October 15, 2017 - Math Circle 2.0

Warm Up

Find the solutions to the following algebraic equations.

5 + x = 8

8 - y = 1

4 + 5 = z + 6

Game Theory

Play the nickel and penny game that we introduced last session with a partner.  Does any player have an advantage?

Remember the game:
A penny is worth one cent and a nickel is worth 5 cents.

There are two players to the game, player one and player two.  Both players will pick a coin in secret to play.  If both coins that are played match, player one wins both coins.  If both coins that are played do not match, player two wins both coins.  

Create payoff matrix for the nickel and penny game that we discussed in the previous math circle.  

What would happen if we added a dime to the mix to create a dime, nickel and penny game?  Let's answer next class.

October 8. 2017 - Math Circle 2.0

Warm Up

Perimeters of squares and circumferences of circles.  

Game Theory

Last class, we learned how to express a the game paper, rock, scissors in a pay-off matrix.  We found that the game was very balanced in which no player really had an advantage over another player.  Let's try another game....  Pennies and nickels.

A penny is worth one cent and a nickel is worth 5 cents.

There are two players to the game, player one and player two.  Both players will pick a coin in secret to play.  If both coins that are played match, player one wins both coins.  If both coins that are played do not match, player two wins both coins.  Play the game at home and see if you can find a winning strategy to win the most money.

Counting

There are three dogs:  Charlotte, Ozzy, and Rosie.  Each dog has a choice of one of four types of snacks:  Bone, Cupcake, Food, and Treats.  There is only one of each snack so two of the dogs cannot receive cupcake, for example.  Using your counting skills, map out the possible combinations of dogs and snacks.  There are 4 x 3 x 2, or 24 combinations.

Here are some combinations to get started:

               Ozzy        Rosie        Charlotte

1.            Bone        Cupcake   Food

2.            Bone        Cupcake   Treats

3.            Bone        Food         Cupcake

4.            Bone        Food         Treats

 

Map the rest of the combinations for homework.

October 1, 2017 - Math Circle 2.0

Homework Review

Using the supplied cup, measure the circumference of the cup.  Next, draw a square that inscribes the circle.  Is the perimeter of the square greater than the circumference of the circle?

Circumference of the circle

Use a string to mesasure the circumference of a set of circles.  Compare the diameter of the circles to the circumference.  In all cases, the circumference of the circle is a little over 3 times the diameter of the circle.  The number that expresses the ratio of diameter to circumference is represented by the greek letter pi.


Rock Paper Scissors

Learn how to play Rock, Paper, Scissors.  

Can you win at Rock Paper Scissors with a strategy?  Create a payoff matrix.

From:  http://shack.typepad.com/shack_in_japan/2012/05/game-theory-for-middle-school-day-1-payoff-matrices-and-rock-paper-scissors.html

September 3, 2017 - Math Circle 2.0

Warm up

Homework review.  
You need to capture precisely 4 gallons of water for some unknown experiment that requires precisely 4 gallons of water.  The tools available to you are a 3 gallon container, a 5 gallon container and an unlimited water source that allows you to fill the 5 and 3 gallon containers.  How would you fill the containers to get precisely 4 gallons of water?  

Make a scale model of classroom

What does scale mean?  Also is the classroom a rectangle or a square?  Draw a scale model of the classroom.  Can we calculate the size of the classroom to the size of the paper?

Number Puzzle

For the puzzle below, use the numbers 1 through 6 only once per position.  Try to have all the sides of the triangle add up to the same number.

For homework, use only the numbers 1, 3, and 5 once to complete the puzzle.  The sides must add to 11.

August 27, 2017 - Math Circle 2.0

Warm Up

Scale of the solar system.  Watch movie:

https://www.youtube.com/watch?v=Kj4524AAZdE

Perimeter

Introduce perimeter of a square and a rectangle.  How do we define a rectangle and a square?  How many side lengths of a rectangle do you need to know to calculate the perimeter?


Puzzle

Cows in a field game


Homework puzzle

You need to capture precisely 4 gallons of water for some unknown experiment that requires precisely 4 gallons of water.  The tools available to you are a 3 gallon container, a 5 gallon container and an unlimited water source that allows you to fill the 5 and 3 gallon containers.  How would you fill the containers to get precisely 4 gallons of water?  

August 20, 2017 - Math Circle 2.0

Warm Up

Welcome back to Math Circle.   Let's discuss the solar eclipse that will happen Monday, August 21, 2017 in the USA.


