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## 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.