Extensions and Generalizations

See the "What's going on here?" page if you don't understand this page.

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!

Questions

For a given modulus and rule:

What is the length of the longest orbit?
What are the lengths of all the orbits, and how many of each are there?

Find patterns for varying moduli, but the same rule:

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?

Ask similar questions for a fixed modulus, but changing rules.

The most important question: WHY?

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