Caterpillar Numbers--Type A
Pick a natural number n
If n is even, divide by 2
If n is odd, add 1
Repeat using your new result as n
The chain of numbers becomes a caterpillar number.
For example, a caterpillar numbers of this type is the sequence 21-22-11-12-6-3-4-2-1.
Some Questions To Explore:
- Are all caterpillar numbers finite in length?
- What is the shortest caterpillar number you can find? Longest?
- Is there a pattern formed by caterpillar numbers of the same length?
- Is there an "even-odd-even...odd..." caterpillar number?
- Can you find a group of caterpillar numbers that end with the sequence 5-6-3-4-2-1?
- Try to build caterpillar numbers going in reverse from a given "tail" (e.g. if the "tail" was 4-2-1, the preceding missing segment could be either an 8 or a 3).
Source: J.M. (Bellingham), who led me to J. Russell" "Caterpillar Collection," Mathematics Teaching, Mar 2005
Hint: You just need to play with this pattern. No real "answer" is the goal....rather the goal is the discovery process itself.
Solution Commentary: Send me your discoveries....I am prepared to include them as part of this commentary.