewhere to the left of the person who is tall.
12. The person who is very tall is the person who owns a Toyota Camry.
13. Peter is directly left of the person who is a pizza lover.
14. The person who loves the stew is not in the third house.
15. There is one house between the person whose birthday is in September and the person who is very short.
16. There is one house between the person whose birthday is in March and the person who is super tall.
17. The person who is tall is Bob.
18. The person whose birthday is in May is somewhere to the right of Alice.
19. The person who is very short is in the fourth house.
20. The person whose birthday is in March is the person who is short.
21. Carol is the person who owns a Tesla Model 3.
22. Eric is the person whose birthday is in January.
 What is the value of attribute Birthday for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


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 Cigar for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: prince


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`, `Eric`, `Peter`, `Alice`
 - Each person has a unique level of education: `master`, `high school`, `associate`, `bachelor`
 - Everyone has something unique for lunch: `stew`, `grilled cheese`, `pizza`, `spaghetti`
 - People have unique favorite sports: `soccer`, `swimming`, `tennis`, `basketball`
 - People have unique hair colors: `blonde`, `black`, `brown`, `red`
 - People have unique favorite book genres: `mystery`, `fantasy`, `science fiction`, `romance`

## Clues:
1. The person who loves the stew is in the first house.
2. The person who loves science fiction books is the person who has black hair.
3. The person who has blonde hair is not in the second house.
4. The person with a master's degree is in the first house.
5. Eric is the person who loves soccer.
6. The person with a master's degree and the person who loves mystery books are next to each other.
7. The person who loves tennis is the person with a high school diploma.
8. The person who loves the spaghetti eater is the person who loves science fiction books.
9. The person who is a pizza lover is the person with an associate's degree.
10. The person who has blonde hair is not in the fourth house.
11. The person with a master's degree is directly left of Eric.
12. The person with a bachelor's degree is directly left of the person who loves fantasy books.
13. The person who has red hair is Peter.
14. The person who loves swimming is directly left of the person who is a pizza lover.
15. Alice is not in the first house.
 What is the value of attribute BookGenre for the person whose attribute HairColor is red? Please reason step by step, and put your final answer within \boxed{}

Answer: fantasy


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`, `Eric`, `Alice`, `Peter`
 - Each person has a unique birthday month: `feb`, `jan`, `april`, `sept`
 - People have unique hair colors: `black`, `blonde`, `brown`, `red`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`, `craftsman`
 - Each person has an occupation: `engineer`, `artist`, `doctor`, `teacher`
 - Everyone has a favorite smoothie: `cherry`, `dragonfruit`, `desert`, `watermelon`

## Clues:
1. The person who is an engineer is the person whose birthday is in February.
2. The person who likes Cherry smoothies is somewhere to the left of the person who has red hair.
3. The person who is a doctor is the person who has blonde hair.
4. The person whose birthday is in January is the person living in a colonial-style house.
5. Alice is the person residing in a Victorian house.
6. Peter is the Desert smoothie lover.
7. Peter is the person whose birthday is in September.
8. The Dragonfruit smoothie lover is the person who is a teacher.
9. The Watermelon smoothie lover is the person residing in a Victorian house.
10. The person living in a colonial-style house is directly left of the person who has black hair.
11. The person whose birthday is in February is in the fourth house.
12. The person who has black hair is not in the third house.
13. Eric is the person in a ranch-style home.
14. The person who has blonde hair is directly left of the person who likes Cherry smoothies.
 What is the value of attribute House for the person whose attribute Smoothie is dragonfruit? 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: `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 Birthday is feb? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


A square is drawn in the Cartesian coordinate plane with vertices at $(2, 2)$ $(-2, 2)$ $(-2, -2)$ $(2, -2)$ . A particle starts at $(0,0)$ . Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is $1/8$ that the particle will move from $(x, y)$ to each of $(x, y + 1)$ $(x + 1, y + 1)$ $(x + 1, y)$ $(x + 1, y - 1)$ $(x, y - 1)$ $(x - 1, y - 1)$ $(x - 1, y)$ , or $(x - 1, y + 1)$ . The particle will eventually hit the square for the first time, either at one of the 4 corners of the square or at one of the 12 lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is $m/n$ , where $m$ and $n$ are relatively prime positive integers. What is $m + n$

Answer: 39


A uniform ladder of mass  $m$  and length  $\mathcal{L}$  is resting on a wall. A man of mass  $m$  climbs up the ladder and is in perfect equilibrium with the ladder when he is  $\frac{2}{3}\mathcal{L}$  the way up the ladder. The ladder makes an angle of  $ \theta = 30^\circ $  with the horizontal floor. If the coefficient of static friction between the ladder and the wall is the same as that between the ladder and the floor, which is  $\mu$ , what is  $\mu$ , expressed to the nearest thousandth?

