There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Carol`, `Peter`, `Arnold`, `Bob`
 - Each person has a unique hobby: `knitting`, `cooking`, `woodworking`, `gardening`, `photography`, `painting`
 - Each person has a unique level of education: `doctorate`, `master`, `bachelor`, `trade school`, `high school`, `associate`
 - People have unique favorite sports: `swimming`, `basketball`, `baseball`, `soccer`, `volleyball`, `tennis`

## Clues:
1. The person who enjoys gardening is somewhere to the right of the person with a high school diploma.
2. Alice is directly left of Bob.
3. The person who loves tennis is directly left of the photography enthusiast.
4. The woodworking hobbyist is the person who loves swimming.
5. The person who loves swimming is Peter.
6. The person with a bachelor's degree is directly left of Arnold.
7. The person who loves basketball is the person who enjoys knitting.
8. Eric is in the sixth house.
9. The person who enjoys gardening is directly left of the person with a master's degree.
10. The person who loves cooking is the person who loves tennis.
11. Arnold is the person who paints as a hobby.
12. The person with a doctorate is in the first house.
13. There are two houses between the person with an associate's degree and the person who loves volleyball.
14. The person with a bachelor's degree is the person who loves baseball.
15. The person with an associate's degree is somewhere to the left of the woodworking hobbyist.
 What is the value of attribute FavoriteSport for the person whose attribute Hobby is painting? Please reason step by step, and put your final answer within \boxed{}

Answer: volleyball


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Carol`, `Peter`, `Arnold`, `Bob`
 - Each person has a unique hobby: `knitting`, `cooking`, `woodworking`, `gardening`, `photography`, `painting`
 - Each person has a unique level of education: `doctorate`, `master`, `bachelor`, `trade school`, `high school`, `associate`
 - People have unique favorite sports: `swimming`, `basketball`, `baseball`, `soccer`, `volleyball`, `tennis`

## Clues:
1. The person who enjoys gardening is somewhere to the right of the person with a high school diploma.
2. Alice is directly left of Bob.
3. The person who loves tennis is directly left of the photography enthusiast.
4. The woodworking hobbyist is the person who loves swimming.
5. The person who loves swimming is Peter.
6. The person with a bachelor's degree is directly left of Arnold.
7. The person who loves basketball is the person who enjoys knitting.
8. Eric is in the sixth house.
9. The person who enjoys gardening is directly left of the person with a master's degree.
10. The person who loves cooking is the person who loves tennis.
11. Arnold is the person who paints as a hobby.
12. The person with a doctorate is in the first house.
13. There are two houses between the person with an associate's degree and the person who loves volleyball.
14. The person with a bachelor's degree is the person who loves baseball.
15. The person with an associate's degree is somewhere to the left of the woodworking hobbyist.
 What is the value of attribute Hobby for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}

Answer: gardening


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Carol`, `Eric`, `Bob`, `Alice`
 - Everyone has something unique for lunch: `stir fry`, `stew`, `grilled cheese`, `soup`, `pizza`, `spaghetti`

## Clues:
1. The person who loves the soup is somewhere to the left of Carol.
2. The person who loves stir fry is in the fifth house.
3. Peter is not in the sixth house.
4. The person who loves the stew is somewhere to the left of Peter.
5. Bob is the person who loves the soup.
6. The person who loves eating grilled cheese is somewhere to the right of the person who loves the spaghetti eater.
7. The person who loves the spaghetti eater is directly left of the person who loves the soup.
8. Arnold is in the second house.
9. The person who loves the stew is directly left of Arnold.
10. Alice is directly left of the person who loves the soup.
 What is the value of attribute Food for the person whose attribute House is 5? Please reason step by step, and put your final answer within \boxed{}

Answer: stir fry


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`, `Bob`, `Alice`, `Eric`, `Peter`
 - People have unique favorite sports: `basketball`, `soccer`, `tennis`, `baseball`, `swimming`
 - The people keep unique animals: `dog`, `fish`, `horse`, `bird`, `cat`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `iphone 13`, `huawei p50`
 - People have unique hair colors: `black`, `red`, `brown`, `blonde`, `gray`
 - Each person lives in a unique style of house: `modern`, `craftsman`, `victorian`, `ranch`, `colonial`

## Clues:
1. The person who has gray hair is the person who loves tennis.
2. The person in a ranch-style home is in the fourth house.
3. The person living in a colonial-style house is the person who loves swimming.
4. The dog owner is somewhere to the right of the person who has gray hair.
5. The fish enthusiast is the person who has gray hair.
6. The person who has black hair is not in the fifth house.
7. The person who uses a Samsung Galaxy S21 is the bird keeper.
8. There is one house between the fish enthusiast and the person who uses a Samsung Galaxy S21.
9. The person residing in a Victorian house is somewhere to the left of the person in a Craftsman-style house.
10. The person who has red hair is not in the fifth house.
11. There is one house between the person who uses a Samsung Galaxy S21 and the person who uses a Huawei P50.
12. The person who uses an iPhone 13 is the person in a Craftsman-style house.
13. Eric is the person who has red hair.
14. The person who has red hair is not in the fourth house.
15. The person who uses a OnePlus 9 is in the first house.
16. The person who loves soccer is the dog owner.
17. The person who keeps horses is Bob.
18. Arnold is the fish enthusiast.
19. The person who loves baseball is Peter.
20. The person who loves soccer is in the fourth house.
21. The person who has blonde hair is directly left of the person who has black hair.
 What is the value of attribute Animal for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: fish


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`, `Alice`
 - People have unique favorite sports: `basketball`, `soccer`, `swimming`, `tennis`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `iphone 13`
 - People own unique car models: `toyota camry`, `ford f150`, `tesla model 3`, `honda civic`
 - They all have a unique favorite flower: `roses`, `daffodils`, `lilies`, `carnations`

## Clues:
1. The person who owns a Honda Civic is the person who uses an iPhone 13.
2. The person who loves tennis is the person who uses an iPhone 13.
3. The person who uses a Google Pixel 6 is somewhere to the right of the person who loves basketball.
4. The person who uses an iPhone 13 is directly left of Eric.
5. Alice is the person who owns a Ford F-150.
6. Arnold is the person who loves a bouquet of daffodils.
7. The person who loves the boquet of lilies is in the third house.
8. The person who loves soccer is Peter.
9. The person who uses a OnePlus 9 is the person who loves the rose bouquet.
10. The person who loves a carnations arrangement is not in the fourth house.
11. Peter is the person who loves the boquet of lilies.
12. The person who owns a Toyota Camry is directly left of the person who owns a Tesla Model 3.
 What is the value of attribute PhoneModel for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: iphone 13


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`, `Alice`, `Arnold`, `Carol`, `Bob`
 - People have unique heights: `very tall`, `very short`, `tall`, `super tall`, `short`, `average`
 - Each person prefers a unique type of vacation: `cruise`, `mountain`, `camping`, `beach`, `city`, `cultural`
 - People have unique hair colors: `blonde`, `red`, `gray`, `brown`, `auburn`, `black`
 - Each person has a unique favorite drink: `tea`, `boba tea`, `milk`, `water`, `coffee`, `root beer`
 - They all have a unique favorite flower: `lilies`, `carnations`, `iris`, `tulips`, `roses`, `daffodils`

## Clues:
1. The person who loves the vase of tulips is the person who has auburn hair.
2. The person who is short is somewhere to the right of Arnold.
3. The person who goes on cultural tours is Arnold.
4. The person who has auburn hair is Bob.
5. The person who is very short is the person who loves a bouquet of daffodils.
6. The person who loves the boquet of iris is the person who is very tall.
7. Peter is somewhere to the right of the root beer lover.
8. The person who loves beach vacations is directly left of the person who prefers city breaks.
9. Bob is directly left of the coffee drinker.
10. The person who is very short is the person who has gray hair.
11. Eric is somewhere to the right of the tea drinker.
12. The person who has red hair is the person who is tall.
13. The person who is tall is somewhere to the right of the person who is short.
14. The person who has black hair is the person who enjoys mountain retreats.
15. The one who only drinks water is the person who prefers city breaks.
16. The person who has brown hair is Carol.
17. The person who has an average height is Arnold.
18. The person who has brown hair is directly left of the person who likes milk.
19. The root beer lover is the person who is very short.
20. The person who is short is somewhere to the left of the coffee drinker.
21. The person who enjoys camping trips is the person who loves a carnations arrangement.
22. The person who is super tall and Arnold are next to each other.
23. There is one house between the person who loves the boquet of lilies and Peter.
24. The person who loves the vase of tulips is directly left of the person who has black hair.
 What is the value of attribute Height for the person whose attribute HairColor is black? Please reason step by step, and put your final answer within \boxed{}

