2.01:_Dots_and_Boxes

<p class="lt-math-9828">Here are some dots; in fact there are nine of them:</p>

<p class="mt-align-center lt-math-9828"><img alt="9dots.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10257/9dots.png?revision=1" /></p>

<p class="lt-math-9828">We&rsquo;re going to a play an &ldquo;exploding dots&rdquo; game. Here&rsquo;s the only rule for the game:</p>

<section class="box-note"><span id="The_1.E2.86.902_Rule"></span><h2 class="box-legend lt-math-9828"><span class="lt-icon-default">The 1&larr;2 Rule</span></h2>

<p class="lt-math-9828">Whenever there are two 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):_Nine_dots_1.E2.86.902_in_the_system"></span><h2 class="box-legend lt-math-9828"><span class="lt-icon-default">Example \(\PageIndex{1}\): Nine dots 1&larr;2 in the system</span></h2>

<p class="lt-math-9828">We start by placing nine dots in the rightmost box.</p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1a-300x87.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10258/Explode1a-300x87.png?revision=1" /></p>

<p class="lt-math-9828">Two dots in that box explode and become one dot in the box to the left.</p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1b-300x83.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10259/Explode1b-300x83.png?revision=1" /></p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1c-300x84.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10260/Explode1c-300x84.png?revision=1" /></p>

<p class="lt-math-9828">Once again, two dots in that box explode and become one dot in the box to the left.</p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1d-300x81.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10261/Explode1d-300x81.png?revision=1" /></p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1e-300x82.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10262/Explode1e-300x82.png?revision=1" /></p>

<p class="lt-math-9828">We do it again!</p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1f-300x86.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10263/Explode1f-300x86.png?revision=1" /></p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1g-300x89.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10264/Explode1g-300x89.png?revision=1" /></p>

<p class="lt-math-9828">Hey, now we have more than two dots in the second box, so those can explode and move!</p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1h-300x82.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10265/Explode1h-300x82.png?revision=1" /></p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1i-300x85.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10266/Explode1i-300x85.png?revision=1" /></p>

<p class="lt-math-9828">And the rightmost box still has more than two dots.</p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1j-300x89.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10267/Explode1j-300x89.png?revision=1" /></p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1k-300x84.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10268/Explode1k-300x84.png?revision=1" /></p>

<p class="lt-math-9828">Keep going, until no box has two dots.</p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1l-300x86.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10269/Explode1l-300x86.png?revision=1" /></p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1m-300x83.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10270/Explode1m-300x83.png?revision=1" /></p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1n-300x86.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10271/Explode1n-300x86.png?revision=1" /></p>

<p class="mt-align-center lt-math-9828"><img alt="Explode1o-300x88.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10272/Explode1o-300x88.png?revision=1" /></p>

<p class="lt-math-9828">After all this, reading from left to right we are left with one dot, followed by zero dots, zero dots, and one final dot.</p>

<dl>
    <dt><strong class="emphasis bold">Answer</strong></dt>
    <dd>
    <p class="lt-math-9828">The 1&larr;2 code for nine dots is: 1001.</p>
    </dd>
</dl>
</section>

<p class="lt-math-9828"><strong>On Your Own. </strong>Here&rsquo;s a diagram showing what happens for seven dots in a 1&larr;2 box. Trace through the diagram, and circle the pairs of dots that &ldquo;exploded&rdquo; at each step.</p>

<p class="mt-align-center lt-math-9828"><img alt="Explode2a-300x85.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10273/Explode2a-300x85.png?revision=1" /></p>

<p class="mt-align-center lt-math-9828"><img alt="Explode2b-300x81.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10274/Explode2b-300x81.png?revision=1" /></p>

<p class="mt-align-center lt-math-9828"><img alt="Explode2c-300x83.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10275/Explode2c-300x83.png?revision=1" /></p>

<p class="mt-align-center lt-math-9828"><img alt="Explode2d-300x84.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10276/Explode2d-300x84.png?revision=1" /></p>

<p class="mt-align-center lt-math-9828"><img alt="Explode2e-300x87.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10277/Explode2e-300x87.png?revision=1" /></p>

<dl>
    <dt><strong class="emphasis bold">Answer</strong></dt>
    <dd>
    <p class="lt-math-9828">The 1&larr;2 code for seven dots is: 111.</p>
    </dd>
</dl>

<section class="box-example"><span id="Problem_1"></span><h2 class="box-legend lt-math-9828"><span class="lt-icon-default">Problem 1</span></h2>

<p class="lt-math-9828">Note: In solving this problem, you don&rsquo;t need to draw on paper; that can get tedious! Maybe you could use buttons or pennies for dots and do this by hand.</p>

<ol>
    <li class="lt-math-9828">Draw 10 dots in the right-most box and perform the explosions. What is the 1&larr;2 code for ten dots?</li>
    <li class="lt-math-9828">Find the 1&larr;2 code for eighteen dots.</li>
    <li class="lt-math-9828">What number of dots has 1&larr;2 code 101?</li>
</ol>
</section>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share"></span><h2 class="box-legend lt-math-9828"><span class="lt-icon-default">Think / Pair / Share</span></h2>

<p class="lt-math-9828">After you worked on the problem, compare your answer with a partner.  Did you both get the same code?  Did you have the same process?</p>
</section>