2.08: Exploration

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.

2. Draw diagrams or use buttons or pennies to find the codes for the following numbers: $$1\; \text{through}\; 20, 24, 27, 30, 33, 36,\; \text{and}\; 39$$
3. 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

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

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