Pinhole Camera

Safety first!  Never look at the sun directly without ISO 12312-2 compliant solar glasses.  You may look at the sun indirectly by making a pinhole camera.

Follow the link below to get the directions.

https://eclipse2017.nasa.gov/sites/default/files/EclipseCerealBoxViewer.pdf


Size of Sun, Earth, and Moon

The following link has a summary of the sizes and distances of the sun, earth, and moon.

http://blair.pha.jhu.edu/scale.html

Almost four diameters of the moon span the earth.
109 diameters of the earth span the sun.

The distance from the earth to the moon is 110 moon diameters
The distance from the earth to the sun is 11759 earth diameters

What does that look like?
How can we create a model of this?
What is a diameter anyway?

Scale - 1 earth diameter = 1 inch
distance from earth to moon is 30 inches
distance from earth to sun is 11759 inches, or 980 feet, or distance to the top of right of the rectangle formed by the sidewalks north of Foellinger auditorium.

aug 20

...2857142857142857143x7=?

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Quadrangles - what is the taxonomy:

Kites (pairs of neighboring sides equal) - Parallelograms (opposite sides <=> opposite angles are equal)-Trapezoids (pairs of neighboring angles are equal)

Rhombi (=diamonds)   Rectangles

Squares

Midsides operations: take midpoint of all 4 sides of a quadrangle Q, get another quadrangle M(Q).

Turns out, - M(Q) is always a parallelogram!

Q-rhombus <=> M(Q) - rectangle, and vice versa.

---

Logics:

Two villages (Knights and Knaves), a person at the fork - which road goes to Knights village?

Larry, Leon, Tom, L's are liars, Tom tells truth. How to find if you speak to Larry?

---

When the angle between the hands is wider: 5 past 11, or 10 past noon?
$x^2 $

May 21, 2017 - Math Circle 2.0

Multiplication review

Use graph paper to solve:

3 x 2 = 
5 x 1 =
8 x 4 =
9 x 6 =

7 x y = 21
solve for y

Number Sequences

Predict the next number in the triangle number sequence and the square number sequence.


Logic Problem One

You are at a picnic and there is a cooler full of juice.  There are five apple juices, three grape juices, and three fruit punches.  You really want an apple juice.  Without looking at package of juice, what is the smallest number of juice packages that you need to take out of the cooler in order to make sure you get one apple juice.  What is the smallest number if you really wanted fruit punch?

Logic Problem Two

From Raymond Smullyan's To Mock a Mockingbird

Suppose I offer one of three prizes: Prize A, Prize B, and Prize C.  Prize A is the best prize.  Prize B is an ok prize, and Prize C is the worst prize.  If you say a true statement to me, for example, 2+2 = 4, I will give you prize A or B.  If you say a false statement, I will give you Prize C.  You really want Prize A.  What could you say to me to force me to give you Prize A, the best prize.

May 14, 2017 - Math Circle 2.0

Warm Up

Using graph paper and a pencil, graphically solve the following multiplication problems:

2 x 3 = 
4 x 3 =
4 x 4 =
5 x 6 =
7 x 7 =


Sphero

Make a square and a triangle with the Sphero.  

May 7, 2017 - Math Circle 2.0

Word problem

Cathy has 6 pair of black gloves and 6 pair of blue gloves in a drawer.  She needs to pick out a matching pair of gloves in complete darkness.  How many gloves does she need to pull in order to make sure she gets a match (one left hand glove, one right hand glove of the same color)?

Logic

Three friends and a missing cookie.  Based on the friends responses, who stole the cookie?

2-variable constraints

In the land of gnomes and dwarfs, there are seven total that came to a party.  Using that the dwarfs ate three more cookies than the gnomes, each being ate one cookie, and some graph paper, how many dwarfs and gnomes were at the party.

Bulls and Cows

April 30, 2017 - Math Circle 2.0

Warm up

4 + e = 10  what is e?
9 - f = 3  what is f?
10 - g = 2 what is g?

1 + 8 = x
x - 5 = y
x + y = z
what is z?

i + 5 = 10
j + j = 14
j - i = k
what is k?

Number sorting

Sort the numbers in ascending order

8 2 10 15 13 4
18 5 22 11 9 40
d a j x s k

Sort the numbers in descending order

9 1 3 11 5 8


Sphero

Let's make a path around the classroom and see if we can program the sphero to navigate the path.  The kids picked a route and programmed the sphero to navigate the classroom.

