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?

for D, 10.20

1/3+1/6=
1/3-1/6=
2/3+1/6=
1/2+1/4+1/8=
1/2+1/4+1/8+1/16=

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A, B and C are looking at a green breeble (a magical creature).

A: B says the breeble has three legs!
B: No, it has five thousand legs.
C: Just one of you is lying.

Indeed, just one of A, B and C is lying. Who?

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What is bigger:
9*9 or 99?
9*9*9 or 999?
5/2 or 7/3?
555/2 or 777/3?

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10x+2y=9
10x-y=0.

3x+y=0
2x-4y=0.

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September 25, 2016 - Math Circle 2.0

Warm-up

In a certain flower garden, each flower was either red, yellow, or blue.  All three colors were in the garden.  One person visited and said that whatever 3 flowers you picked, one was going to be yellow. Another person visited and said that whatever 3 flowers you picked, one was going to be red.  Is it true that for every 3 flowers you pick, one will be blue as well?  Why?

Fibonacci numbers 

We write all possible sequences of black and white dots, such that no two black next to each other. How many different sequences are there?

2  of length 1 of them; 

3 of length 2; 

5 of length 3... 

Writing them one under another - if longer sequence extends the shorter one - helps.

The students extended this picture and found that there were 8 sequences of length 3.

Battleship

The kids formed two teams of two and used coordinates to find each others' hidden battleships.