*(Proposed by Ahaan Rungta)*

Answer: 0.678


In a book written in 628 AD, there is also such a problem: "An experienced architect built a magnificent palace for the king, which has eight doors. Each time, one door, or two doors, or three doors... are opened. How many different ways are there to open the doors?"

Answer: 255


[The triangle formed by the bases of two altitudes and a vertex]

The side of the triangle is $\sqrt{2}$, and the angles adjacent to it are $75^{\circ}$ and $60^{\circ}$.

Find the segment connecting the bases of the altitudes drawn from the vertices of these angles.

Answer: 1


4・68 In a competition with $n(n \geqslant 2)$ participants lasting $k$ days, each participant's score each day is exactly $1,2, \cdots, n$ (no two participants have the same score), and after $k$ days, each participant's total score is 26. Try to find all possible values of $(n, k)$.

Answer: (3, 13)


## Task 3 - 030733

The number $30 \star 0 \star 03$ should be divisible by 13. The $\star$ should each be replaced by one of the digits from 0 to 9. (The same digit does not necessarily have to be used for both stars.)

Give all the numbers that have the required property!

Answer: 3000803


B2. Karel has a large quantity of apples and pears. He wants to choose ten pieces of fruit from these and arrange them in a row. This must be done in such a way that there is never a pear between two apples. The rows AAAAAAAAAA and AAPPPPPPPP are allowed, for example, but the rows AAPPPPPPPA and APPPPPPPAA are not.

How many rows can Karel make like this?

Answer: 56


6.1. On 19 cards, the numbers $15,16,17, \ldots, 33$ are written respectively (one number per card). Members of the math club Vasya, Petya, and Misha decided to divide all these cards among themselves so that each of them gets at least one card and no one ends up with a pair of cards where the difference between the numbers is odd. How many ways are there to distribute the cards?

Answer: 4596


5. (20 points) In an acute-angled triangle $\mathrm{ABC}$, a point $\mathrm{D}$ is chosen on side $\mathrm{BC}$ such that $\mathrm{CD}: \mathrm{DB}=2: 1$, and a point $\mathrm{K}$ is chosen on segment $\mathrm{AD}$ such that $\mathrm{AK}=\mathrm{CD}+\mathrm{DK}$. A line is drawn through point $\mathrm{K}$ and vertex $\mathrm{B}$, intersecting side $\mathrm{AC}$ at point $\mathrm{E}$. Triangle $\mathrm{AEK}$ is isosceles ($\mathrm{AE}=\mathrm{EK}$). Find the measure of angle $\mathrm{ADC}$ in degrees.

Answer: 60


1. Let $\lfloor x\rfloor$ denote the greatest integer not exceeding the real number $x$. Find all positive integers $M$ such that the sequence defined by
$$
\begin{array}{l}
a_{0}=\frac{2 M+1}{2}, \\
a_{k+1}=a_{k}\left\lfloor a_{k}\right\rfloor(k=0,1, \cdots)
\end{array}
$$

contains at least one integer term.

Answer: M \geq 2


Problem 10.1. In each cell of a $5 \times 5$ table, a natural number is written in invisible ink. It is known that the sum of all the numbers is 200, and the sum of three numbers inside any $1 \times 3$ rectangle is 23. What is the central number in the table?

![](https://cdn.mathpix.com/cropped/2024_05_06_6da73bfd3e09e8b55e3fg-39.jpg?height=309&width=311&top_left_y=340&top_left_x=575)

Answer: 16


6. Viewers rate a movie with an integer number of points from 0 to 10. At any given time, the movie's rating is calculated as the sum of all the given ratings divided by their number. At some point in time $T$, the rating was an integer, and then with each new voting viewer, it decreased by one. What is the maximum number of viewers who could have voted after moment $T$? (O. Dmitriev, R. Zhenodarov)

Answer: 6


## Task B-1.1.

Let $a>1$. The points $(x, y)$ for which the coordinates satisfy $|x|+y=a, x^{2}+y=a|x|$ determine a figure in the coordinate plane with an area of 120 square units. Calculate the real number $a$.

