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 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.)
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).
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
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
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.