Summer 2014 – Week 9 Solutions

I hope you enjoyed your summer math problems.  Here are the solutions to the final set.

Level 1

Jennifer just went back to school shopping.  She bought 7 shirts, 5 pairs of pants, 4 pairs of fancy socks, sneakers and sandals.  How many different outfits can Jennifer put together if she always wears one shirt, one pair of pants, one pair of shoes and she always wears socks with sneakers but never wears socks with sandals?

Spoiler Inside: Level 1 Solution SelectShow

Level 2

George is taking five classes this year: Math, Science, History, English, and Spanish.  He needs a spiral notebook and a pocket folder for each class.  Spiral notebooks come in 10 colors (black, blue, red, green, lime, purple, pink, orange, aqua, and yellow).  Pocket folders come in six colors (blue, red, green, yellow, purple, and orange).
a) How many different ways can George choose the notebooks and folders for his classes if he doesn’t buy any duplicate items?
b) How many ways can George choose the notebooks and folders if he wants to have the same color notebook and folder for a certain class?

Spoiler Inside: Level 2 Solution SelectShow

 

Level 3

George is going to arrange his books, folders, and notebooks on the bookshelf.  For each of five subjects he has a textbook, a folder, and a spiral notebook.  He wants to be able to find things so he is considering different ways of organizing them.
a) How many ways can he arrange them if he keeps all three items from one subject together (the three math items are together in any order, the three history items, etc.)?
b) How many ways can he arrange them if he keeps all the textbooks together, all the spiral notebooks together, and all the folders together?

Spoiler Inside: Level 3 Solution SelectShow

Summer 2014 Week 8 Solutions

Here are this week’s solutions.

Knights and Knaves

In the land of knights and knaves, everyone is either a knight or a knave. Knights always tell the truth, while knaves always tell lies. It is impossible to tell apart knights and knaves by appearance.

 Level 1

In the land of knights and knaves, the king’s crown is stolen, and four suspects are questioned.

Suspect 1 says: “Suspect 3 stole the crown”

Suspect 2 says: “I did not steal the crown”

Suspect 3 says: “Suspect 4 stole the crown”

Suspect 4 says:  “Suspect 3 is a knave”

If exactly three of the suspects are knaves, who stole the crown?

Spoiler Inside: Level 1 Solution SelectShow

 

Level 2

In the land of knights and knaves, there are three round tables with the same number of people sitting around each of them. At the first table, each person claims that they are sitting next to a knight and a knave. At the second table, each person claims that the three people to the left of them are all knaves. At the third table, each person claims that the person two seats to the left is a liar.

What is the smallest possible number of seats at each table?

Spoiler Inside: Level 2 Solution SelectShow

 

Level 3

In the land of knights and knaves, you encounter five people and you ask each of them how many of the other four are knights and how many are knaves.

Person 1 says: “There are three knights and one knave”

Person 2 says: “There are no knights and four knaves”

Person 3 says: “There is one knight and three knaves”

Person 4 says: “There are four knights and no knaves”

Before you can ask person 5, he says “it’s my birthday today!”

Is today person 5’s birthday?  Don’t just guess.  Determine who is a knight and who is a knave.

Spoiler Inside: Level 3 Solution SelectShow

Let’s see which statements lead to contradictions.

If Person 1 is telling the truth, all other statements must be true, but it is not possible for 1 and 2 to both be true so Person 4 is a knave.

If Person 1 is telling the truth, Persons 2, 3, and 5 must be telling the truth, but Person 2 contradicts Person 1 so Person 1 is a knave.

If Person 2 is telling the truth, then everyone else is lying.  Person 3 would be telling the truth when he said there was 1 knight, referring to Person 2, so Person 2 is a knave.

If Person 3 is telling the truth, there is one knight.  We have already eliminated Persons 1, 2, and 4, so the only other knight is Person 5 and it is his birthday.

[\spoiler]

Summer 2014 – Week 9

This is the last set of problems for the summer.  Please submit all solutions by August 10. I am looking forward to seeing you all at the kickoff on August 14.

Level 1

Jennifer just went back to school shopping.  She bought 7 shirts, 5 pairs of pants, 4 pairs of fancy socks, sneakers and sandals.  How many different outfits can Jennifer put together if she always wears one shirt, one pair of pants, one pair of shoes and she always wears socks with sneakers but never wears socks with sandals?

Level 2

George is taking five classes this year: Math, Science, History, English, and Spanish.  He needs a spiral notebook and a pocket folder for each class.  Spiral notebooks come in 10 colors (black, blue, red, green, lime, purple, pink, orange, aqua, and yellow).  Pocket folders come in six colors (blue, red, green, yellow, purple, and orange).
a) How many different ways can George choose the notebooks and folders for his classes if he doesn’t buy any duplicate items?
b) How many ways can George choose the notebooks and folders if he wants to have the same color notebook and folder for a certain class?

Level 3

George is going to arrange his books, folders, and notebooks on the bookshelf.  For each of five subjects he has a textbook, a folder, and a spiral notebook.  He wants to be able to find things so he is considering different ways of organizing them.
a) How many ways can he arrange them if he keeps all three items from one subject together (the three math items are together in any order, the three history items, etc.)?
b) How many ways can he arrange them if he keeps all the textbooks together, all the spiral notebooks together, and all the folders together?

 

 

Summer 2014 – Week 8

A special thanks to Timothy for this week’s problems.

Knights and Knaves

In the land of knights and knaves, everyone is either a knight or a knave. Knights always tell the truth, while knaves always tell lies. It is impossible to tell apart knights and knaves by appearance.