Answer: 11


2. If real numbers $x, y$ satisfy
$$
\begin{array}{l}
\frac{x}{2^{10}+5^{3}}+\frac{y}{2^{10}+6^{3}}=1, \\
\frac{x}{3^{10}+5^{3}}+\frac{y}{3^{10}+6^{3}}=1,
\end{array}
$$

then $x+y=$ $\qquad$ (the result should be written in the form of a power).

Answer: 2^{10} + 3^{10} + 5^{3} + 6^{3}


Given a cube $A B C D A 1 B 1 C 1 D 1$. A sphere touches the edges $A D, D D 1, C D$ and the line $B C 1$. Find the radius of the sphere, if the edge of the cube is 1.

Answer: 2\sqrt{2} - \sqrt{5}


Given the tetrahedron \(P Q R S\) with \(P Q = 4\), \(S R = 6\), \(\angle Q R S = \angle P S R = 50^\circ\), and \(\angle Q S R = \angle P R S = 40^\circ\). A sphere is circumscribed around the tetrahedron. Consider the set of all points on this sphere where the sum of the spherical distances from these points to \(P\), \(Q\), \(R\), and \(S\) is no less than \(6\pi\). What is the area of this set? Note that the spherical distance between two points on a sphere is the length of the shortest arc of a great circle connecting these points.

Answer: 18\pi


The diagram below shows an $8$x$7$ rectangle with a 3-4-5 right triangle drawn in each corner. The lower two triangles have their sides of length 4 along the bottom edge of the rectangle, while the upper two
triangles have their sides of length 3 along the top edge of the rectangle. A circle is tangent to the hypotenuse of each triangle. The diameter of the circle is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find m + n.
For diagram go to http://www.purplecomet.org/welcome/practice, go to the 2015 middle school contest questions, and then go to #20

Answer: 47


Five first-graders stood in a row holding 37 flags. To the right of Tata, there are 14 flags; to the right of Yasha, there are 32 flags; to the right of Vera, there are 20 flags; and to the right of Maxim, there are 8 flags. How many flags does Dasha have?

Answer: 8


Example 6 Let positive integers $a, b, c, d$ satisfy $\frac{a}{b}+\frac{c}{d} < 1$ and $a+c=20$. Find the maximum value of $\frac{a}{b}+\frac{c}{d}$.

Answer: \dfrac{1385}{1386}


8. Given the set
$$
\begin{array}{c}
A_{k}=\left\{2^{k-1}+\sum_{i=0}^{k-2} a_{i} 2^{a_{i}+i} \mid a_{i} \in\{0,1\},\right. \\
i=0,1, \cdots, k-2\} .
\end{array}
$$

Let $n_{k}$ denote the sum of all elements in the set $A_{k}$. Then $\sum_{k=1}^{2015} n_{k}=$ $\qquad$

Answer: \dfrac{(2^{2016} - 1)(2^{2016} - 2)}{6}


205. It is required to find a number which, when multiplied by itself, added to two, then doubled, added to three again, divided by 5, and finally multiplied by 10, results in 50.

Answer: 3


5. 155 birds are perched on a circle $C$. If $\overparen{P_{i} P_{j}} \leqslant 10^{\circ}$, we say $P_{i}$ and $P_{j}$ form a mutually visible pair of birds. Find the minimum number of mutually visible pairs of birds (assuming multiple birds can occupy the same position).

 untranslated text:
(assuming a position can simultaneously have several birds).

