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(01/07/97) Happy New Year!

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MathMagic Cycle 23: Level K-3 Regular

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Last Christmas, while cutting trees form colored acetates (see-through

"plastic" sheets) I ended up lining five trees next to each other, their sides overlapping like this:

/\ /\ /\ /\ /\

/ \ / \ / \ / \ / \

/ # x # x # x # x # \

/ /#\ /#\ /#\ /#\ \

/----/---\/---\/---\/---\----\

|| || || || ||

For decorations, I cut nine circular bulbs out of shinny paper, wrote numbers one through nine and was able to place them on the "#" spots so that the sum of all the digits in any two adjacent (next to each other) trees was 20. Working with your NTPs, can you tell us how they should be arranged? Make sure to copy down the steps you take when sharing your answers with us.

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MathMagic Cycle 23: Level K-3 Advanced

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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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Good luck

MrH