Answer: very tall


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `photography`
 - People own unique car models: `honda civic`, `toyota camry`, `tesla model 3`, `ford f150`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`, `spaghetti`, `stew`
 - Each person has an occupation: `doctor`, `artist`, `engineer`, `teacher`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`

## Clues:
1. The person who loves the stew is Eric.
2. The person who is a teacher is directly left of Peter.
3. The person who loves a carnations arrangement is the person who is an artist.
4. The person who paints as a hobby is the person who loves the spaghetti eater.
5. There is one house between the person who is an artist and the photography enthusiast.
6. The person who enjoys gardening is in the first house.
7. The person who loves the spaghetti eater is somewhere to the right of the person who owns a Honda Civic.
8. The person who is a pizza lover is directly left of the photography enthusiast.
9. The person who loves a carnations arrangement is Alice.
10. The person who owns a Tesla Model 3 is directly left of Eric.
11. The person who is an engineer is the person who loves a bouquet of daffodils.
12. The person who loves the boquet of lilies is somewhere to the left of the person who loves the stew.
13. There is one house between the person who enjoys gardening and the person who owns a Toyota Camry.
14. The person who is a teacher is in the third house.
 What is the value of attribute Food for the person whose attribute CarModel is honda civic? Please reason step by step, and put your final answer within \boxed{}

Answer: grilled cheese


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Peter`, `Bob`
 - People have unique heights: `average`, `tall`, `short`, `very short`, `very tall`
 - Each person has a unique hobby: `cooking`, `knitting`, `photography`, `gardening`, `painting`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `mountain`, `beach`, `camping`
 - People own unique car models: `ford f150`, `toyota camry`, `tesla model 3`, `honda civic`, `bmw 3 series`
 - Each person has a unique level of education: `bachelor`, `doctorate`, `master`, `high school`, `associate`

## Clues:
1. The person who is short is directly left of the person with a high school diploma.
2. The person who enjoys mountain retreats is the person who paints as a hobby.
3. The person who loves cooking is the person with a doctorate.
4. The person who owns a Honda Civic is in the third house.
5. The person who owns a Honda Civic is the person who has an average height.
6. Bob is the person who paints as a hobby.
7. The person with a master's degree is Eric.
8. The person who enjoys knitting is directly left of the person with a high school diploma.
9. The person who owns a Toyota Camry and the person who enjoys camping trips are next to each other.
10. Peter is not in the fifth house.
11. The person who prefers city breaks is the person who owns a BMW 3 Series.
12. The person who owns a Honda Civic is Eric.
13. The person who paints as a hobby is the person with a bachelor's degree.
14. Arnold is not in the second house.
15. The person who owns a Ford F-150 is not in the second house.
16. The person who enjoys gardening is the person who likes going on cruises.
17. The person who loves beach vacations is the person who is very short.
18. The person who owns a BMW 3 Series is not in the first house.
19. The person who owns a Toyota Camry is directly left of the person who is tall.
20. Arnold is not in the fifth house.
 What is the value of attribute Hobby for the person whose attribute Education is bachelor? Please reason step by step, and put your final answer within \boxed{}

Answer: painting


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`, `Arnold`, `Bob`, `Peter`, `Alice`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `blends`, `pall mall`, `prince`
 - Each person has an occupation: `lawyer`, `artist`, `engineer`, `doctor`, `teacher`
 - People own unique car models: `ford f150`, `toyota camry`, `honda civic`, `bmw 3 series`, `tesla model 3`
 - Everyone has a favorite smoothie: `watermelon`, `cherry`, `dragonfruit`, `desert`, `lime`
 - Everyone has something unique for lunch: `stew`, `spaghetti`, `grilled cheese`, `pizza`, `stir fry`

## Clues:
1. There are two houses between the person who owns a BMW 3 Series and the person who owns a Tesla Model 3.
2. The person who is an artist is Bob.
3. The person who owns a Ford F-150 is the person who smokes Blue Master.
4. The person who loves the spaghetti eater is in the second house.
5. The person who is an engineer is the Watermelon smoothie lover.
6. Alice is the person who drinks Lime smoothies.
7. The person who is an artist is the Dunhill smoker.
8. The person who likes Cherry smoothies is not in the fifth house.
9. Peter is not in the fourth house.
10. The Desert smoothie lover is the person who loves stir fry.
11. The person who likes Cherry smoothies is somewhere to the right of the person who smokes many unique blends.
12. The person who is a teacher is the person who loves eating grilled cheese.
13. Arnold is the person who is a lawyer.
14. The Dragonfruit smoothie lover is the person who owns a Honda Civic.
15. Peter is somewhere to the right of the Desert smoothie lover.
16. The Prince smoker is the Watermelon smoothie lover.
17. There is one house between the person who owns a Honda Civic and the person who loves the stew.
18. The person who owns a Tesla Model 3 is the person who likes Cherry smoothies.
19. The person who owns a Ford F-150 is directly left of the person who is a doctor.
20. The person who is an artist is the Desert smoothie lover.
 What is the value of attribute Cigar for the person whose attribute Occupation is engineer? 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: `Peter`, `Eric`, `Alice`, `Arnold`
 - Each person has a unique hobby: `photography`, `cooking`, `painting`, `gardening`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `prince`, `pall mall`

## Clues:
1. The person who loves cooking is not in the fourth house.
2. The person who paints as a hobby is somewhere to the left of the photography enthusiast.
3. The person partial to Pall Mall is Arnold.
4. There is one house between Peter and the photography enthusiast.
5. The Dunhill smoker is directly left of the person who loves cooking.
6. The person partial to Pall Mall is directly left of Alice.
7. The person partial to Pall Mall is somewhere to the left of the person who smokes Blue Master.
8. The Prince smoker is the photography enthusiast.
 What is the value of attribute Cigar for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall


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`
 - Each person has a unique birthday month: `april`, `sept`, `jan`, `feb`
 - Everyone has a unique favorite cigar: `dunhill`, `prince`, `pall mall`, `blue master`
 - People use unique phone models: `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `iphone 13`
 - They all have a unique favorite flower: `lilies`, `roses`, `carnations`, `daffodils`
 - People have unique heights: `average`, `short`, `tall`, `very short`

## Clues:
1. The person who loves a bouquet of daffodils is Arnold.
2. The person whose birthday is in April is directly left of the person who smokes Blue Master.
3. The person who is short is the person who uses an iPhone 13.
4. The person who uses a Samsung Galaxy S21 is the person who is tall.
5. Eric is the person who uses a OnePlus 9.
6. The person who uses a Samsung Galaxy S21 is directly left of the person who uses a Google Pixel 6.
7. The person whose birthday is in February is Alice.
8. The Prince smoker is somewhere to the right of the person who is short.
9. The Prince smoker is the person whose birthday is in February.
10. The person who uses a OnePlus 9 is directly left of the person who loves the rose bouquet.
11. The person partial to Pall Mall is somewhere to the left of the person who loves the boquet of lilies.
12. The person who has an average height and the person whose birthday is in September are next to each other.
13. The person who loves a bouquet of daffodils is in the fourth house.
 What is the value of attribute PhoneModel for the person whose attribute Flower is carnations? Please reason step by step, and put your final answer within \boxed{}

Answer: oneplus 9


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

Answer: 2


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Arnold`, `Peter`
 - People have unique favorite sports: `basketball`, `soccer`, `tennis`, `swimming`
 - Each person lives in a unique style of house: `craftsman`, `victorian`, `ranch`, `colonial`
 - People have unique favorite book genres: `mystery`, `science fiction`, `romance`, `fantasy`

## Clues:
1. The person in a Craftsman-style house is somewhere to the left of the person who loves science fiction books.
2. The person residing in a Victorian house is somewhere to the right of the person living in a colonial-style house.
3. Eric is somewhere to the left of Arnold.
4. The person who loves basketball is the person who loves romance books.
5. The person who loves soccer is somewhere to the right of Arnold.
6. Alice is the person who loves tennis.
7. The person who loves science fiction books is the person in a ranch-style home.
8. The person who loves tennis is somewhere to the left of the person in a Craftsman-style house.
9. Arnold is directly left of the person who loves mystery books.
 What is the value of attribute HouseStyle for the person whose attribute FavoriteSport is soccer? Please reason step by step, and put your final answer within \boxed{}

Answer: victorian


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

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

Answer: photography


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `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 House for the person whose attribute FavoriteSport is volleyball? 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: `Arnold`, `Alice`, `Eric`, `Bob`, `Peter`
 - People have unique hair colors: `gray`, `blonde`, `red`, `brown`, `black`
 - Each person has an occupation: `engineer`, `artist`, `lawyer`, `teacher`, `doctor`