Answer: 270


4. Let's fix a point $O$ on the plane. Any other point $A$ is uniquely determined by its radius vector $\overrightarrow{O A}$, which we will simply denote as $\vec{A}$. We will use this concept and notation repeatedly in the following. The fixed point $O$ is conventionally called the initial point. We will also introduce the symbol $(A B C)$ - the simple ratio of three points $A, B, C$ lying on the same line; points $A$ and $B$ are called the base points, and point $C$ is the dividing point. The symbol $(A B C)$ is a number determined by the formula:

$$
\overrightarrow{C A}=(A B C) \cdot \overrightarrow{C B}
$$

Furthermore, for two collinear vectors $\vec{a}$ and $\vec{b}$, there is a unique number $\lambda$ such that $\vec{a}=\lambda \vec{b}$. We introduce the notation $\lambda=\frac{\vec{a}}{\vec{b}}$ (the ratio of collinear vectors). In this case, we can write:

$$
(A B C)=\frac{\overrightarrow{C A}}{\overrightarrow{C B}}
$$

Remark. The notation for the ratio of vectors is not related to the scalar product of vectors! (For example, the relation $\frac{\vec{a}}{\vec{b}}=\frac{\vec{c}}{\vec{d}}$ implies $\vec{a}=\lambda \vec{b}$ and $\vec{c}=\lambda \vec{d}$, from which $\vec{a} \cdot \vec{d}=(\lambda \vec{b}) \cdot \left(\frac{1}{\lambda} \vec{c}\right)=\vec{b} \cdot \vec{c}$, but the converse is not true - from the relation $\vec{a} \cdot \vec{d}=\vec{b} \cdot \vec{c}$ it does not follow that $\frac{\vec{a}}{\vec{b}}=\frac{\vec{c}}{\vec{d}}$ : vectors $\vec{a}$ and $\vec{b}$ do not have to be collinear. Provide a corresponding example.)

Derive a formula expressing the vector $\vec{C}$ in terms of the vectors $\vec{A}$ and $\vec{B}$, given that $(A B C)=k$.

Answer: \dfrac{\vec{A} - k\vec{B}}{1 - k}


4. Determine all pairs of natural numbers $(a, b)$ such that $(a+6)(b+5)=3ab$.

Find all pairs of natural numbers $(a, b)$ that satisfy the equation $(a+6)(b+5)=3ab$.

Answer: (4, 25)


The centers \(O_{1}, O_{2},\) and \(O_{3}\) of three non-intersecting circles of equal radii are located at the vertices of a triangle. Tangents are drawn from points \(O_{1}, O_{2},\) and \(O_{3}\) to these circles as shown in the figure. It is known that these tangents intersect to form a convex hexagon whose edges are alternately colored red and blue. Prove that the sum of the lengths of the red segments is equal to the sum of the lengths of the blue segments.

Answer: \text{The sum of the lengths of the red segments is equal to the sum of the lengths of the blue segments.}


[ Pythagorean Theorem (direct and inverse) $]$ [ Circle, sector, segment, etc. $\quad]$

In the circular sector $OAB$, the central angle of which is $45^{\circ}$, a rectangle $KMPT$ is inscribed. The side $KM$ of the rectangle lies on the radius $OA$, the vertex $P$ is on the arc $AB$, and the vertex $T$ is on the radius $OB$. The side $KT$ is 3 units longer than the side $KM$. The area of the rectangle $KMPT$ is 18. Find the radius.

Answer: 3\sqrt{13}


An integer $n \ge 2$ is called *delicious* if there exist $n$ positive integers that sum up to 2014 and have distinct remainders when divided by $n$. Determine the smallest delicious integer.

Answer: 4


A magician has $300$ cards, each with a unique number from $1$ to $300$. These cards are arranged in a $3 \times 100$ rectangle, with one card per unit square. The cards are placed such that consecutive numbers are in neighboring squares, but the numbers are not visible. The magician then turns over $k$ cards of his choice. What is the smallest value of $k$ such that the revealed cards can determine the exact positions of all other cards?

Answer: 2


Let $n$ be a positive integer. Define $f(n)$ as the probability that if divisors $a, b, c$ of $n$ are selected uniformly at random with replacement, then \( \gcd(a, \text{lcm}(b, c)) = \text{lcm}(a, \gcd(b, c)) \). Let $s(n)$ be the sum of the distinct prime divisors of $n$. If \( f(n) < \frac{1}{2018} \), compute the smallest possible value of $s(n)$.

