Number Bracelets: Extensions

# Extensions and Generalizations

See the "What's going on?" 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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