## Clues:
1. The person who has black hair is the person who is a teacher.
2. The person who is a doctor is somewhere to the right of the person who is an artist.
3. There is one house between Eric and Alice.
4. Eric is somewhere to the left of Alice.
5. The person who has brown hair and Alice are next to each other.
6. Alice is the person who has red hair.
7. There are two houses between the person who has brown hair and the person who has blonde hair.
8. Alice is the person who is an engineer.
9. Arnold is the person who has blonde hair.
10. The person who is a doctor is Peter.
11. The person who has brown hair is somewhere to the left of the person who is a teacher.
 What is the value of attribute HairColor for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: gray


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `iphone 13`, `huawei p50`, `samsung galaxy s21`
 - Each person has a favorite color: `blue`, `red`, `yellow`, `green`, `white`
 - The mothers' names in different houses are unique: `Janelle`, `Penny`, `Holly`, `Aniya`, `Kailyn`
 - The people are of nationalities: `german`, `norwegian`, `swede`, `dane`, `brit`
 - Everyone has something unique for lunch: `spaghetti`, `grilled cheese`, `stew`, `pizza`, `stir fry`

## Clues:
1. The person who uses a OnePlus 9 and Arnold are next to each other.
2. There are two houses between the person who uses a OnePlus 9 and The person whose mother's name is Kailyn.
3. The person whose mother's name is Penny is somewhere to the left of the Swedish person.
4. The person who loves yellow is the person who uses a Google Pixel 6.
5. The person who loves eating grilled cheese is The person whose mother's name is Holly.
6. Arnold is the person who loves the spaghetti eater.
7. Bob is the person who uses a Huawei P50.
8. The German is the person who is a pizza lover.
9. Eric is directly left of the person who loves blue.
10. Alice is The person whose mother's name is Holly.
11. The person who loves stir fry is the British person.
12. The person whose mother's name is Penny is not in the third house.
13. Arnold is directly left of the person who uses a Huawei P50.
14. The British person is somewhere to the left of the Dane.
15. The person who uses a Google Pixel 6 is Arnold.
16. The person whose mother's name is Aniya is the person who uses a Samsung Galaxy S21.
17. The person who loves blue is the Norwegian.
18. The person who loves white is the person who uses a Huawei P50.
19. The person whose favorite color is red is directly left of the person who loves blue.
 What is the value of attribute Food for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: spaghetti


10. (10 points) There is a cup filled with a 15% saltwater solution. There are large, medium, and small iron balls, with volume ratios of 10: 5: 3. First, the small ball is submerged in the saltwater cup, causing 10% of the saltwater to overflow. The small ball is then removed. Next, the medium ball is submerged in the saltwater cup and then removed. Finally, the large ball is submerged in the saltwater cup and then removed. After that, pure water is added to the cup until it is full. What is the concentration of the saltwater in the cup at this point?

Answer: 10


The periods of two sequences are 7 and 13. What is the maximum length of the initial segment that can match in both sequences?

Answer: 18


A \(20 \times 19\) rectangle is plotted on the Cartesian plane with one corner at the origin and with sides parallel to the coordinate axes. How many unit squares do the two diagonals of this rectangle pass through?

Answer: 74


A semicircle with diameter \( AB \) and center at point \( O \) is divided by points \( C \) and \( D \) into three parts such that point \( C \) lies on arc \( AD \). From point \( D \), perpendiculars \( DE \) and \( DF \) are drawn to segments \( OC \) and \( AB \) respectively. It turns out that \( DE \) is the angle bisector of triangle \( ADC \), and \( DO \) is the angle bisector of triangle \( ADF \). Find the angle \( CAD \).

Answer: 20


Consider the equation

$$
(x-1)(x-2) \ldots(x-2016) = (x-1)(x-2) \ldots(x-2016)
$$

written on a board. What is the smallest integer $k$ such that we can erase $k$ factors among these 4032 factors so that there is at least one factor remaining on each side of the equation and the equation no longer has any real solutions?

Answer: 2016


Edge \( A_{1} A \) of the parallelepiped \( A B C D A_{1} B_{1} C_{1} D_{1} \) is perpendicular to its face \( A B C D \). Sphere \( \Omega \) touches the edges \( B B_{1}, B_{1} C_{1}, C_{1} C, C B, C D \), and touches the edge \( C D \) at a point \( K \) such that \( C K = 9 \) and \( K D = 1 \).

a) Find the length of edge \( A_{1} A \).

b) Additionally, suppose the sphere \( \Omega \) touches the edge \( A_{1} D_{1} \). Find the volume of the parallelepiped \( A B C D A_{1} B_{1} C_{1} D_{1} \) and the radius of the sphere \( \Omega \).

Answer: 3\sqrt{10}


Consider an equilateral triangular grid \( G \) with 20 points on a side, where each row consists of points spaced 1 unit apart. Specifically, there is a single point in the first row, two points in the second row, ..., and 20 points in the last row, for a total of 210 points. Let \( S \) be a closed non-selfintersecting polygon which has 210 vertices, using each point in \( G \) exactly once. Find the sum of all possible values of the area of \( S \).

Answer: 52\sqrt{3}


André is hiking on the paths shown in the map. He is planning to visit sites \(A\) to \(M\) in alphabetical order. He can never retrace his steps and he must proceed directly from one site to the next. What is the largest number of labelled points he can visit before going out of alphabetical order?
(A) 6
(B) 7
(C) 8
(D) 10
(E) 13

Answer: 10


In a notebook, a triangular grid is drawn (see the figure). Tanya placed integers at the grid nodes. We define two numbers as close if they are located at adjacent grid nodes. It is known that:

- The sum of all ten numbers is 43.
- The sum of any three numbers, such that any two of them are close, is 11.

What is the value of the central number?

Answer: 10


The sum and difference of 44 and 18 consist of the same digits, just in reverse order. How many pairs of two-digit numbers have this property?

Answer: 9


Five people stand in a line, each wearing a different hat numbered $1, 2, 3, 4, 5$. Each person can only see the hats of the people in front of them. Xiao Wang cannot see any hats; Xiao Zha can only see hat $4$; Xiao Tian does not see hat $3$, but can see hat $1$; Xiao Yan can see three hats, but not hat $3$; Xiao Wei can see hat $3$ and hat $2$. What number hat is Xiao Wei wearing?

Answer: 5


There are 30 students in a class: top achievers, average students, and underachievers. Top achievers always answer questions correctly, underachievers always get answers wrong, and average students answer questions in a strict alternating pattern of correct and incorrect. Each student was asked three questions: "Are you a top achiever?", "Are you an average student?", "Are you an underachiever?". 19 students responded "Yes" to the first question, 12 to the second question, and 9 to the third question. How many average students are in this class?

Answer: 20


Giuseppe has a sheet of plywood that measures $22 \times 15$. Giuseppe wants to cut out as many rectangular pieces of $3 \times 5$ as possible from it. How can he do this?

Answer: 22


As shown in the figure, in the rabbit shape formed by tangram pieces, the area of the rabbit's ears (shaded part) is 10 square centimeters. Then, the area of the rabbit shape is $\qquad$ square centimeters.

Answer: 80


A regular triangular prism \(ABC A_{1}B_{1}C_{1}\) is inscribed in a sphere with the base \(ABC\) and lateral edges \(AA_{1}\), \(BB_{1}\), \(CC_{1}\). The segment \(C_{1}D\) is the diameter of this sphere, and point \(K\) is the midpoint of edge \(CC_{1}\). Find the volume of the prism if \(DK=2\) and \(DA=\sqrt{6}\).

Answer: 2


3. In the equation $A \times B \times \overline{B B C}=\overline{A D A A}$, the same letters represent the same digits, different letters represent different digits, and $D$ is the smallest natural number, $A$ is the smallest prime number. Using the digits $A$, $B$, and $C$ to form natural numbers without repeated digits, the number of prime numbers formed is $\qquad$.

Answer: 6


Given a convex quadrilateral \(ABCD\) with only \(\angle D\) being an obtuse angle, we want to divide this quadrilateral into \(n\) obtuse triangles using some straight line segments. Additionally, apart from vertices \(A\), \(B\), \(C\), and \(D\), there should be no vertices of the resulting obtuse triangles on the boundary of the quadrilateral. Prove that the necessary and sufficient condition for \(n\) is \(n \geq 4\).

Answer: n \geq 4


The sequence \(\left\{u_{n}\right\}\) is defined as follows:
\[
u_{1}=1, \ u_{2}=1, \ u_{n}=u_{n-1}+2u_{n-2}, \ n=3,4, \cdots.
\]

Prove that for any natural number \(n\) and \(p\) (where \(p>1\)), \(u_{n+p}=u_{n+1}u_{p}+2u_{n}u_{p-1}\).

