The Number Bracelets Game
Contents.
Introduction and Rules
This is a game that lots of kids are playing (or so I've heard). You only
need to be able to add whole numbers to play it, but there are interesting
variations and extensions for people who like to think about mathematical
patterns (6th graders, high school students, math majors, graduate students,
...)
The origin of this game, in this form, is an activity by Marilyn Burns
(About Teaching Mathematics, a K8 Resource, Math Solutions Publications,
1992.) Another version can be found in Joan Cotter's book of Math
Card Games.
Imagine that you have lots of beads, numbered from 0 through 9,
as many as you want of each kind.
Here are the rules for making a number bracelet:

Pick a first and a second bead. They can have the same number.

To get the third bead, add the numbers on the first and second beads. If
the sum is more than 9, just use the last (ones) digit of the sum.
To get the next bead, add the numbers on the last two beads you used,
and use only the ones digit. So to get the fourth bead, add the numbers
on the second and third beads, and use the ones digit.
Keep going until you get back to the first and second beads, in that
order.

How long (or short) a bracelet can you make?
Example
Choose 2 and 6 for the first and second beads:
The third bead is 2 + 6 = 8:
To get the fourth bead, add 6 and 8, then use only the ones digit:
6 + 8 = 14; use 4:
8 + 4 = 12; use 2:
4 + 2 = 6:
But the last two beads are the same as the first two, so instead of
making a long string, use 2, 6, 8, and 4 in a loop, or bracelet.
Another Example
Choose 1 and 3 for the first and second beads:
The third bead is 1 + 3 = 4:
3 + 4 = 7:
4 + 7 = 11; use 1:
7 + 1 = 8:
1 + 8 = 9:
8 + 9 = 17; use 7:
9 + 7 = 16; use 6:
7 + 6 = 13; use 3:
6 + 3 = 9:
3 + 9 = 12; use 2:
9 + 2 = 11; use 1:
2 + 1 = 3:
But the last two beads are the same as the first two, so pop them off
and make a bracelet:
Questions
Think about these before looking at the answers. Some of the questions
are easier to answer if you work with other people and make lots of different
bracelets.

How long (or short) a number bracelet can you make?
Answers

Will a number bracelet always loop back to the beginning, or can you have
a string of beads that never repeats? Answers

How many different starting pairs of beads are there? Answers

How many different number bracelets are there? Answers

If you start with the same two beads, but in the opposite order, do you
get the same bracelet? Do you get the same bracelet in reverse? Answers
To Extensions and Generalizations.