Answer: 58


Let $\mathbb{N}_0$ be the set of non-negative integers. Consider a triple $(f,a,b)$, where $f$ is a function from $\mathbb{N}_0$ to $\mathbb{N}_0$ and $a,b \in \mathbb{N}_0$ satisfying the following conditions:

1. $f(1) = 2$
2. $f(a) + f(b) \leq 2\sqrt{f(a)}$
3. For all $n > 0$, $f(n) = f(n-1)f(b) + 2n - f(b)$

Find the sum of all possible values of $f(b+100)$.

Answer: 10201


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`, `Arnold`, `Peter`
 - The people keep unique animals: `cat`, `fish`, `horse`, `bird`
 - Each person has a favorite color: `yellow`, `green`, `white`, `red`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `beach`, `mountain`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `stew`, `grilled cheese`
 - People have unique favorite sports: `soccer`, `swimming`, `basketball`, `tennis`

## Clues:
1. The person whose favorite color is red is the person who loves the spaghetti eater.
2. The person who loves the stew is the cat lover.
3. The person who loves yellow is not in the fourth house.
4. Alice is somewhere to the right of Arnold.
5. The person who loves the spaghetti eater is somewhere to the right of the person who loves soccer.
6. The person who loves white is the person who loves tennis.
7. The fish enthusiast is in the second house.
8. Arnold is the person who loves the stew.
9. The person who prefers city breaks is the person who is a pizza lover.
10. The person who prefers city breaks is directly left of the person who likes going on cruises.
11. The person who loves swimming is the person who loves beach vacations.
12. The person who keeps horses and the person whose favorite color is red are next to each other.
13. The person whose favorite color is green is directly left of Eric.
14. The person who prefers city breaks is somewhere to the left of the cat lover.
 What is the value of attribute FavoriteSport for the person whose attribute Name is Alice? 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`, `Arnold`, `Bob`, `Eric`, `Peter`, `Alice`
 - People use unique phone models: `iphone 13`, `huawei p50`, `oneplus 9`, `google pixel 6`, `xiaomi mi 11`, `samsung galaxy s21`
 - Everyone has a unique favorite cigar: `pall mall`, `yellow monster`, `prince`, `blends`, `dunhill`, `blue master`
 - Each person prefers a unique type of vacation: `cultural`, `beach`, `city`, `camping`, `cruise`, `mountain`
 - People have unique hair colors: `brown`, `red`, `gray`, `black`, `blonde`, `auburn`
 - Each person has a unique favorite drink: `coffee`, `boba tea`, `milk`, `water`, `root beer`, `tea`

## Clues:
1. The person who prefers city breaks is Bob.
2. The person who has brown hair is the one who only drinks water.
3. Eric is directly left of the person who uses a Samsung Galaxy S21.
4. There is one house between the person who loves beach vacations and the person who enjoys camping trips.
5. The one who only drinks water is somewhere to the left of the person who uses a Xiaomi Mi 11.
6. The person who has brown hair is the Dunhill smoker.
7. The person who has red hair is Arnold.
8. The person who has auburn hair is the person who uses an iPhone 13.
9. The person who has gray hair is not in the sixth house.
10. The person who likes milk is the person who has auburn hair.
11. The person who smokes Blue Master is Alice.
12. The person who uses a OnePlus 9 is the person who likes going on cruises.
13. The coffee drinker is the Prince smoker.
14. The coffee drinker is the person who uses a Huawei P50.
15. Eric is the tea drinker.
16. The Prince smoker is in the third house.
17. The person who smokes Yellow Monster is somewhere to the right of the coffee drinker.
18. The person who enjoys mountain retreats is Peter.
19. The person who uses a Huawei P50 is the person who has blonde hair.
20. There are two houses between the person who loves beach vacations and the person who uses a Google Pixel 6.
21. The one who only drinks water is in the fourth house.
22. Eric is the person partial to Pall Mall.
23. Bob is the boba tea drinker.
 What is the value of attribute PhoneModel for the person whose attribute HairColor is red? 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: `Eric`, `Arnold`, `Alice`, `Peter`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `victorian`, `colonial`
 - Each person has a unique favorite drink: `water`, `tea`, `milk`, `coffee`

