Thursday,
April 13, 2006
Infinity and Rates
Things that grow large:
The Tower of /Hanoi
(Click on this to go to Java website)
The general problem: (illustrated with three
objects)
Move objects that have an order (size) from one place to another using
only
a third place for "storage". No larger object can be placed on top of a
smaller
object during the move. Move only one object at a time!
Get experience going over the solution for the 3,4 and 5 tower puzzles.
Elimination tournament for doing the puzzle fast. Prizes awarded!?
A Solution for the 3 Tower of Hanoi Puzzle.
(Using playing cards 1,2,3)
Card-
Post Changes to cards 0-1 Changes to
cards
0 for even numbers ,
1
for
odd numbers
(0, 0, 0)
(0, 0, 0)
1. 1 →
B (1, 0, 0)
(1, 0, 0)
2. 2 → C
(1, 1, 0)
(1, 1, 0)
3. 1 →
C (2, 1, 0)
(0, 1, 0)
4. 3 →
B (2, 1, 1)
(0, 1, 1)
5. 1 →
A (3, 1, 1)
(1, 1, 1)
6. 2 →
B (3, 2, 1)
(1, 0, 1)
7. 1 → B
(4, 2, 1)
(0, 0, 1)
Notice the last column give a hamiltonian tour of the vertices of the
cube.
Now try the same organization for the 4 tower puzzle.
Solution of the 4 Tower of Hanoi Puzzle.
Card→Post
Changes to cards
0-1 Switches to cards
(0, 0, 0, 0)
(0,
0, 0, 0)
1. 1 → B
(1, 0, 0, 0)
(1, 0, 0, 0)
2. 2 → C
(1, 1, 0, 0)
(1, 1, 0, 0)
3. 1 → C
(2, 1, 0, 0)
(0, 1, 0, 0)
4. 3 → B
(2, 1, 1, 0)
(0, 1, 1, 0)
5. 1 → A (3, 1, 1,
0)
(1, 1, 1, 0)
6. 2 → B
(3, 2, 1, 0)
(1, 0, 1, 0)
7. 1 → B
(4, 2, 1, 0)
(0, 0, 1, 0)
8.
9.
10
11.
12.
13.
14.
15.
2. Discuss how to solve the 5-tower puzzle.
Move 4, then 1, then 4... so
How many moves would it take to solve the 5-tower puzzle?
15 + 1 + 15 =31 moves.
How many moves would it take to solve the 6-tower puzzle?
31 + 31 +1= 63
Based on the actual time it takes you now to do the 4-tower, how long
do
you think it would take you to do the 8-tower puzzle?
If we used 10 seconds for 15 moves.
2 *2 *2* 2* 2* 2* 2* 2 -1=255 moves
255 *10/ 15 = about 170 second = about 3minute
Arithmetic
growth: 2, 7, 12, 17, 22, ...
Examples?
Formula: A (x) = 5x + 2; Y(x) = mx + b, m>0
Geometric growth: 2, 10, 50 , 250, 1250, 6250,
Examples?
Formula: P(n) = 2*5n ; P(x) = A rx , r>1.
For each type of growth, when x >
> 0.
Tales
from the Arabian Nights: A month's wages.
Malthus -economics and gloomy predictions: Agricultural food
supplies grow arithmetically. Populations
grow geometrically.
Either war or famine are inevitable.
Example:
The dragon curve
What is a
curve?
Curves and approximations of curves by line segments and polygons.
Regular Polygons and the circle.
The length of a curve.
Infinite curves: The snowflake.
Activity on snowflakes.
WinFeed (old version)
Space
filling curves.
Other examples:measurements of curves and surfaces:
Archimedes: The area of the circle- the area of a triangle.
Kepler: The volume of a torus- the volume of a cylinder.
Growth Rates: m - linear, r -
exponential.
Graphing and rates: slopes of lines, approximating curves with
lines.
slope: m = (change in y)/(change in x)
Motion: time: t ; position:s
velocity: v = (change in s)/(change in t)
"zooming"
Example: What line does the graph of y = x2 look like near (3,9).