2.02:_Other_Rules

<p class="lt-math-9829">Let&rsquo;s play the dots and boxes game, but change the rule.</p>

<p class="lt-math-9829">Whenever there are three dots in single box, they &ldquo;explode,&rdquo; disappear, and become one dot in the box to the left.</p>
</section>

<section class="box-example"><span id="Example_.5C(.5CPageIndex.7B1.7D.5C):_Fifteen_dots_in_the_1.E2.86.903_system"></span><h2 class="box-legend lt-math-9829"><span class="lt-icon-default">Example \(\PageIndex{1}\): Fifteen dots in the 1&larr;3 system</span></h2>

<p class="lt-math-9829">Here&rsquo;s what happens with fifteen dots:</p>

<dl>
    <dt><strong class="emphasis bold">Answer</strong></dt>
    <dd>
    <p class="lt-math-9829">The 1&larr;3 code for fifteen dots is: 120.</p>
    </dd>
</dl>
</section>

<section class="box-example"><span id="Problem_2"></span><h2 class="box-legend lt-math-9829"><span class="lt-icon-default">Problem 2</span></h2>

<ol>
    <li class="lt-math-9829">Show that the 1&larr;3 code for twenty dots is 202.</li>
    <li class="lt-math-9829">What is the 1&larr;3 code for thirteen dots?</li>
    <li class="lt-math-9829">What is the 1&larr;3 code for twenty-five dots?</li>
    <li class="lt-math-9829">What number of dots has 1&larr;3 code 1022?</li>
    <li class="lt-math-9829">Is it possible for a collection of dots to have 1&larr;3 code 2031? Explain your answer.</li>
</ol>
</section>

<section class="box-example"><span id="Problem_3"></span><h2 class="box-legend lt-math-9829"><span class="lt-icon-default">Problem 3</span></h2>

<ol>
    <li class="lt-math-9829">Describe how the 1&larr;4 rule would work.</li>
    <li class="lt-math-9829">What is the 1&larr;4 code for thirteen dots?</li>
</ol>
</section>

<section class="box-example"><span id="Problem_4"></span><h2 class="box-legend lt-math-9829"><span class="lt-icon-default">Problem 4</span></h2>

<ol>
    <li class="lt-math-9829">What is the 1&larr;5 code for the thirteen dots?</li>
    <li class="lt-math-9829">What is the 1&larr;5 code for five dots?</li>
</ol>
</section>

<section class="box-example"><span id="Problem_5"></span><h2 class="box-legend lt-math-9829"><span class="lt-icon-default">Problem 5</span></h2>

<ol>
    <li class="lt-math-9829">What is the 1&larr;9 code for thirteen dots?</li>
    <li class="lt-math-9829">What is the 1&larr;9 code for thirty dots?</li>
</ol>
</section>

<section class="box-example"><span id="Problem_6"></span><h2 class="box-legend lt-math-9829"><span class="lt-icon-default">Problem 6</span></h2>

<ol style="list-style-type:lower-alpha;">
    <li class="lt-math-9829">What is the 1&larr;10 code for thirteen dots?</li>
    <li class="lt-math-9829">What is the 1&larr;10 code for thirty-seven dots?</li>
    <li class="lt-math-9829">What is the 1&larr;10 code for two hundred thirty-eight dots?</li>
    <li class="lt-math-9829">What is the 1&larr;10 code for five thousand eight hundred and thirty-three dots?</li>
</ol>
</section>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share"></span><h2 class="box-legend lt-math-9829"><span class="lt-icon-default">Think / Pair / Share</span></h2>

<p class="lt-math-9829">After you have worked on the problems on your own, compare your ideas with a partner.  Can you describe what&rsquo;s going on in Problem 6 and why?</p>
</section>

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