April 9, 2017 - Math Circle 2.0

Warm up

Calculating the number of triangles in an object....

There are 4 math circle students:
Count the total number of items that the students have:

Shirts?
Shoes?
Fingers?

Max was able to bake cookies.  He put a the cookies in 3 rows, with 4 cookies in each row.  How many cookies did he make? Are there other problems that can be solved like this?

Cabbage Patch Problem:

Divide the cabbage patch into 6 sections that contain only two cabbages in each section using only 3 straight lines.

Speed and Time

The sphero robot needs three parameters to travel:

• Direction (in the coordinate system we learned)
• Speed 
• Time (in seconds)

Program sphero to make start at a point, move to a second point, and then move to a final point at speed of 50.

April 2, 2017 - Math Circle 2.0

Warm up

Sequences with numbers and letters.

2 4 6 8 _
B D F H _

Review equality and letter substitution:

6 = x + x
10 = y + y
3 + n = 8
10 - z = 2

Try 2 equations:

a + b = 7
2 + a = 4
Solve for a and b

Speed and Time

The sphero robot needs three parameters to travel:

• Direction (in the coordinate system we learned)
• Speed 
• Time (in seconds)

Program sphero to hit a target in the room.  Set sphero at a fixed speed, choose a direction, and then figure out the amount of time sphero needs to travel to hit the target.


Sequence problem

Bulls and Cows

Always a favorite, we wrapped up math circle with a nice game of bulls and cows.

april 4

- Find:

• 2x4-1x5=3;
• 3x5-2x6=
• 4x6-3x7=
• ....
• 322x324-321x325=
• ....
• (a+1)(a+3)-a(a+4)=

- Look up the lunar eclipses over the past 20 years. On what months do they happen?


- Puzzle: guess what's in the picture?

Try to make your own picture of something that is hard to guess...

March 12, 2017 - Math Circle 2.0

Warm up

Focus on equality and operation sign:

Does 5 + 7 = 11?
Why not?

5 = _ + _
10 = _ + _
2 = _ - _


Substitute a symbol

6 = x + x
10 = y + y

Game

Bulls and Cows - mathematical game.  There is a secret code n digit code with all the digits  being different.  The kids then guess the code.  If they have a digit that is present in the code, it is a cow.  It they have a digit that is both present and in the correct position, they have a bull.  The number of bulls and cows are reported after each guess.  The game stops when the code is guessed correctly.

The kids picked a code for Sanjay and Darrin to solve that was 3 digits long.  The kids then stumped the adults with a 4 digit code.

March 5, 2017 - Math Circle 2.0

Warm up

Focus on equality and operation sign:

* + 0 = _
**** +* = _
**** - * = _
** + *** =  * + _
* + ****** =  *** + _

Game

Bulls and Cows - mathematical game.  There is a secret code 2 digit code with all the digits  being different.  The kids then guess the code.  If they have a digit that is present in the code, it is a cow.  It they have a digit that is both present and in the correct position, they have a bull.  The number of bulls and cows are reported after each guess.  The game stops when the code is guessed correctly.

Coordinates

Homework review:  How do we get an eight pointed star using squares?  

New homework:  Can we make a 10 pointed star using squares or triangles?

3.5

2564103*13
142857143*7=
14285714286*7=
142857142857*7
2564103*13

p-adic arithmetic:

...6666666666667*3=1

...666666667*9=3

....6666666667^2=?

Bulls and cows

2.26

multiplication

666667x3

37x3

divisibility by 3 rule

142857x7

p-adic arithmetic

...142857143x7

Opening the brackets

1001x107

(a+b)x(c+d)

Bulls and cows

Dragon curves

Phases of the moon

February 26, 2017 - Math Circle 2.0

Warm up

1 + 0 = _
4 +1 = _
2 + 3 =  1 + _
1 + 6 =  3 + _
6 + _ = 2 + 8
6 - 5 = 1 + _
7 - 2 = 2 + _

Coordinates

Coordinate review























Homework review - make a 6 pointed star using the coordinate system.  How can we make the star bigger?  

Homework - How do we make a 8 pointed star?


Logic question


There are three brothers named John, James, and William.  John and James always lie, but William always tells the truth.  The three are triplets and you cannot tell them apart.  You meet one of the three on the street and you want to find out if he is John.  You can only ask him one yes or no question.  The question you ask him can only have three words.  What do you ask him?

