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.