Number Bracelets: Extensions
Extensions and Generalizations
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Change the clock
Play the number bracelets game on a clock with a different
number of hours. (The number of hours on the clock
is called the modulus.) For instance, you could use a 12-hour
clock, on which 12 = 0, 13 = 1, etc.
Here's a worked-out example for the game on a 4-hour clock.
There are 4 different beads: 0, 1, 2, and 3.
For any sums over 3, subtract 4 until the result is one of
the four beads.
Starting pair (0,0): orbit 0 0; length 1.
Starting pair (0,1): orbit 0 1 1 2 3 1; length 6.
Starting pair (0,2): orbit 0 2 2; length 3.
Starting pair (0,3): orbit 0 3 3 2 1 3; length 6.
Since there are 16 ordered pairs and the total of the lengths
of the orbits listed is 16, we have found all the orbits.
Change the rule
The original number bracelets game used the Fibonacci sequence
rule: add the last two numbers to get the next one.
Try a different rule, such as adding twice the second number to
get the next number, or add the three previous numbers to get the
next number. Use your imagination.
For a given modulus and rule:
Find patterns for varying moduli, but the same rule:
- What is the length of the longest orbit?
- What are the lengths of all the orbits, and how many of each
Ask similar questions for a fixed modulus, but changing rules.
- How is the length of the longest orbit related to the modulus?
- Is there a way to know how to generate the longest orbit
in advance, without writing out all the orbits?
- How are the lengths of all the orbits related to the length
of the longest orbit?
The most important question: WHY?
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