We made a truth table:

feb 19

Recall powers of 10.

Multiplication rules - explain distributivity (bracket opening).

Other usages for bracket opening: explains -x-=+.

Puzzle - ?

Knots/unknots?

February 19, 2017 - Math Circle 2.0

Warm up
Speed review.

The kids ran 83 feet from the steps of Altgeld hall to the Alma Mater statue.  The kids ran the distance in the following times: 6 sec, 6 sec, 7 sec, 10 sec, 11 sec.  How do these times compare with certain animals:

A tortoise will travel the same distance in 56 seconds
A cheetah will travel the same distance in 1 second


Coordinates with graph paper

Re-affirm degree coordinate system.  

Homework answer - Calculate the corresponding degrees of the southwest direction and the northwest direction.
Southwest - 225
Northwest - 315

At defined starting point on graph paper, draw lines using the following direction.

0 deg - 3 spaces
90 deg - 10 spaces
180 deg - 3 spaces
270 deg - 10 spaces
0 deg - 3 spaces
90 deg -2 spaces
45 deg - 2 spaces
90 deg - 2 spaces
135 deg - 2 spaces

What did you make if you add wheels to the bottom?

Homework - make a 6 pointed star using the coordinate system.  List the instructions that you used to make the star.

February 12, 2017 - Math Circle 2.0

Warm-up - Last week's Logic Problem

From http://brainden.com/logic-puzzles.htm

There are three switches downstairs. Each corresponds to one of the three light bulbs in the attic. You can turn the switches on and off and leave them in any position. 

How would you identify which switch corresponds to which light bulb, if you are only allowed one trip upstairs?  Hint:  The lightbulbs generate heat.


Drawing on Graph Paper

Re-affirm degree coordinate system.  

At defined starting point on graph paper, draw lines using the following direction.

90 deg - 10 spaces
0 deg - 10 spaces
270 deg - 10 spaces
180 deg - 10 spaces

where do you end up?

If the square that we built was a house, how would you draw the roof?  Have the kids figure out the degrees in between.

0 deg - 10 spaces (to return to top)
45 deg - 5 spaces
135 deg - 5 spaces

Homework - Calculate the corresponding degrees of the southwest direction and the northwest direction.

Speed for now ... because velocity is a vector

Are we faster than animals?
A tortoise moves less than 1 mph 
A cat can run at 29 mph
A horse can run at 30 mph
A dog can run at 45 mph (Greyhound)
A cheetah can move at 70 mph

How fast can you run?  How can we calculate it?  Hint:  The distance from the steps of Altgeld Hall to the Alma Mater statue is roughly 80 feet.  Using this distance and a stopwatch, the kids ran between 5 mph and 8 mph.

feb 12

Multiplication

Powers (compare 2^(3^2) and (2^3)^2 etc). More about bracket forming.
Guess number trick:
- think of a number;
- add 5;
- double the result;
- subtract the original number;
- add 10;
- subtract the original number again.

Solar eclipse (start at 0:45)

Lunar eclipse (start at 0:30)

Homework:

1. Invent your own number trick
2. Compare: 2²²²²² and 3⁽³³⁾
3. Find: 1111111 * 1111111; 11111111*11111111
4. What is the phase of moon during the lunar eclipse? solar eclipse?

February 5, 2017 - Math Circle 2.0

Warm up - City of Truth Answer... from last class

You are at an unmarked intersection ... one way is the City of Lies and another way is the City of Truth. 

Citizens of the City of Lies always lie. 

Citizens of the City of Truth always tell the truth. 

A citizen of one of those cities (you don't know which) is at the intersection. What question could you ask to them to find the way to the City of Truth?

Multiplication Problems

Anya made delicious chocolate cupcakes using a pan that had 6 spaces for cupcakes.  She made four pans.  How many cupcakes did she make?

Willa was having a huge party and made hamburgers.  She had 12 people coming over.  Each person gets one hamburger.   She went to the store to buy hamburger buns that come 8 in a package.  How many packages does she need?  How many buns will she have left over?

Max is doing a flooring project again.  He bought 28 tiles at the flooring store.  He has to tile a square area that is 6 tiles wide be 6 tiles long.  Did he buy enough tiles.  Does he need another trip to the tile store again?