Find the greatest common divisor of \(u_{n}\) and \(u_{n+3}\).

Answer: 3


$A$ walks $30 \mathrm{~km}$ per day from $P$ to $Q$. After walking $72 \mathrm{~km}$, $B$ starts from $Q$ towards $P$ on the same road, and walks $\frac{1}{10}$ of the entire distance each day. After $B$ has walked for as many days as the $\frac{1}{8}$ of the distance he walks per day, he meets $A$. What is the distance between $P$ and $Q$?

Answer: 320


5. In triangle $ABC$, the median from vertex $A$ is perpendicular to the bisector of angle $B$, and the median from vertex $B$ is perpendicular to the bisector of angle $A$. It is known that side $AB=1$. Find the perimeter of triangle $ABC$.

Answer: 5


1. Given the sequence $\left\{\frac{n+1}{n+2}\right\}$, for any $e>0$, take $\mathbf{N}=$ $\qquad$ $\sim$, then when $n>N$, it always holds that $\qquad$ $<\varepsilon$, so the limit of the sequence $\left\{\frac{n+1}{n+2}\right\}$ is 1.

Answer: 1


Example 13. Given that the characteristic of $\lg x^{\frac{3}{2}}$ is the solution to the inequality
$$
a^{2}-8 a+15<0
$$

and the mantissa is the sum of the solutions to the equation
$$
\frac{1}{4}\left|y+\frac{1}{2}\right|+\frac{6}{121} \sqrt{z-1}=0
$$

find the value.

Answer: 1000


12.133. The arc $AB$ of sector $AOB$ contains $\alpha$ radians. A line is drawn through point $B$ and the midpoint $C$ of radius $OA$. In what ratio does this line divide the area of the sector?

Answer: \dfrac{\sin \alpha}{2\alpha - \sin \alpha}


1. Fill two $a$s and two $b$s into the 16 cells shown in Figure 3, with at most one letter per cell. If the same letters must not be in the same row or column, find the number of different ways to fill the cells.
(2007, National High School Mathematics Competition)

Answer: 3960


3. Given the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a, b>0)$ with left and right foci $F_{1}$ and $F_{2}$, respectively, and the hyperbola $C$ intersects the circle $x^{2}+y^{2}=r^{2}(r>0)$ at a point $P$. If the maximum value of $\frac{\left|P F_{1}\right|+\left|P F_{2}\right|}{r}$ is $4 \sqrt{2}$, then the eccentricity of the hyperbola $C$ is $\qquad$

Answer: 2\sqrt{2}


5、Insert $n$ real numbers between 1 and 100, such that these $n+2$ numbers form an increasing geometric sequence. Let the product of these $n+2$ numbers be denoted as $T_{n}$, and let $a_{n}=\lg T_{n}, n \geq 1$. Define $b_{n}=\tan a_{n} \cdot \tan a_{n+1}$, then the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$ is $S_{n}=$ $\qquad$

Answer: \dfrac{\tan(n + 3) - \tan 3}{\tan 1} - n


Chekanov Yu.V.

A square board is divided by seven lines parallel to one side of the board and seven lines parallel to the other side of the board into 64 rectangular cells, which are painted in white and black in a checkerboard pattern. The distances between adjacent lines are not necessarily the same, so the cells can be of different sizes. However, it is known that the ratio of the area of each white cell to the area of any black cell is no more than 2. Find the greatest possible ratio of the total area of the white cells to the total area of the black cells.

Answer: \dfrac{5}{4}


## [ Properties and characteristics of an isosceles triangle. ]

 [ The inscribed angle is half of the central angle ] [ The product of the lengths of the segments of the chords and the lengths of the segments of the secants ] [ Inscribed quadrilaterals (other) ]
[ Auxiliary similar triangles ]

From the vertex of the obtuse angle $A$ of triangle $A B C$, a height $A D$ is dropped. A circle with center $D$ and radius $D A$ is drawn, which intersects sides $A B$ and $A C$ again at points $M$ and $N$ respectively. Find $A C$, if $A B=c, A M=m$ and $A N=n$.

Answer: \dfrac{cm}{n}


Example 2 For any natural number $k$, if $k$ is even, divide it by 2, if $k$ is odd, add 1 to it, this is called one operation. Let the number of numbers that become 1 after exactly $n$ operations be $a_{n}$, try to find $a_{15}$.

Answer: 610


Danka and Janka received two identical white cubes for their birthdays, each made up of 125 small cubes as shown in the picture. To distinguish the cubes, they decided to paint them. Danka took a brush and painted three of the six faces of her cube red. Janka painted three faces of her cube green. After some time, both cubes were cut into the individual small cubes they were made of. When they did this, they were surprised to find that the number of small cubes with at least one red face differed from the number of small cubes with at least one green face. Determine this difference.

(E. Novotná)

![](https://cdn.mathpix.com/cropped/2024_04_17_1af6648ff2d5da4e2253g-1.jpg?height=426&width=434&top_left_y=838&top_left_x=820)

Answer: 4


2. Two math teachers are conducting a geometry test, checking each 10th-grade student's ability to solve problems and their knowledge of theory. The first teacher spends 5 and 7 minutes per student, respectively, while the second teacher spends 3 and 4 minutes per student. What is the minimum time they will need to interview 25 students?

Answer: 110


6. For the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{16}=1$, the left and right foci are $A, B$, and $P$ is a point on the hyperbola. If the incenter of $\triangle P A B$ is $(3,1)$, then the radius of the circumcircle of $\triangle P A B$ is $\qquad$ .

Answer: \dfrac{65}{12}


5. For the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1(0<b<4)$, the left focus is $F$, and line $l$ intersects the ellipse at points $A$ and $B$, with $|A B|$ $=6$. Then the maximum value of $|F A| \cdot|F B|$ is $\qquad$

Answer: 25


13. In $\triangle A B C$, $\angle B=\frac{\pi}{4}, \angle C=\frac{5 \pi}{12}, A C$ $=2 \sqrt{6}, A C$'s midpoint is $D$. If a line segment $P Q$ of length 3 (point $P$ to the left of point $Q$) slides on line $B C$, then the minimum value of $A P+D Q$ is $\qquad$.

Answer: \dfrac{3\sqrt{10} + \sqrt{30}}{2}


During a tennis tournament, a commentator who is a great fan of Zhang Shuai is keeping track of several statistics. Zhang played 5 matches and won 3 of them. In each match she won, she scored 6 games per set, and each match had 2 sets. In the matches she lost, Zhang scored 4 games per set, and each match also had 2 sets. How many total games did Zhang Shuai win during the tournament?

Answer: 52


A local historian and tour guide is preparing a special walking tour focusing on the religious history of Dublin. The tour includes visits to 5 historic churches, each representing different architectural styles. Each church is located 1 kilometer apart, and the tour begins at the historian's office. After visiting the fifth church, the tour concludes back at the office. If the historian spends 30 minutes at each church and walks at a speed of 4 kilometers per hour, how long will the entire tour take in hours, including both walking and visiting time?

Answer: 4


As a seasoned executive at a mainstream headphone company, Alex is skeptical about the potential sales of a new line of niche gaming headphones. The company currently sells 100,000 units of their mainstream headphones every month. They are considering launching the niche gaming headphones and project that they could sell 5% of the mainstream units in the first month.

If the company decides to launch the gaming headphones, they plan to increase production by 2% each month, based on the initial projected sales of the gaming headphones. Calculate how many units of the gaming headphones the company expects to produce in the third month after the launch.

Answer: 5200


A young apprentice is training to become an auctioneer like their mentor. During an auction, the apprentice is tasked with keeping track of the bids for a beautiful painting. The first bid is $150. The next four bids increase sequentially by $25 each time. However, the fifth bid jumps by $50 more than the last bid. After these five bids, the apprentice announces the total amount of the bids so far. What is the total amount of the bids announced by the apprentice?

Answer: 1050


Alex is a decisive IT project leader who is currently managing 3 different IT projects. For Project A, Alex has a budget of $2,000 and has already spent $1,200. For Project B, the budget is $3,500 and $2,100 has been spent. Project C has a budget of $4,000, with $2,800 already used. Alex wants to ensure that the total remaining budget for all projects combined is at least $3,500. How much more can Alex spend across all three projects while still keeping the total remaining budget at or above $3,500?

Answer: 0


Maria is a devoted Catholic congregant living in Bermuda. Each Sunday, she attends Mass at her local church, which is located 3 kilometers from her home. After Mass, Maria often spends time volunteering at the church's community center, which is an additional 2 kilometers from the church. She then returns home directly from the community center. One weekend, Maria attends Mass and volunteers on both Saturday and Sunday. How many kilometers does Maria travel in total over the weekend for these activities?

