2.08:_Exploration

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<section class="box-example"><span id="Problem_44"></span><h2 class="box-legend lt-math-10361"><span class="lt-icon-default">Problem 44</span></h2>

<p class="lt-math-10361">Jay decides to play with a system that follows a 1&larr;1 rule. He puts one dot into the right-most box. What happens?</p>

<section class="box-example"><span id="Problem_45"></span><h2 class="box-legend lt-math-10361"><span class="lt-icon-default">Problem 45</span></h2>

<p class="lt-math-10361">Poindexter decides to play with a system that follows the rule 2&larr;3.</p>

<ol>
    <li class="lt-math-10361">Describe what this rule does when there are three dots in the right-most box.</li>
</ol>

<ol start="2">
    <li class="lt-math-10361">Draw diagrams or use buttons or pennies to find the codes for the following numbers: $$1\; \text{through}\; 20, 24, 27, 30, 33, 36,\; \text{and}\; 39$$</li>
    <li class="lt-math-10361">Can you find (and <i>explain</i>) any patterns?</li>
</ol>
</section>

<section class="box-example"><span id="Problem_46"></span><h2 class="box-legend lt-math-10361"><span class="lt-icon-default">Problem 46</span></h2>

<p class="lt-math-10361">Repeat problem 45 for your own rule. Choose two numbers $a \neq 1$ and <img alt="b" height="13" title="Rendered by QuickLaTeX.com" width="8" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/21767/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.svg?revision=1" />.  For each of the numbers</p>

<p class="lt-math-10361">\[1\; \text{through}\; 20, 24, 27, 30, 33, 36,\; \text{and}\; 39 \nonumber \]</p>

<p class="lt-math-10361">figure out the $a \leftarrow b$ code.  Look for patterns, and explain them if you can!</p>
</section>

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