The shortest bracelet has length 1 and starts with (0,0). If you add one time, you get back to the first 0. But the bracelet has two beads. (Every bracelet has to have at least two beads to start.)

The next shortest bracelet starts with (0,5), and has length 3:

0 5 5

There is a bracelet of length 4 that starts with (2,6) (the
first example on the main page):

2 6 8 4

There is a bracelet of length 12 that starts with (1,3) (the second
example on the main page):

1 3 4 7 1 8 9 7 6 3 9 2

There is a bracelet of length 20 that starts with
(0,4)

0 4 4 8 2 0 2 2 4 6 0 6 6 2 8 0 8 8 6 4

There is a bracelet of length 60 that starts with (0,1)

(I'll let you have the fun of writing all the beads.)

0 1 2 3 4 5 6 7 8 9

For each of these choices there are the same 10 choices for the second bead. For example, if the first bead is 0, the possibilities for the first two beads are

(0,0) (0,1) (0,2) (0,3) (0,4) (0,5) (0,6) (0,7) (0,8) (0,9) [10 choices]

So there are 10 x 10 = 100 pairs in all.

The order of the numbers is important,
so we have to count *ordered pairs*: pairs of numbers in
a particular order. In mathematics, ordered pairs are written
in parentheses with a comma in between: (2,6).

6 8 4 2,

which, if written in a circle, is the same as the (2,6) bracelet:

2 6 8 4

A bracelet of length *L* has *L* different starting pairs on
it. For example, the (2,6) bracelet has the starting pairs
(2,6), (6,8), (8,4), and (4,2)

on it. If a starting pair occurs twice in the same bracelet,
then you made a mistake: the pattern would repeat exactly the same
after the second repeat of the pair as after the first repeat.
So you can't actually have the same pair twice in the same bracelet.

There are 100 different starting pairs
(see above).
The bracelets listed at the
top of this page have lengths

1, 3, 4, 12, 20, and 60,

so they account for

1 + 3 + 4 + 12 + 20 + 60 = 100

pairs, so we have found all the bracelets.

But could it loop back to another pair in the bracelet, without ever
returning to the
original pair?

There is a rule for going backwards on the bracelet. (Can you
discover the rule?) If the bracelet looked like the picture above,
there would be two rules at the branch point, one for each branch,
and no way to choose
which rule to use. So this can't happen, because there is a single
rule for going backwards.

Example: The (2,6) bracelet has length 4. But the (6,2) bracelet is the same as the (0,4) bracelet, which has length 20.

Back to main page.