Answer: 20


Maria, an experienced pyrotechnician, is preparing a fireworks display for the grand opening of a new skyscraper in Dubai. For this event, she plans to use 250 fireworks, and she knows from her past projects that each firework will last for 5 seconds. If the fireworks need to be launched in groups of 10 to create a synchronized effect, and each group is launched consecutively without any delay between groups, how long will the entire fireworks display last in seconds?

Answer: 125


A film director is planning a classic western movie scene that involves a high-stakes poker game in an old saloon. The director has decided that each player at the table will start with 150 poker chips. There are 5 players in total. During the game, Player A wins 30 chips, Player B loses 20 chips, Player C gains 40 chips, Player D loses 10 chips, and Player E neither wins nor loses any chips. How many total poker chips are left on the table at the end of the game?

Answer: 750


Mr. Johnson owns a small grocery store in a quaint town and is known for his friendly smile rather than keeping up with global news. Each week, he orders 250 apples from a local farm to sell in his store. Due to a recent change in the farm's pricing, which Mr. Johnson only found out about from a neighbor, the cost per apple has increased from $0.40 to $0.55. If Mr. Johnson still sells each apple for $1.00, how much more profit will he make in a week after the price increase, compared to before the price increase?

Answer: -37.50


A curious and passionate student is researching the impact of Georgetown University on local communities. The student discovers that the university hosts 150 community events annually. Each event attracts an average of 75 local residents. If the student finds that 60% of these residents participate in more than one event, how many unique local residents attend at least one community event hosted by Georgetown University each year?

Answer: 7031


Ana is an enthusiastic Croatian librarian who specializes in collecting and preserving local comics. She recently acquired a new collection of 150 unique Croatian comics. Ana wants to organize these comics into special archival boxes. Each archival box can hold 10 comics. If she wants to keep 5 comics aside for a special exhibition display, how many full archival boxes will she need to store the remaining comics?

Answer: 14


A famous actor has recently been accused of plagiarism in their latest screenplay. The screenplay has 120 pages in total. The plagiarism allegations claim that 20% of the screenplay's content was copied from another source. The actor decided to rewrite those plagiarized portions and added an additional 5 new pages of original content. How many pages of the screenplay are now completely original?

Answer: 125


Mr. Thompson is a computer science educator who insists that his students submit their assignments on paper. Each week, he teaches three classes with 25 students each. Every student must submit a 5-page paper assignment. Mr. Thompson needs to staple all the pages together before grading. He uses a stapler that can hold 50 staples at a time.

How many times will Mr. Thompson need to refill his stapler to staple all the assignments for one week?

Answer: 1


The coordinates of the vertices of a quadrilateral are $A(0 ; 0), B(16 ; 0), C(8 ; 8), D(0,8)$. Write the equation of the line parallel to $AC$ that divides the area of the quadrilateral in half.

Answer: y = x + 8\sqrt{3} - 16


Exercise 14. Consider a grid of size $2019 \times 2019$. On this grid, pebbles are placed. A configuration is said to be beautiful if there does not exist a parallelogram formed by four pebbles $A B C D$, such that $A, B, C$, and $D$ are not all aligned.

What is the maximum number of pebbles that can be placed on the grid?

Answer: 4037


34. For a large cube with integer edge length, several faces are fully painted, and then it is divided into small cubes with edge length 1. At this point, the ratio of the number of small cubes with at least one face painted to the number of small cubes with no faces painted is $1: 3$. What is the minimum edge length of the original large cube? $\qquad$ .

Answer: 4


5. On the sides of a square with length 1, isosceles trapezoids are constructed outward such that the vertices of all trapezoids are also vertices of a regular dodecagon. What is the perimeter of this dodecagon?

## City competition:2606, I. Faured,-B wkarijentan-: solutions to problems

Each correctly solved problem is worth 20 points.1. Which digit is in the 2006th place after the decimal point in the decimal representation of the number $\frac{469}{1998} ?$

Answer: 3


Problem 1. Find all triples $(x, y, z)$ of natural numbers such that $y$ is a prime number, $y$ and 3 do not divide $z$, and $x^{3}-y^{3}=z^{2}$.

Nikolay Nikolov

Answer: (8, 7, 13)


A polynomial product of the form \[(1-z)^{b_1}(1-z^2)^{b_2}(1-z^3)^{b_3}(1-z^4)^{b_4}(1-z^5)^{b_5}\cdots(1-z^{32})^{b_{32}},\] where the $b_k$ are positive integers, has the surprising property that if we multiply it out and discard all terms involving $z$ to a power larger than $32$, what is left is just $1-2z$.  Determine, with proof, $b_{32}$.

Answer: 134215680


Example 16 Try to express $\sum_{k=0}^{n} \frac{(-1)^{k} \mathrm{C}_{n}^{k}}{k^{3}+9 k^{2}+26 k+24}$ in the form $\frac{P(n)}{Q(n)}$. Here $P(n) 、 Q(n)$ are two polynomials with integer coefficients.

Answer: \dfrac{1}{2(n + 3)(n + 4)}


4. To celebrate Barbara's birthday, Alberto proposes the following game: given the set of numbers $0,1, \ldots, 1024$, Barbara removes $2^{9}$ numbers from this set. In the next step, Alberto removes $2^{8}$ numbers from the remaining ones. It's Barbara's turn again, and she removes $2^{7}$ numbers from the remaining ones, and so on, until only 2 numbers $a$ and $b$ are left. At this point, Alberto must pay Barbara $|a-b|$ euros.

Determine the maximum amount of euros that Barbara is sure to collect, regardless of the strategy adopted by Alberto.

Answer: 32


Seven people of seven diﬀerent ages are attending a meeting. The seven people leave the meeting one at a
time in random order. Given that the youngest person leaves the meeting sometime before the oldest
person leaves the meeting, the probability that the third, fourth, and ﬁfth people to leave the meeting do so in order of their ages (youngest to oldest) is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$.

Answer: 25


1. The smallest natural number whose square ends with three fours is 38, since $38^{2}=1444$. What is the next smallest natural number with this property?

Answer: 462


Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$ -axis. What is the least positive
integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \cdots, T_n$ returns the point $(1,0)$ back to itself?

Answer: 359


921. Find the mass of a hemisphere if the surface density at each of its points is numerically equal to the distance of this point from the radius perpendicular to the base of the hemisphere.

Answer: \dfrac{\pi^2}{2} R^3


4. Two circles are concentric, with radii $R$ and $r$ ($R>r$). The vertices of the convex quadrilateral $ABCD$ are such that vertex $A$ is on the smaller circle, and vertices $B$, $C$, and $D$ are on the larger circle. The maximum area of the convex quadrilateral $ABCD$ is $\qquad$

Answer: R(R + r)


3. On a line, seven points A, B, C, D, E, F, G are marked in the given order. It turned out that $A G=23 \mathrm{~cm}, B F=17 \mathrm{~cm}$ and $\mathrm{CE}=9 \mathrm{~cm}$. Find the sum of the lengths of all segments with endpoints at these points.

Answer: 224


8,9  Two circles touch each other externally at point $A$, and a third circle at points $B$ and $C$. The extension of chord $A B$ of the first circle intersects the second circle at point $D$, the extension of chord $A C$ intersects the first circle at point $E$, and the extensions of chords $B E$ and $C D$ intersect the third circle at points $F$ and $G$ respectively. Find $B G$, if $B C=5$ and $B F=12$.

Answer: 13


4. Let's call a rectangular parallelepiped typical if all its dimensions (length, width, and height) are different. What is the smallest number of typical parallelepipeds into which a cube can be cut? Don't forget to prove that this is indeed the smallest number.

Answer: 4


14th Mexico 2000 Problem B1 Given positive integers a, b (neither a multiple of 5) we construct a sequence as follows: a 1 = 5, a n+1 = a a n + b. What is the largest number of primes that can be obtained before the first composite member of the sequence?

Answer: 5


Find the smallest positive integer $k$ such that $ \underbrace{11\cdots 11}_{k\text{ 1's}}$ is divisible by $9999$.

Let $T = TNYWR$. Circles $\omega_1$ and $\omega_2$ intersect at $P$ and $Q$.  The common external tangent $\ell$ to the two circles closer to $Q$ touches $\omega_1$ and $\omega_2$ at $A$ and $B$ respectively.  Line $AQ$ intersects $\omega_2$ at $X$ while $BQ$ intersects $\omega_1$ again at $Y$.  Let $M$ and $N$ denote the midpoints of $\overline{AY}$ and $\overline{BX}$, also respectively.  If $AQ=\sqrt{T}$, $BQ=7$, and $AB=8$, then find the length of $MN$.

Answer: 36


7. Given $O$ is the circumcenter of $\triangle A B C$, $A D$ is the altitude on $B C$, $\angle C A B=66^{\circ}, \angle A B C=44^{\circ}$. Then, $\angle O A D=$ $\qquad$ .