 Level 1

In the land of knights and knaves, the king’s crown is stolen, and four suspects are questioned.

Suspect 1 says: “Suspect 3 stole the crown”

Suspect 2 says: “I did not steal the crown”

Suspect 3 says: “Suspect 4 stole the crown”

Suspect 4 says:  “Suspect 3 is a knave”

If exactly three of the suspects are knaves, who stole the crown?

Level 2

In the land of knights and knaves, there are three round tables with the same number of people sitting around each of them. At the first table, each person claims that they are sitting next to a knight and a knave. At the second table, each person claims that the three people to the left of them are all knaves. At the third table, each person claims that the person two seats to the left is a liar.

What is the smallest possible number of seats at each table?

Level 3

In the land of knights and knaves, you encounter five people and you ask each of them how many of the other four are knights and how many are knaves.

Person 1 says: “There are three knights and one knave”

Person 2 says: “There are no knights and four knaves”

Person 3 says: “There is one knight and three knaves”

Person 4 says: “There are four knights and no knaves”

Before you can ask person 5, he says “it’s my birthday today!”

Is today person 5’s birthday?  Don’t just guess.  Determine who is a knight and who is a knave.

 

Summer 2014 – Week 7 Solutions

Thank you to Timothy for writing this week’s solutions.  Since the Level1 and Level 2 problems are related, their solutions will be together.

Level 1

Three cuts can divide a circle into 7 pieces as shown in the figure below.  3 cuts can divide a sphere into 8 pieces.  What is the maximum number of pieces of a circle that can be obtained with 4 cuts?  What is the maximum number of pieces of a sphere that can be obtained with 4 cuts?

Circle with 3 lines sphere

Level 2

What is the maximum number of pieces of a circle that can be obtained with 5 cuts?

What is the maximum number of pieces of a sphere that can be obtained with 5 cuts?

Spoiler Inside: Level 1 & 2 Solution SelectShow

Level 3

A spherical cap is the region of a sphere that lies above or below a given plane as shown in the diagram.  If the sphere has a radius of 5 m, and the cap has a height of 3 m, what is the volume of the cap?

spherical cap

 

 

Summer 2014 – Week 7

Level 1

Three cuts can divide a circle into 7 pieces as shown in the figure below.  3 cuts can divide a sphere into 8 pieces.  What is the maximum number of pieces of a circle that can be obtained with 4 cuts?  What is the maximum number of pieces of a sphere that can be obtained with 4 cuts?

Circle with 3 lines sphere

Level 2

What is the maximum number of pieces of a circle that can be obtained with 5 cuts?  What is the maximum number of pieces of a sphere that can be obtained with 5 cuts?

Level 3

A spherical cap is the region of a sphere that lies above or below a given plane as shown in the diagram.  If the sphere has a radius of 5 m, and the cap has a height of 3 m, what is the volume of the cap?

spherical cap

 

Summer 2014 – Week 6 Solutions

Level 1

At midnight the hour and minute hands of a clock are directly on top of one another.  How many times are the hands on top of each other from midnight Monday to midnight Wednesday?  (Count both the starting and stopping time.)

Spoiler Inside: Level 1 Solution SelectShow

 

Level 2

What is the smaller angle between the hands of a clock at 4:37?

Spoiler Inside: Level 2 Solution SelectShow

Level 3

Mr. Smith has two antique clocks that do not keep very good time.  His fast clock gains 3 minutes every hour and his slow clock loses 4 minutes every hour.  He sets both of the clocks correctly at 8:00 a.m.  What time does the fast clock read when the slow clock reads 1:30 that afternoon?

Spoiler Inside: Level 3 Solution SelectShow

Summer 2014 – Week 6

Level 1

At midnight the hour and minute hands of a clock are directly on top of one another.  How many times are the hands on top of each other from midnight Monday to midnight Wednesday?  (Count both the starting and stopping time.)

Level 2

What is the smaller angle between the hands of a clock at 4:37?

Level 3

Mr. Smith has two antique clocks that do not keep very good time.  His fast clock gains 3 minutes every hour and his slow clock loses 4 minutes every hour.  He sets both of the clocks correctly at 8:00 a.m.  What time does the fast clock read when the slow clock reads 1:30 that afternoon?

 

Summer 2014 – Week 4/5 Solution

 

Level 1

Find the area in the figure below.  All lines are straight and all angles are right.

Area 1

 

Spoiler Inside: Level 1 Solution SelectShow

Level 2

The figure below consists of a 6 x 8 inch rectangle and four semicircles.  Find the area of the shaded region.  You may leave your answer in terms of π.

Area 2

 

Spoiler Inside: Level 2 Solution SelectShow

Level 3

The figure below consists of a 60° arc with a radius of 6 cm and a semicircular arc with a diameter of 6 cm.  Find the area of the shaded region. You may leave your answer in terms of π.

Area 3

Spoiler Inside: Level 3 Solution SelectShow

 

 

 

Summer 2014 – Week 4/5

This week we will be doing some geometry problems.  Answers will be posted on July 6.

Level 1

Find the area in the figure below.  All lines are straight and all angles are right.

Area 1

Level 2

The figure below consists of a 6 x 8 inch rectangle and four semicircles.  Find the area of the shaded region.  You may leave your answer in terms of π.

Area 2

Level 3

The figure below consists of a 60° arc with a radius of 6 cm and a semicircular arc with a diameter of 6 cm.  Find the area of the shaded region. You may leave your answer in terms of π.

Area 3