## Clues:
1. Eric is the one who only drinks water.
2. The person who likes milk is the person residing in a Victorian house.
3. The person in a ranch-style home is somewhere to the left of the person in a Craftsman-style house.
4. Eric is in the third house.
5. Arnold is somewhere to the right of Alice.
6. The person in a ranch-style home is Arnold.
7. Eric and the person residing in a Victorian house are next to each other.
8. The tea drinker is not in the first house.
 What is the value of attribute House for the person whose attribute HouseStyle is victorian? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


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 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 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`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `blends`, `blue master`, `dunhill`
 - The people are of nationalities: `brit`, `norwegian`, `dane`, `swede`, `german`
 - Each person has a unique birthday month: `april`, `jan`, `sept`, `feb`, `mar`
 - Each person has a unique type of pet: `fish`, `bird`, `hamster`, `cat`, `dog`
 - The people keep unique animals: `cat`, `dog`, `bird`, `fish`, `horse`

## Clues:
1. The British person is Bob.
2. Peter is the bird keeper.
3. The person who smokes Blue Master is the Swedish person.
4. The fish enthusiast is Bob.
5. The Dunhill smoker is the German.
6. The cat lover is the person partial to Pall Mall.
7. The person with a pet hamster is the person whose birthday is in March.
8. The person who keeps horses is somewhere to the right of the person whose birthday is in April.
9. The fish enthusiast is the person whose birthday is in January.
10. Alice is the dog owner.
11. The person whose birthday is in February is in the first house.
12. The person who owns a dog is somewhere to the left of Peter.
13. The Dane is the person who smokes many unique blends.
14. Peter and Arnold are next to each other.
15. The person with an aquarium of fish is Peter.
16. The Dunhill smoker is the person who has a cat.
17. The Swedish person is directly left of the person whose birthday is in January.
18. The person who owns a dog is in the third house.
 What is the value of attribute House for the person whose attribute Pet is dog? 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`, `Peter`, `Alice`, `Arnold`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`
 - Each person has a unique birthday month: `feb`, `sept`, `jan`, `april`
 - The mothers' names in different houses are unique: `Aniya`, `Janelle`, `Kailyn`, `Holly`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `blue master`, `dunhill`
 - They all have a unique favorite flower: `lilies`, `daffodils`, `carnations`, `roses`

## Clues:
1. The person who loves yellow is directly left of The person whose mother's name is Kailyn.
2. The person whose birthday is in September and Peter are next to each other.
3. The person whose favorite color is red is not in the fourth house.
4. The person who smokes Blue Master and Arnold are next to each other.
5. The person whose mother's name is Holly is the Dunhill smoker.
6. The person whose birthday is in January is the person who loves a bouquet of daffodils.
7. There are two houses between the person who loves the boquet of lilies and the person whose birthday is in January.
8. The person whose mother's name is Janelle is the person who loves a carnations arrangement.
9. The person whose birthday is in September and the person whose favorite color is red are next to each other.
10. The person whose birthday is in February is the person whose favorite color is green.
11. The person whose favorite color is green is directly left of The person whose mother's name is Janelle.
12. There is one house between the person whose birthday is in February and the person partial to Pall Mall.
13. The person who smokes Blue Master is The person whose mother's name is Aniya.
14. Eric is the person whose birthday is in February.
 What is the value of attribute Birthday for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: feb


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 House for the person whose attribute PhoneModel is xiaomi mi 11? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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 favorite sports: `swimming`, `tennis`, `basketball`, `soccer`
 - Each person prefers a unique type of vacation: `cruise`, `beach`, `mountain`, `city`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `stew`, `pizza`

## Clues:
1. The person who loves the spaghetti eater is somewhere to the left of the person who likes going on cruises.
2. The person who loves the stew is directly left of the person who loves eating grilled cheese.
3. The person who is a pizza lover is in the fourth house.
4. Peter is somewhere to the left of the person who enjoys mountain retreats.
5. The person who likes going on cruises is in the third house.
6. Arnold is the person who loves soccer.
7. The person who loves basketball is not in the first house.
8. There are two houses between the person who prefers city breaks and Eric.
9. The person who loves tennis is the person who likes going on cruises.
10. Alice is in the first house.
 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: stew