Answer: 26^\circ


3. $k 4$ ** A natural number that can be expressed as the difference of the squares of two natural numbers is called a "wisdom number". For example, $16=5^{2}-3^{2}$, so 16 is a "wisdom number". In the sequence of natural numbers starting from 1, which is the 1990th "wisdom number"? Please explain your reasoning.

Answer: 2656


A column of cars 300 m long is driving on the highway at a speed of 60 km/h. Passing by a police checkpoint, each car reduces its speed to 40 km/h. What will be the length of the column when all cars have passed the checkpoint?

#

Answer: 200


Example 6 There are 1994 matches on the table, two children, A and B, take turns to take 1, 2 or 3 matches each time, the one who can take the last match wins. Now A takes first, which child will win? How should he play to win this game?

Answer: 2


Find the number of positive integers $n$ less than $2017$ such that \[1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+\frac{n^4}{4!}+\frac{n^5}{5!}+\frac{n^6}{6!}\] is an integer.
Please give the answer directly without any intermediate steps.

Answer: 134


21. There is 1 long enough rectangular strip of paper, one side of the strip is black, and the other side is red. If 1 strip of paper is folded to form a red and black alternating "cross" pattern, then the minimum number of folds required to form the "cross" pattern is $\qquad$ times (folding is allowed to overlap, and excess parts and small gaps do not need to be considered).

Answer: 3


13. Given 10 points of the World Team, where 5 of these points lie on a straight line, and no three points lie on another straight line besides these, the number of distinct rays that can be drawn through any 2 of these 10 points is $\qquad$.

Answer: 78


10.3. Given a right triangle $A B C$. A point $M$ is taken on the hypotenuse $A C$. Let $K, L$ be the centers of the circles inscribed in triangles $A B M$ and $C B M$ respectively. Find the distance from point $M$ to the midpoint of $K L$, if the radius $R$ of the circle circumscribed around triangle $B K L$ is known.

Answer: \dfrac{R}{\sqrt{2}}


8. (15 points) The seven members of the Running Family need to complete the challenge by passing through 7 doors in the figure; the first person can choose any door to activate, and after completing the challenge, it will activate the adjacent doors on both sides; the next person can choose any unchallenged door from the activated doors, and after completing the challenge, it will activate the unactivated adjacent doors on both sides; and so on. In the end, all seven members of the Running Family completed the challenge. Sorting the numbers of the seven doors according to the order they were challenged will result in a seven-digit number, and there are $\qquad$ different possible seven-digit numbers.

Answer: 64


In an isosceles triangle $A B C(A B=B C)$, the altitudes $A A_{1}, B B_{1}$ and $C C_{1}$ are drawn.

Find the ratio of the area of triangle $A_{1} B_{1} C_{1}$ to the area of triangle $A B C$, if ${ }^{A B} / A_{1} B_{1}=\sqrt{3}$.

Answer: \dfrac{2}{9}


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Bob`, `Peter`
 - Each person has a unique favorite drink: `root beer`, `water`, `coffee`, `tea`, `milk`
 - Each person has a unique birthday month: `jan`, `april`, `sept`, `feb`, `mar`
 - Each person has an occupation: `teacher`, `lawyer`, `engineer`, `doctor`, `artist`
 - Each person prefers a unique type of vacation: `city`, `cruise`, `mountain`, `beach`, `camping`

## Clues:
1. The root beer lover is Bob.
2. The person who is an engineer is Bob.
3. The coffee drinker is not in the first house.
4. Peter is not in the second house.
5. The one who only drinks water is not in the fifth house.
6. The person who is a doctor is directly left of the person who prefers city breaks.
7. The person who likes going on cruises is somewhere to the left of the person whose birthday is in April.
8. The person whose birthday is in February is the person who loves beach vacations.
9. Eric is somewhere to the right of Arnold.
10. The tea drinker is the person who enjoys mountain retreats.
11. The person who enjoys mountain retreats is not in the second house.
12. The person whose birthday is in April is not in the second house.
13. The tea drinker is somewhere to the left of the person who enjoys camping trips.
14. The person who is an engineer is the person who enjoys camping trips.
15. The person who is an artist is the person whose birthday is in April.
16. The person whose birthday is in March is not in the first house.
17. Arnold is the coffee drinker.
18. The coffee drinker is the person whose birthday is in February.
19. The person who is a lawyer is somewhere to the right of the person who is a teacher.
20. The coffee drinker is not in the second house.
21. The person whose birthday is in April is the one who only drinks water.
22. The person whose birthday is in January is the root beer lover.
 What is the value of attribute Vacation for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: mountain


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`, `Bob`, `Peter`
 - Each person lives in a unique style of house: `victorian`, `colonial`, `ranch`, `modern`, `craftsman`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `oneplus 9`, `samsung galaxy s21`, `huawei p50`
 - Everyone has a unique favorite cigar: `blue master`, `dunhill`, `pall mall`, `blends`, `prince`
 - People have unique favorite book genres: `mystery`, `science fiction`, `biography`, `romance`, `fantasy`

## Clues:
1. There is one house between the person residing in a Victorian house and the person who uses a Samsung Galaxy S21.
2. The person who smokes many unique blends is Eric.
3. The person who smokes many unique blends is the person who loves fantasy books.
4. Alice is somewhere to the right of the person who uses a Samsung Galaxy S21.
5. Bob is directly left of Eric.
6. The person who loves mystery books is directly left of the person who loves fantasy books.
7. Bob is somewhere to the left of the person who uses an iPhone 13.
8. The person who uses a Google Pixel 6 is somewhere to the right of Peter.
9. The person who loves romance books is directly left of the person in a ranch-style home.
10. The person living in a colonial-style house is in the fourth house.
11. The person who loves fantasy books is in the fourth house.
12. The person partial to Pall Mall is Peter.
13. The person in a modern-style house and the person who smokes Blue Master are next to each other.
14. The person who loves science fiction books is the Dunhill smoker.
15. Eric is the person who uses a Samsung Galaxy S21.
16. The person who uses a Huawei P50 and the person who loves fantasy books are next to each other.
 What is the value of attribute BookGenre for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: mystery


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`
 - People have unique favorite sports: `basketball`, `soccer`, `swimming`, `tennis`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `iphone 13`
 - People own unique car models: `toyota camry`, `ford f150`, `tesla model 3`, `honda civic`
 - They all have a unique favorite flower: `roses`, `daffodils`, `lilies`, `carnations`

## Clues:
1. The person who owns a Honda Civic is the person who uses an iPhone 13.
2. The person who loves tennis is the person who uses an iPhone 13.
3. The person who uses a Google Pixel 6 is somewhere to the right of the person who loves basketball.
4. The person who uses an iPhone 13 is directly left of Eric.
5. Alice is the person who owns a Ford F-150.
6. Arnold is the person who loves a bouquet of daffodils.
7. The person who loves the boquet of lilies is in the third house.
8. The person who loves soccer is Peter.
9. The person who uses a OnePlus 9 is the person who loves the rose bouquet.
10. The person who loves a carnations arrangement is not in the fourth house.
11. Peter is the person who loves the boquet of lilies.
12. The person who owns a Toyota Camry is directly left of the person who owns a Tesla Model 3.
 What is the value of attribute PhoneModel for the person whose attribute House is 2? 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: `Alice`, `Eric`, `Bob`, `Arnold`, `Peter`, `Carol`
 - People have unique favorite music genres: `classical`, `rock`, `jazz`, `pop`, `country`, `hip hop`
 - Each person has an occupation: `teacher`, `artist`, `nurse`, `engineer`, `lawyer`, `doctor`
 - Each person prefers a unique type of vacation: `cruise`, `camping`, `city`, `cultural`, `mountain`, `beach`
 - Each person has a favorite color: `blue`, `white`, `red`, `yellow`, `purple`, `green`

## Clues:
1. The person who loves beach vacations is Arnold.
2. The person who is an engineer is the person who enjoys mountain retreats.
3. Peter is not in the second house.
4. The person whose favorite color is red is in the third house.
5. The person who loves beach vacations and the person who is a teacher are next to each other.
6. The person who is a nurse is Carol.
7. The person who goes on cultural tours is the person who loves pop music.
8. The person who loves rock music and the person who loves country music are next to each other.
9. The person whose favorite color is green is in the second house.
10. The person whose favorite color is red is the person who enjoys camping trips.
11. Peter is the person who loves pop music.
12. The person who loves classical music is the person who loves purple.
13. The person who loves blue is directly left of Alice.
14. The person who loves rock music is in the fourth house.
15. The person who loves white is the person who is a doctor.
16. The person who loves jazz music is the person whose favorite color is red.
17. Carol is somewhere to the right of the person who prefers city breaks.
18. The person who loves rock music is the person who is an artist.
19. The person who enjoys mountain retreats is in the first house.
20. Bob is the person who loves classical music.
 What is the value of attribute Name for the person whose attribute Vacation is cultural? 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: `Eric`, `Peter`, `Arnold`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`
 - They all have a unique favorite flower: `carnations`, `lilies`, `daffodils`
 - The people keep unique animals: `cat`, `horse`, `bird`
 - Each person has a unique hobby: `photography`, `cooking`, `gardening`

