Thursday,  April  13, 2006
Infinity and Rates

Things that grow large:

The Tower of /(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)
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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

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 r
x ,   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).