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`, `Peter`, `Eric`, `Carol`, `Alice`, `Bob`
 - The people are of nationalities: `swede`, `brit`, `chinese`, `dane`, `german`, `norwegian`
 - People have unique favorite book genres: `science fiction`, `mystery`, `fantasy`, `biography`, `historical fiction`, `romance`
 - Everyone has a favorite smoothie: `blueberry`, `dragonfruit`, `desert`, `watermelon`, `cherry`, `lime`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `huawei p50`, `xiaomi mi 11`, `iphone 13`, `google pixel 6`

## Clues:
1. Alice is the British person.
2. There is one house between Eric and the Dragonfruit smoothie lover.
3. The Chinese is somewhere to the left of Alice.
4. The person who loves historical fiction books is directly left of the person who loves biography books.
5. The person who loves biography books is somewhere to the right of Carol.
6. The person who uses an iPhone 13 is somewhere to the right of the Desert smoothie lover.
7. The person who loves mystery books is the British person.
8. Bob is somewhere to the right of the person who uses a OnePlus 9.
9. Carol is the person who drinks Blueberry smoothies.
10. There are two houses between the person who loves romance books and the Watermelon smoothie lover.
11. The person who likes Cherry smoothies is the Swedish person.
12. The person who loves science fiction books is the person who uses a Google Pixel 6.
13. The person who drinks Lime smoothies is Eric.
14. Bob is in the fifth house.
15. Carol is in the second house.
16. The Norwegian and Alice are next to each other.
17. The Swedish person is not in the fifth house.
18. The German and the Norwegian are next to each other.
19. The person who loves fantasy books is the Dragonfruit smoothie lover.
20. The British person is somewhere to the left of Peter.
21. The person who loves romance books is the person who uses a Samsung Galaxy S21.
22. Alice is the person who uses a Xiaomi Mi 11.
23. The person who loves romance books is Eric.
 What is the value of attribute House for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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: `soccer`, `basketball`, `tennis`
 - Each person lives in a unique style of house: `victorian`, `colonial`, `ranch`
 - Each person has a unique type of pet: `fish`, `dog`, `cat`
 - Each person has a unique birthday month: `sept`, `april`, `jan`
 - The people are of nationalities: `swede`, `dane`, `brit`

## Clues:
1. The person who loves soccer is the person residing in a Victorian house.
2. The person who owns a dog is the person whose birthday is in April.
3. The person who owns a dog is the person residing in a Victorian house.
4. The person living in a colonial-style house is Arnold.
5. The person living in a colonial-style house is the person whose birthday is in January.
6. Arnold is directly left of the person whose birthday is in April.
7. The British person is the person who loves basketball.
8. The Swedish person is the person whose birthday is in April.
9. The Dane is Eric.
10. The person who has a cat is the person whose birthday is in January.
11. The person who loves soccer is in the third house.
 What is the value of attribute Birthday for the person whose attribute FavoriteSport is tennis? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


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`, `Bob`, `Arnold`, `Eric`, `Carol`
 - People own unique car models: `tesla model 3`, `honda civic`, `toyota camry`, `chevrolet silverado`, `bmw 3 series`, `ford f150`

## Clues:
1. Alice is not in the fifth house.
2. Carol is somewhere to the left of the person who owns a Chevrolet Silverado.
3. The person who owns a Toyota Camry is Bob.
4. The person who owns a BMW 3 Series is not in the second house.
5. The person who owns a Honda Civic is somewhere to the right of Arnold.
6. There is one house between the person who owns a BMW 3 Series and the person who owns a Chevrolet Silverado.
7. There is one house between Arnold and Bob.
8. The person who owns a Tesla Model 3 is not in the sixth house.
9. The person who owns a Toyota Camry is not in the third house.
10. The person who owns a Tesla Model 3 is Peter.
11. The person who owns a Honda Civic and the person who owns a Toyota Camry are next to each other.
12. The person who owns a Toyota Camry is not in the fifth house.
13. The person who owns a Chevrolet Silverado is not in the second house.
 What is the value of attribute CarModel for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: honda civic


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`
 - People own unique car models: `toyota camry`, `ford f150`, `tesla model 3`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `victorian`
 - Each person has a unique type of pet: `cat`, `dog`, `fish`
 - Each person has an occupation: `engineer`, `doctor`, `teacher`
 - Each person prefers a unique type of vacation: `city`, `mountain`, `beach`