## Clues:
1. The person who keeps horses and the photography enthusiast are next to each other.
2. The bird keeper is the person who likes Cherry smoothies.
3. The person who loves cooking is the Desert smoothie lover.
4. The person who enjoys gardening is the person who loves a carnations arrangement.
5. The person who loves cooking is directly left of Peter.
6. The person who loves a bouquet of daffodils is the Desert smoothie lover.
7. The Watermelon smoothie lover is the person who keeps horses.
8. The photography enthusiast is Eric.
 What is the value of attribute Name for the person whose attribute Flower is carnations? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 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`, `Arnold`, `Peter`, `Bob`, `Alice`
 - Each person has a favorite color: `red`, `blue`, `white`, `green`, `yellow`
 - Everyone has a unique favorite cigar: `blends`, `blue master`, `pall mall`, `prince`, `dunhill`

## Clues:
1. There are two houses between the person partial to Pall Mall and the person who loves blue.
2. The Dunhill smoker is not in the fourth house.
3. The person who smokes many unique blends is somewhere to the right of Bob.
4. Alice is somewhere to the left of Bob.
5. The person whose favorite color is red is not in the fourth house.
6. The Prince smoker is not in the fifth house.
7. The Dunhill smoker is not in the fifth house.
8. Bob is somewhere to the right of the person whose favorite color is green.
9. The person whose favorite color is red is not in the first house.
10. The Dunhill smoker is somewhere to the right of Eric.
11. The person whose favorite color is red is not in the fifth house.
12. The person who loves yellow and the person who loves blue are next to each other.
13. Alice is directly left of the person who loves yellow.
14. There is one house between the person who smokes Blue Master and Peter.
15. The person who loves blue is somewhere to the left of Arnold.
 What is the value of attribute House for the person whose attribute Cigar is blue master? 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 House for the person whose attribute Hobby is gardening? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Bob`, `Carol`, `Arnold`, `Eric`
 - Each person has an occupation: `engineer`, `doctor`, `lawyer`, `artist`, `nurse`, `teacher`
 - Everyone has something unique for lunch: `soup`, `stew`, `grilled cheese`, `spaghetti`, `pizza`, `stir fry`
 - Everyone has a favorite smoothie: `cherry`, `dragonfruit`, `watermelon`, `lime`, `blueberry`, `desert`
 - Each person prefers a unique type of vacation: `camping`, `cultural`, `city`, `cruise`, `beach`, `mountain`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `oneplus 9`, `samsung galaxy s21`, `xiaomi mi 11`, `huawei p50`

## Clues:
1. The person who likes Cherry smoothies is directly left of the person who enjoys camping trips.
2. The person who is a lawyer is the person who uses a Xiaomi Mi 11.
3. The person who is a doctor is Eric.
4. The person who is a doctor and the Desert smoothie lover are next to each other.
5. The person who is an artist is the person who prefers city breaks.
6. The person who loves beach vacations is Alice.
7. There are two houses between the person who is a pizza lover and the person who uses a Huawei P50.
8. The person who uses a Xiaomi Mi 11 is the person who enjoys camping trips.
9. The person who is an artist is directly left of the person who loves eating grilled cheese.
10. The person who uses a Google Pixel 6 is in the first house.
11. There is one house between the person who is a teacher and the person who loves stir fry.
12. There are two houses between the person who is a pizza lover and the person who is an engineer.
13. The person who is a pizza lover is somewhere to the left of the person who loves eating grilled cheese.
14. The Watermelon smoothie lover is the person who uses a Xiaomi Mi 11.
15. The person who loves eating grilled cheese is Carol.
16. Alice is the person who drinks Blueberry smoothies.
17. The person who likes Cherry smoothies is the person who loves the stew.
18. The person who loves the soup is the person who uses a Samsung Galaxy S21.
19. The person who uses an iPhone 13 is directly left of the Dragonfruit smoothie lover.
20. The person who loves the stew is somewhere to the left of Peter.
21. Arnold is somewhere to the right of the Desert smoothie lover.
22. The person who likes going on cruises is the person who uses a Google Pixel 6.
23. The person who goes on cultural tours is not in the fifth house.
24. The person who is a teacher is Alice.
 What is the value of attribute Food for the person whose attribute Occupation is nurse? Please reason step by step, and put your final answer within \boxed{}

Answer: grilled cheese


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Bob`, `Peter`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `mountain`, `camping`, `beach`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Meredith`, `Samantha`, `Timothy`
 - Each person has a unique type of pet: `bird`, `fish`, `dog`, `hamster`, `cat`
 - People have unique favorite sports: `basketball`, `swimming`, `tennis`, `baseball`, `soccer`
 - Each person has a unique favorite drink: `tea`, `coffee`, `water`, `milk`, `root beer`

## Clues:
1. The person who keeps a pet bird is the person who loves baseball.
2. The person with a pet hamster is not in the fourth house.
3. The person's child is named Bella is in the third house.
4. The person who prefers city breaks is directly left of the person's child is named Samantha.
5. The person's child is named Meredith is the person who enjoys camping trips.
6. There is one house between the person who prefers city breaks and Peter.
7. The person who loves tennis is the person's child is named Fred.
8. The person who owns a dog is Bob.
9. Bob is the person who loves beach vacations.
10. Bob is directly left of the person who has a cat.
11. The person's child is named Bella is the root beer lover.
12. The person who likes going on cruises is the person who likes milk.
13. The coffee drinker is not in the fourth house.
14. The one who only drinks water is the person who loves swimming.
15. The person who loves basketball is the person who has a cat.
16. The person who loves swimming is not in the fourth house.
17. The person who loves baseball is Eric.
18. There are two houses between Bob and the person's child is named Meredith.
19. The person who likes milk is the person who is the mother of Timothy.
20. The person's child is named Fred is directly left of Arnold.
21. The person who keeps a pet bird is directly left of Bob.
 What is the value of attribute House for the person whose attribute Pet is hamster? 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`, `Alice`, `Bob`, `Peter`, `Arnold`
 - Each person prefers a unique type of vacation: `cruise`, `city`, `mountain`, `camping`, `beach`

## Clues:
1. Bob is in the fourth house.
2. The person who enjoys mountain retreats is Arnold.
3. The person who likes going on cruises is Peter.
4. The person who likes going on cruises is in the fifth house.
5. Eric is directly left of the person who loves beach vacations.
6. There is one house between the person who prefers city breaks and Peter.
7. Alice is not in the first house.
 What is the value of attribute House for the person whose attribute Vacation is camping? 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: `Eric`, `Arnold`, `Carol`, `Bob`, `Alice`, `Peter`
 - Each person prefers a unique type of vacation: `mountain`, `camping`, `beach`, `cultural`, `city`, `cruise`
 - Everyone has something unique for lunch: `spaghetti`, `soup`, `stir fry`, `stew`, `grilled cheese`, `pizza`
 - Each person has a unique favorite drink: `boba tea`, `root beer`, `water`, `milk`, `coffee`, `tea`
 - People have unique heights: `tall`, `average`, `very tall`, `short`, `very short`, `super tall`

## Clues:
1. The person who enjoys mountain retreats and the person who prefers city breaks are next to each other.
2. Alice is in the fourth house.
3. The coffee drinker is the person who is a pizza lover.
4. The person who is very tall is directly left of the person who enjoys mountain retreats.
5. Peter is not in the first house.
6. The boba tea drinker is in the fourth house.
7. The person who loves stir fry is somewhere to the left of Eric.
8. The person who likes milk is Bob.
9. The person who loves the soup is not in the second house.
10. The person who loves the spaghetti eater is not in the second house.
11. The person who loves the stew is Eric.
12. The root beer lover is the person who is short.
13. The person who loves beach vacations is in the first house.
14. The person who loves stir fry is the person who enjoys mountain retreats.
15. The person who loves the soup is somewhere to the left of the person who loves the spaghetti eater.
16. The boba tea drinker is somewhere to the left of the person who is a pizza lover.
17. Arnold is the person who likes going on cruises.
18. The person who is very short is somewhere to the right of the person who has an average height.
19. The person who is tall is not in the fourth house.
20. The person who prefers city breaks is the coffee drinker.
21. Bob and the one who only drinks water are next to each other.
22. The person who loves the stew is the person who enjoys camping trips.
23. Arnold is the person who is very short.
24. Carol is in the fifth house.
 What is the value of attribute Food for the person whose attribute Vacation is beach? Please reason step by step, and put your final answer within \boxed{}

