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?