## Clues:
1. The person with an aquarium of fish is in the first house.
2. The person who owns a Toyota Camry is in the second house.
3. The person who enjoys mountain retreats is not in the second house.
4. The person who prefers city breaks is not in the second house.
5. The person in a ranch-style home is somewhere to the left of Peter.
6. The person who owns a Toyota Camry is directly left of the person living in a colonial-style house.
7. Arnold is the person who has a cat.
8. Eric is somewhere to the left of the person who enjoys mountain retreats.
9. The person who is an engineer is not in the third house.
10. The person who owns a Tesla Model 3 is somewhere to the left of the person who is a teacher.
11. The person who owns a dog is the person who is an engineer.
 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 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`, `Alice`, `Arnold`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `lime`, `desert`, `dragonfruit`
 - Each person has a unique hobby: `gardening`, `photography`, `painting`, `knitting`, `cooking`
 - Each person has a unique type of pet: `cat`, `fish`, `hamster`, `dog`, `bird`

## Clues:
1. Bob is somewhere to the right of the photography enthusiast.
2. Alice is somewhere to the left of the person who loves cooking.
3. The person who paints as a hobby is the person with a pet hamster.
4. The person with an aquarium of fish is the person who enjoys knitting.
5. The person who likes Cherry smoothies is somewhere to the right of the Watermelon smoothie lover.
6. The person who keeps a pet bird is the person who drinks Lime smoothies.
7. The photography enthusiast and the person who owns a dog are next to each other.
8. There are two houses between the person with an aquarium of fish and the Desert smoothie lover.
9. Alice is the person who owns a dog.
10. The person who enjoys knitting is not in the fourth house.
11. Eric is somewhere to the right of Peter.
12. The Dragonfruit smoothie lover is in the second house.
13. Arnold is the person who drinks Lime smoothies.
14. The photography enthusiast is the Dragonfruit smoothie lover.
 What is the value of attribute Smoothie for the person whose attribute Pet is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: lime


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`, `Alice`, `Arnold`
 - People have unique hair colors: `brown`, `black`, `blonde`, `red`
 - People have unique favorite sports: `tennis`, `basketball`, `soccer`, `swimming`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `dragonfruit`, `desert`
 - Each person has a unique birthday month: `april`, `sept`, `feb`, `jan`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `iphone 13`, `google pixel 6`

## Clues:
1. The person whose birthday is in January is somewhere to the left of the person who loves basketball.
2. The person whose birthday is in February is directly left of the person who has red hair.
3. Eric is the person who uses a OnePlus 9.
4. The person whose birthday is in April is Arnold.
5. The person who uses a Samsung Galaxy S21 and the person who likes Cherry smoothies are next to each other.
6. The person who has blonde hair and Peter are next to each other.
7. The person who has black hair is somewhere to the right of the person who has brown hair.
8. The person who has black hair is the person whose birthday is in February.
9. The Dragonfruit smoothie lover is in the second house.
10. The person who loves soccer is somewhere to the right of the person who uses a OnePlus 9.
11. There are two houses between the person who loves tennis and the person who loves basketball.
12. The person who has black hair is the person who uses a Samsung Galaxy S21.
13. Alice is the person who likes Cherry smoothies.
14. The Desert smoothie lover is the person who uses a Google Pixel 6.
 What is the value of attribute Smoothie for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


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 House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


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 MusicGenre for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: rock


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`, `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `iris`, `tulips`, `lilies`
 - The people are of nationalities: `norwegian`, `dane`, `chinese`, `brit`, `german`, `swede`

## Clues:
1. The Dane is the person who loves a carnations arrangement.
2. The Swedish person is Arnold.
3. The Chinese is the person who loves the rose bouquet.
4. The Norwegian and the person who loves the rose bouquet are next to each other.
5. The person who loves the vase of tulips is Arnold.
6. There are two houses between Bob and the person who loves the boquet of iris.
7. The person who loves a carnations arrangement is directly left of the person who loves a bouquet of daffodils.
8. The Dane is in the second house.
9. The person who loves a carnations arrangement is directly left of Alice.
10. The person who loves a bouquet of daffodils and the Swedish person are next to each other.
11. The German is somewhere to the right of the person who loves a carnations arrangement.
12. The Norwegian is somewhere to the right of Peter.
13. The Norwegian is Eric.
 What is the value of attribute Flower for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: tulips