Answer: soup


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

## Clues:
1. Alice is not in the fifth house.
2. Eric is the person who is the mother of Timothy.
3. Alice and Bob are next to each other.
4. The person who loves blue is not in the fourth house.
5. The person's child is named Samantha is the person who loves the vase of tulips.
6. The person who loves the vase of tulips is Alice.
7. The person's child is named Fred is the person whose favorite color is red.
8. The person's child is named Bella is in the first house.
9. There is one house between the person who loves blue and the person who loves yellow.
10. The person who loves yellow is the person who loves the rose bouquet.
11. The person's child is named Meredith is somewhere to the left of Bob.
12. The person who loves white is not in the second house.
13. The person who loves a carnations arrangement is not in the second house.
14. The person who loves white is Arnold.
15. The person who loves the boquet of lilies is in the first house.
 What is the value of attribute Color for the person whose attribute Children is Bella? Please reason step by step, and put your final answer within \boxed{}

Answer: white


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 Drink for the person whose attribute HairColor is gray? Please reason step by step, and put your final answer within \boxed{}

Answer: water


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

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

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Bob`, `Alice`, `Carol`, `Peter`
 - The people are of nationalities: `german`, `brit`, `dane`, `chinese`, `swede`, `norwegian`
 - People have unique favorite music genres: `pop`, `jazz`, `hip hop`, `classical`, `country`, `rock`
 - Each person has a unique level of education: `doctorate`, `associate`, `master`, `bachelor`, `high school`, `trade school`

## Clues:
1. The Chinese is the person who loves pop music.
2. The person with a bachelor's degree is directly left of Arnold.
3. The person with a master's degree is in the first house.
4. Carol is somewhere to the left of Peter.
5. The Dane is the person with a doctorate.
6. The person who loves classical music is the person with a master's degree.
7. The person with a bachelor's degree is in the fifth house.
8. Eric is somewhere to the right of Bob.
9. Carol is the British person.
10. The person with a doctorate is somewhere to the right of the Norwegian.
11. The person with a doctorate is the person who loves rock music.
12. The Norwegian is somewhere to the right of the German.
13. The person who attended trade school is somewhere to the left of the person with a high school diploma.
14. The person who attended trade school and the Chinese are next to each other.
15. The person who loves jazz music is Arnold.
16. The person who loves hip-hop music is directly left of Alice.
 What is the value of attribute Nationality for the person whose attribute MusicGenre is classical? Please reason step by step, and put your final answer within \boxed{}

Answer: german


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Carol`, `Arnold`, `Peter`, `Alice`
 - Everyone has a unique favorite cigar: `blends`, `prince`, `dunhill`, `yellow monster`, `blue master`, `pall mall`
 - People have unique hair colors: `auburn`, `gray`, `blonde`, `black`, `red`, `brown`
 - Each person has a unique favorite drink: `boba tea`, `root beer`, `water`, `tea`, `coffee`, `milk`
 - Each person has an occupation: `artist`, `nurse`, `teacher`, `doctor`, `engineer`, `lawyer`

## Clues:
1. Bob is the tea drinker.
2. The Dunhill smoker is not in the second house.
3. There is one house between Carol and Arnold.
4. The Prince smoker is the person who has black hair.
5. The person who has auburn hair is not in the third house.
6. Arnold is in the fourth house.
7. The Prince smoker is not in the fourth house.
8. The person who is an artist is the person partial to Pall Mall.
9. The person who is an engineer is directly left of the root beer lover.
10. The boba tea drinker is the person who has brown hair.
11. The person who is a lawyer is in the fifth house.
12. The person who is an engineer is the person who smokes Blue Master.
13. The person who likes milk is the person who has auburn hair.
14. Alice is somewhere to the left of the person who is a teacher.
15. The person who has blonde hair is the person who smokes Yellow Monster.
16. The person who has black hair is the person who is a doctor.
17. Arnold is somewhere to the right of the person who has gray hair.
18. The person who has auburn hair is not in the first house.
19. The one who only drinks water is the person who smokes many unique blends.
20. The person who smokes many unique blends is somewhere to the right of the person who has brown hair.
21. Arnold is the root beer lover.
22. The person who has brown hair is not in the third house.
23. The person who smokes many unique blends is somewhere to the left of Peter.
24. The person who likes milk is somewhere to the left of the person who has blonde hair.
 What is the value of attribute House for the person whose attribute Drink is water? 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: `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 Birthday for the person whose attribute Nationality is norwegian? Please reason step by step, and put your final answer within \boxed{}

Answer: feb


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

Answer: master


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Bob`, `Eric`, `Peter`
 - Each person has a unique hobby: `painting`, `photography`, `cooking`, `gardening`, `knitting`
 - Each person has a unique birthday month: `jan`, `sept`, `mar`, `feb`, `april`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `baseball`, `soccer`
 - Each person has an occupation: `engineer`, `lawyer`, `teacher`, `doctor`, `artist`
 - Each person has a unique level of education: `bachelor`, `master`, `high school`, `doctorate`, `associate`

## Clues:
1. The person with a high school diploma is directly left of the person whose birthday is in February.
2. The person who loves soccer is directly left of the photography enthusiast.
3. Arnold is the person whose birthday is in January.
4. The person who is a doctor is the person who loves baseball.
5. The person who is an artist is somewhere to the left of Bob.
6. The photography enthusiast is somewhere to the right of the person with an associate's degree.
7. The photography enthusiast is directly left of the person who paints as a hobby.
8. Eric is in the fifth house.
9. The person whose birthday is in February is directly left of the person who loves tennis.
10. The person who loves baseball is somewhere to the right of the person whose birthday is in January.
11. The person who is an engineer is not in the first house.
12. Alice is not in the second house.
13. The person who enjoys gardening is somewhere to the left of the person whose birthday is in September.
14. The person who loves swimming is not in the first house.
15. The person who is a teacher is Bob.
16. Arnold is the person who enjoys knitting.
17. The person who is an artist is the person whose birthday is in March.
18. Arnold is the person with a master's degree.
19. The person whose birthday is in March is directly left of the person who is a doctor.
20. The person whose birthday is in September is the person with a bachelor's degree.
 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: photography


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`, `Alice`
 - The mothers' names in different houses are unique: `Holly`, `Kailyn`, `Aniya`, `Janelle`
 - People have unique favorite book genres: `science fiction`, `mystery`, `fantasy`, `romance`
 - Each person has a favorite color: `green`, `red`, `white`, `yellow`
 - Each person prefers a unique type of vacation: `mountain`, `cruise`, `beach`, `city`

## Clues:
1. The person who loves romance books and the person who likes going on cruises are next to each other.
2. Arnold is The person whose mother's name is Janelle.
3. The person who prefers city breaks is the person who loves yellow.
4. The person who loves science fiction books is directly left of Peter.
5. The person who loves science fiction books is somewhere to the left of The person whose mother's name is Kailyn.
6. The person who loves beach vacations is not in the first house.
7. There is one house between Alice and the person who likes going on cruises.
8. The person who loves mystery books is the person who loves white.
9. Peter is the person who prefers city breaks.
10. The person whose favorite color is green is in the third house.
11. The person whose mother's name is Aniya is Peter.
12. The person whose favorite color is green is The person whose mother's name is Janelle.
 What is the value of attribute Mother for the person whose attribute Vacation is mountain? Please reason step by step, and put your final answer within \boxed{}

Answer: Holly


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`
 - Each person has a unique level of education: `master`, `high school`, `associate`, `bachelor`
 - The people keep unique animals: `bird`, `horse`, `cat`, `fish`
 - Each person has a unique type of pet: `bird`, `fish`, `dog`, `cat`
 - Each person has a unique favorite drink: `water`, `tea`, `milk`, `coffee`

## Clues:
1. Peter is directly left of the fish enthusiast.
2. The person with a master's degree is in the second house.
3. The coffee drinker is the person who owns a dog.
4. Arnold is the person with an associate's degree.
5. The person with a bachelor's degree is the person who owns a dog.
6. The one who only drinks water is the bird keeper.
7. Alice is the person with an aquarium of fish.
8. The person who keeps horses is in the first house.
9. The person with a master's degree is the tea drinker.
10. The person with an associate's degree is the person who keeps a pet bird.
11. The fish enthusiast is the coffee drinker.
12. The person with an associate's degree is not in the fourth house.
13. Eric is not in the fourth house.
 What is the value of attribute Drink for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: water