(01/07/97)

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MathMagic Cycle 23: Level 4-6 Regular 

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Professor Susan Addington from the Geometry Center at the University of
Minnesota (www.geom.umn.edu) has done some wonderful work with discrete
dynamical systems (fancy name for fun stuff.)

Her "Number Bracelets Game" is described below. It can be as simple as
working with the beads only, but it can quickly lead into higher math.

Begin by having lots of "beads" numbered zero through nine, with as many
as you want of each kind:

          (0)  (1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)

Now follow these rules:

a) Pick a first and second bead (can be the same number):
          (2)  (6)

b) To choose the third bead, add the first and the second beads. If the
   number is more than 9, just use the last (ones) digit of the sum:
          (2)  (6)  (8)

c) To get the next bead add the last two digits and use only the ones
   digit:     (2)  (6)  (8)  (4)

d) Keep doing this until you get back to the first and second beads in
   that order:
          (2)  (6)  (8)  (4)  (2)  (6)
          --------            --------
          first 2             repeated                        (2)
                                                             /    \
Since you don't want repeated numbers in your bracelet,   (4)      (6)
the bracelet will only have these four digits:               \    /
Questions:                                                    (8)
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1- How many different starting pairs of beads are there?
2- What is the longest bracelet you can make?
3- Is there a string of beads that never repeats?
4- If you start with the same two beads, but in opposite order, do you get
   the same bracelet? Do you get the same bracelet in reverse?

Remember to write down your steps and conclussions.

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MathMagic Cycle 23: Level 4-6 Advanced 

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The Old Annoying Building

  A            This diagram shows a very old building that has four
 | |-------    entrances, A,B,C and D, and four elevators, E,F,G and H.
 |        B    The problem is that the building is so old that the 
 | E---F  -    elevators can only operate between adjacent floors. So to
 | |   | |     go to a fifth floor (ground floor is considered first) a
 | |   | |     patron can go A, E(1-2), F(2-3), G(3-4) and H(4-5), having
-  H---G |     to change elevators every floor...
D        |
-------| |     Under these rather annoying circumstances, please discuss
        C      with your NTPs the following:

a) How many ways are there to go to a second floor office?

b) How many ways into a third floor office?

c) How many ways (without repeating elevators) into a fourth floor office?

d) How does C) above change when you CAN repeat an elevator, BUT not on
   consecutive floors? (say E(1-2) and E(3-4))

e) How many ways into a fifth floor office, both repeating elevators in 
   non-consecutive floors, and not repeating elevators at all?

f) What patterns have you and your NTPs developed? Please share with us
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Good luck
MrH