Logic Puzzle - Light Bulbs - for homework

From http://brainden.com/logic-puzzles.htm

There are three switches downstairs. Each corresponds to one of the three light bulbs in the attic. You can turn the switches on and off and leave them in any position. 

How would you identify which switch corresponds to which light bulb, if you are only allowed one trip upstairs?

jan 22

Homework:

• Compare:
• 2¹¹ and 4⁵, 
• 8⁴ and 4⁵ ,
• 3¹² and 9⁶, 
• 5¹⁰ and 25⁵
• Compare 2³⁰ and 1,000,000,000. 
•  Look at the globe: if you walk 100 miles North, 100 miles East, 100 miles South and 100 miles West, would you return to the same point where you started? 
•  If you look at the Earth from above the North pole, is it rotating around its axis clockwise, or counterclockwise?

January 22, 2017 - Math Circle 2.0

Warm up - sequence of numbers

Continue the sequence:

1, 2, 3, .....
2, 4, 6, .....

During this segment, we discussed how a sequence of numbers differed from a repeating pattern.


Multiplication Problems

Anya made delicious chocolate cupcakes using a pan that had 6 spaces for cupcakes.  She made three pans.  How many cupcakes did she make?

Ellie was going to give out all her math circle friends 5 crackers.  How many crackers would she be giving out in total?

Max was doing a flooring project that was 5 tile squares wide by 5 tile squares long.  How many tiles does he need to complete the project?

For next time (did not get to it during the class....)

Willa was having a huge party and made hamburgers.  She had 12 people coming over.  Each person gets one hamburger.   She went to the store to buy hamburger buns that come 8 in a package.  How many packages does she need?  How many buns will she have left over.

Directions to City of Truth - discussed at the end of class but not solved

From http://www.mathisfun.com
You are at an unmarked intersection ... one way is the City of Lies and another way is the City of Truth. 

Citizens of the City of Lies always lie. 

Citizens of the City of Truth always tell the truth. 


A citizen of one of those cities (you don't know which) is at the intersection. What question could you ask to them to find the way to the City of Truth?

jan 15

We worked with "nonthreatening rook diagrams" and proved our first theorem ever...

Consider the family of boards of growing sizes as shown below. We will call their length to be the number of exposed NW squares (so the sizes here are 2 and 3).

Now, how many rook diagrams for these boards are there? Let us call R(n) the number of rook diagrams in such a board of length n. We computed:
R(1)=2, R(2)=3, R(3)=5... It does remind us of Fibonacci numbers!

So, let's remember that hte Fibonacci numbers satisfy F(n+1)=F(n)+F(n-1), and F(3)=2, F(4)=3, F(5)=5... So if we can prove that R(n+1)=R(n)+R(n-1), then knowing that R and F start similarly, we get that always R(n)=F(n+2)...

To prove that R(n+1)=R(n)+R(n-1) we look at the ways to place a rook into the first row:

As the picture shows, these two ways yield R(n) and R(n-1) as numbers to place the rooks - QED.

What about the rook number for this board?

One more question: here are two star pictures: which is taken closer to the equator?

 

jan 8

This semester: 

Big numbers. Infinities. Set theory.
Astronomy as practical geometry.

1. Rooks 

On 2x2 how many ways to place 2 rooks not threatening ea other

3x3 and 3 rooks

A recursive solution

With 3x3, once you have one rook down the others follow a 2x2 solution

2. How many cubes? How high the stack?

Start with a 1x1x1ft cube

Cut it with with n cuts across each side into... how many cubes?

One cut for each side -> you split it into 2x2x2 cubes.
Three cuts -> into 4x4x4 cubes.
And so on...

Then stack the (n+1)^3 blocks in a stack; how tall is it?

16ft for 4x4x4 split. And in general?

3. What is a number?

We have
Six chalks
Six chairs - these numbers are the same as we can match the object, one-to-one.

Sometimes we can match the objects without even knowing the numbers: the number of tails on the US pennies in circulation is the same as the number of tails on these pennies!

Some big numbers - we have to think what does this mean. The name gives us a description of the number, not the number itself. But we know how to operate with the description anyway...

4. Cosmology 

Camera open for a few hours at night
Based on picture, how long was it open?

Homework:

• how many ways are there to place 4 nonthreatening rooks on the 4x4 board?

 

and what about this board?

• think of two numbers that you don't know, but such that one is two times bigger than the other.
• how long this camera was open?