December 8

Still cutting the paper pieces

 

With 1 cut, - 2 pieces, 2 cuts - 4 pieces, 3 - 7, 4 - 11, 5 - 16...

(Recall triangular numbers: 1, 3, 6, 10, 15,... - do you see any connections?)

Even more cuttings

Can we cut a square with straight cuts? How about pentagon? pentagram?

Of course we cannot cut the pentagram! It is not convex! Only convex figures can be done with straight cuts.

But there is a trick: fold paper! can you cut non-convex stars?

Other triangular numbers

More gallery guards: 

how many you need to guard these galleries?

2, 2, 3 and 2!

December 1

Cutting again - rectangular pieces with 4 cuts

What if we cut through 2 shapes? 2 circles?

Gallery guards

Divisibility by 2, 3, 4, 5. Which triangular numbers are divisible by 2, 3, 4? Game of exchange

SET!

Magic trick with SpotIt cards

Experiment with magnets.

November 24

lines in rectangle

Still counting how into how many pieces one can cut a sheet of paper by straight cuts. Can one with 3 cuts gets 4 pieces? 5? 6? 7? 8?

3 cuts, 6 pieces

OK, next task: with 4 cuts, can you get 4, 5, 6, 7, 8, 9, 10, 11 or 12 pieces?

Euler characteristic, again

summing up these vertices, edges and pieces leads us again to Euler characteristics...

Is it still 1?

12 Vertices + 6 pieces = 17 edges + 1 (this is Euler characteristic)

It still is...

Gallery guards

How many of them do you need to keep an eye on all the walls of this crazy gallery?

SET

Catching those sets - about 1 in 4 correct!

experiment du jour: ice and the level of the water

when our little iceberg melts, will the water level go up?

No... 

November 17

triangulations

we will continue drawing triangulations, and also will be counting triangles, edges and vertices there. We'll do a table, and then try to find some relations between these numbers:

V-E+T=1

This is called Euler characteristic...

triangular and square numbers

we will continue playing with triangular and square numbers

0, 1, 3, 6, 10, 15, 21, ...

0, 1, 4, 9, 16, 25, ...

folding paper, drawing lines.

if you fold a sheet of paper once, the crease creates two parts. If you fold it twice, the creases partition the sheet into four pieces. How many parts you will get if you fold it three times? fours times?

if you draw a straight line on a sheet of paper, it creates two parts. Two lines give you fours parts. Three lines? Four lines?

SET

for experiments 

we will do paper plane competition.

November 10

triangulations

If you draw non-intersecting straight lines connecting the dots, so that no more lines can be added, your polygon is now split into triangles. We call this partition a triangulation... There are many triangulations of a polygon with the dots inside, but they all seem to have the same number of straight lines there. Why?

triangular and square numbers

triangular numbers: how many dots in a triangular pile? If we count empty triangle and start with 0, we get

0, 1, 3, 6, 10, 15, 21...

What happens if you take the differences between the neighboring triangular numbers? Try it:

1, 2, 3, 4, 5, 6,...

Seems like a pattern. Can one explain it?

One can also form square numbers:

0, 1, 4, 9, 16, 25,...

What happens if we take sums of the neighboring triangular numbers? Here's the beginning of the sequence:

0+1, 1+3, 3+6, 6+10, 10+15,...

Looks familiar? can one explain this?

cutting rectangles and finding g.c.d.

did not really work well. Postpone till later.

ball rotations

experiments: 

tying shoelaces in one sleek movement!

November 3

We continue change game: how one can pay with fancy coins. Say, if one has only coins of 6 and 7 cents, can one pay 20c? 21c? 22c? 23c? 24c?

The game of criss-cross. We will count the triangles, and segments, and see what results.

More games! a game of guessing: do you always win if you guess right?

Games again! we'll play SET.

For the experiments, electronics.

October 27

Less with more

We have two jugs, 4 and 3 gallons, and plenty of water: how can one obtain 1 gallon of water?

Same question if the jugs volumes are 5 and 2? 7 and 2? 5 and 3? 

What about 6 and 3? 6 and 2?

Paying your dues

In Oddland, they have coins of 3 and 7 cents only. Can one pay 8 cents in exact change? 9? 10? 11 cents? 

In Evenland, they have coins of 2 and 6 cents. Can one pay 8 cents? 11?

Criss-cross

In the game of "cross-cross" on needs to add straight segments connecting points. No intersections allowed! Whoever makes the last possible move, wins. Who would be that?

Pascal triangle

robotic adventures continue: we are counting the numbers of ways to get from start to each of the crossings. 

These numbers have a name: Binomial coefficients, and are typically arranged as the

Pascal triangle:

start

1   1

1   2   1

1   3   3   1

1   4   6   4   1

...

SET

we will try the real game

As an experiment, we'll try the constant width (a.k.a. Reuleaux) rollers...

October 20

Finding all the trajectories along the grid (leading to Pascal triangle)

symmetries in n different lines

shape-o-metry

SET - three features the same

October 13

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.

Can you extend this picture? write all the sequences - without two black dots nearby! - of lengths 4... 5...

Robots still wandering 

how many ways to walk from the starting point to each of the crossings? Try to draw all the paths (or write them as instructions: North-West-North...

Find the number of paths to each crossing!

Eulerian graphs.

SET

Still the baby version: put the card that has three features the same as the one on the table.

Experiment 

rope on two pegs; one peg removed - rope slides off.

October 6

Latin squares

 Continuing latin square filling: A,B,C and D only, each once in each row and each column...

Three cities, each has 2 roads out.

Graphs. 

Can we have a road map with 4 cities, each with 2 roads out (roads just connect the cities, or intersect in a regular way, no T-crossings!). 

Can we have 5 cities, each with 2 roads (easy)? 4 cities with 3 roads each? 

What about 4 cities of which 3 cities with have 3 roads each, and one city has just two roads? Is this possible at all?

Robots on the map: How many ways to go?

To recall, we program robots to go from one shape to another. Our starting point is the red triangle, and the robots go one block if you told them the direction. We use only North and West as the direction (otherwise robots will waste their precious charge).

Some place can be reached in more than one way: say from red triangle to yellow pentagon one can go in three different ways:

North West West

West North West

West West North

Try to find for each of the crossings the number of ways to go there (might be tricky for far away crossings!) - and write the numbers on the map. We'll discuss what results later.

Eulerian graphs.

The task is to trace these graphs without retracing and taking your pencil off the paper. The graphs for which this is possible are called  Eulerian. One of these graphs is not Eulerian - which one?

Experiments

Understanding skin conductivity using advanced microelectronics from Radio Shack!

September 29

Problem 1: Cyclists and Bikes

One person rides a bicycle, two persons ride a tandem. 

Six cyclists ride on 5 bikes. How many of them are on usual bicycles? how many are riding tandems? 

What if one has 7 cyclists on 5 bikes? 8 on 4?

Problem 2: Robots in Urbana

One needs to program a robot to go from one place to another. Robot can go one block on its own, if you tell the direction. A program to go from, say, 5-star to yellow triangle is:

North-West-North

Create a plan for a robot going from the 5-star to each of the colored polygons.
Are there other ways to go from 5-star to yellow triangle (do not waste your robot time; do not make it go South or East)? find all of them! Do they all have the same number of steps?

Problem 3: latin squares

Fill the latin squares (the rule is the letters in each column and each row are different - A, B, C if you have 3-by-3 square, A, B, C, D for 4-by-4 square and so on)...

Game 4: baby set game

Game 5: overhang

Game 6: labyrinth with weasel balls