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2.8: Exploration

• • Contributed by Michelle Manes
• Professor (Mathematics) at University of Hawaii

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Problem 44

Jay decides to play with a system that follows a 1←1 rule. He puts one dot into the right-most box. What happens? Problem 45

Poindexter decides to play with a system that follows the rule 2←3.

1. Describe what this rule does when there are three dots in the right-most box. 1. Draw diagrams or use buttons or pennies to find the codes for the following numbers: $$1\; \text{through}\; 20, 24, 27, 30, 33, 36,\; \text{and}\; 39$$
2. Can you find (and explain) any patterns?

Problem 46

Repeat problem 45 for your own rule. Choose two numbers $$a \neq 1$$ and . For each of the numbers

$$1\; \text{through}\; 20, 24, 27, 30, 33, 36,\; \text{and}\; 39$$

figure out the $$a \leftarrow b$$ code. Look for patterns, and explain them if you can!