7.06:_Painted_Cubes

<p class="lt-math-9868">You can build up squares from smaller squares:</p>

<table class="mt-responsive-table">
    <tbody>
        <tr>
            <td class="mt-align-center mt-noheading lt-math-9868"><img alt="1x1sq.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/11723/1x1sq.png?revision=1" /></td>
            <td class="mt-align-center mt-noheading lt-math-9868"><img alt="2x2sq-225x221.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/11724/2x2sq-225x221.png?revision=1" /></td>
            <td class="mt-align-center mt-noheading lt-math-9868"><img alt="3x3q-300x300.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/11725/3x3q-300x300.png?revision=1" /></td>
        </tr>
        <tr>
            <td class="mt-align-center mt-noheading lt-math-9868">1 &times; 1 square</td>
            <td class="mt-align-center mt-noheading lt-math-9868">2 &times; 2 square</td>
            <td class="mt-align-center mt-noheading lt-math-9868">3 &times; 3 square</td>
        </tr>
    </tbody>
</table>

<p class="lt-math-9868">In a similar way, you can build up cubes from smaller cubes:</p>

<table class="mt-responsive-table">
    <tbody>
        <tr>
            <td class="mt-align-center mt-noheading lt-math-9868"><img alt="512px-Face_colored_cube.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/11726/512px-Face_colored_cube.png?revision=1" /></td>
            <td class="mt-align-center mt-noheading lt-math-9868"><img alt="Pocket_cube_solved.jpg" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/11727/Pocket_cube_solved.jpg?revision=1" /></td>
            <td class="mt-align-center mt-noheading lt-math-9868"><img alt="Rubiks_cube_scrambled.jpg" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/11728/Rubiks_cube_scrambled.jpg?revision=1" /></td>
        </tr>
        <tr>
            <td class="mt-align-center mt-noheading lt-math-9868">1 &times; 1 &times; 1 cube<u><sup>[1]</sup></u></td>
            <td class="mt-align-center mt-noheading lt-math-9868">2 &times; 2 &times; 2 cube<u><sup>[2]</sup></u></td>
            <td class="mt-align-center mt-noheading lt-math-9868">3 &times; 3 &times; 3 cube<u><sup>[3]</sup></u></td>
        </tr>
    </tbody>
</table>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share"></span><h2 class="box-legend lt-math-9868"><span class="lt-icon-default">Think / Pair / Share</span></h2>

<p class="lt-math-9868">We call a 1 &times; 1 &times; 1 cube a <b>unit cube</b>.</p>

<ul>
    <li class="lt-math-9868">How many unit cubes are in a 2 &times; 2 &times; 2 cube?</li>
    <li class="lt-math-9868">How many unit cubes are in a 3 &times; 3 &times; 3 cube?</li>
    <li class="lt-math-9868">How many unit cubes are in a <em>n</em> &times; <em>n</em> &times; <em>n</em> cube?</li>
</ul>

<p class="lt-math-9868">Explain your answers.</p>
</section>

<section class="box-example"><span id="Problem_10"></span><h2 class="box-legend lt-math-9868"><span class="lt-icon-default">Problem 10</span></h2>

<p class="lt-math-9868">Imagine you build a 3 &times; 3 &times; 3 cube from 27 small white unit cubes. Then you take your cube and dip it into a bucket of bright blue paint. After the cube dries, you take it apart, separating the small unit cubes.</p>

<ol>
    <li class="lt-math-9868">After you take the cube apart, some of the unit cubes are still all white (no blue paint). How many? How do you know you are right?</li>
    <li class="lt-math-9868">After you take the cube apart, some of the unit cubes have blue paint on just one face. How many? How do you know you are right?</li>
    <li class="lt-math-9868">After you take the cube apart, some of the unit cubes have blue paint on two faces. How many? How do you know you are right?</li>
    <li class="lt-math-9868">After you take the cube apart, some of the unit cubes have blue paint on three faces. How many? How do you know you are right?</li>
    <li class="lt-math-9868">After you take the cube apart, do any of the unit cubes have blue paint on more than three faces? How many? How do you know you are right?</li>
</ol>
</section>

<section class="box-example"><span id="Problem_11"></span><h2 class="box-legend lt-math-9868"><span class="lt-icon-default">Problem 11</span></h2>

<p class="lt-math-9868">Generalize your work on Problem 10. What if you started with a 2 &times; 2 &times; 2 cube? Answer the same questions. What about a 4 &times; 4 &times; 4 cube? How about an <em>n</em> &times; <em>n</em> &times; <em>n</em> cube? Be sure to justify what you say.</p>
</section>

<hr />
<ol>
    <li class="lt-math-9868">Image by Robert Webb's <u><a title="en:Stella (software)" href="https://en.Wikipedia.org/wiki/Stella_(software)" target="_blank" rel="external noopener nofollow" class="link-https">Stella software</a></u>: <u><a href="http://www.software3d.com/Stella.php" target="_blank" rel="external noopener nofollow" class="external">http://www.software3d.com/Stella.php</a></u>, via Wikimedia Commons. <u>&crarr;</u></li>
    <li class="lt-math-9868">Image by Mike Gonzalez (TheCoffee) (Work by Mike Gonzalez (TheCoffee)) [<u><a href="https://creativecommons.org/licenses/by-sa/3.0" target="_blank" rel="external noopener nofollow" class="link-https">CC BY-SA 3.0</a></u>], via Wikimedia Commons. <u>&crarr;</u></li>
    <li class="lt-math-9868">Image by Mike Gonzalez (TheCoffee) (Work by Mike Gonzalez (TheCoffee)) [CC BY-SA 3.0], via Wikimedia Commons. <u>&crarr;</u></li>
</ol>

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