http://mathpuzzlewiki.com/index.php?action=history&feed=atom&title=Patriotic_chameleons 2026-04-05T07:14:26Z Revision history for this page on the wiki MediaWiki 1.44.0 http://mathpuzzlewiki.com/index.php?title=Patriotic_chameleons&diff=742&oldid=prev 2010-12-01T15:40:49Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 09:40, 1 December 2010</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This is a problem I saw on someone's Putnam Exam review sheet.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This is a problem <ins style="font-weight: bold; text-decoration: none;">based on one </ins>I saw on someone's Putnam Exam review sheet.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Puzzle==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Puzzle==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>For <del style="font-weight: bold; text-decoration: none;">last </del>year's 4th of July party, you <del style="font-weight: bold; text-decoration: none;">bought 2010 </del>chameleons: <del style="font-weight: bold; text-decoration: none;">an equal number colored </del>red, white<del style="font-weight: bold; text-decoration: none;">, </del>and blue.  These chameleons have the strange property that whenever two of different colors meet, they both change into the third color.  <del style="font-weight: bold; text-decoration: none;">After </del>the <del style="font-weight: bold; text-decoration: none;">party, </del>you <del style="font-weight: bold; text-decoration: none;">worry </del>that <del style="font-weight: bold; text-decoration: none;">while </del>in <del style="font-weight: bold; text-decoration: none;">storage for next year's party</del>, the chameleons will all change <del style="font-weight: bold; text-decoration: none;">into a single </del>color, <del style="font-weight: bold; text-decoration: none;">thus making it impossible </del>to <del style="font-weight: bold; text-decoration: none;">recreate the patriotic mix </del>of colors.  <del style="font-weight: bold; text-decoration: none;">To remedy this problem</del>, <del style="font-weight: bold; text-decoration: none;">you go ahead an buy one more </del>chameleon.  <del style="font-weight: bold; text-decoration: none;">Will </del>the <del style="font-weight: bold; text-decoration: none;">2011 </del>chameleons <del style="font-weight: bold; text-decoration: none;">ever </del>be all the <del style="font-weight: bold; text-decoration: none;">same color?  </del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For <ins style="font-weight: bold; text-decoration: none;">next </ins>year's 4th of July party, you <ins style="font-weight: bold; text-decoration: none;">buy 2011 </ins>chameleons: <ins style="font-weight: bold; text-decoration: none;">some </ins>red, <ins style="font-weight: bold; text-decoration: none;">some </ins>white and <ins style="font-weight: bold; text-decoration: none;">some </ins>blue.  These chameleons have the strange property that whenever two of different colors meet, they both change into the third color.  <ins style="font-weight: bold; text-decoration: none;">Must there be a color among red, white and blue for which it is impossible for all of </ins>the <ins style="font-weight: bold; text-decoration: none;">chameleons to simultaneously become that color? </ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Help==</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{Hint | You know you have some chameleons of each color.  Do </ins>you <ins style="font-weight: bold; text-decoration: none;">know </ins>that <ins style="font-weight: bold; text-decoration: none;">there must be two colors with a different number of chameleons </ins>in <ins style="font-weight: bold; text-decoration: none;">each?  What can that difference be?  What happens to that difference when two chameleons meet?}}</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{Answer | Yes, there must always be such a color.}}</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{Solution | Since 2011 is not divisible by 3, there must be two colors (say red and white) which are not both multiples of 3.  That is</ins>, the <ins style="font-weight: bold; text-decoration: none;">difference in the number of red and white </ins>chameleons <ins style="font-weight: bold; text-decoration: none;">is either 1 or 2 modulo 3.  We </ins>will <ins style="font-weight: bold; text-decoration: none;">show that it is impossible for the chameleons to </ins>all change <ins style="font-weight: bold; text-decoration: none;">to the ''other'' </ins>color <ins style="font-weight: bold; text-decoration: none;">(in our case</ins>, <ins style="font-weight: bold; text-decoration: none;">blue).  </ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Let &lt;m&gt;D = R - W&lt;/m&gt; be the difference between the number of red and white chameleons.  Now consider what happens </ins>to <ins style="font-weight: bold; text-decoration: none;">&lt;m&gt;D&lt;/m&gt; when two chameleons </ins>of <ins style="font-weight: bold; text-decoration: none;">different </ins>colors <ins style="font-weight: bold; text-decoration: none;">meet:</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:Case 1. A red and white chameleon meet: both chameleons change to blue</ins>.  <ins style="font-weight: bold; text-decoration: none;">The number of red and white chameleons both drop by 1, so &lt;m&gt;D&lt;/m&gt; remains constant.</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:Case 2. A red and and a blue chameleon meet: both chameleons change to white. The number of red chameleons drop by 1</ins>, <ins style="font-weight: bold; text-decoration: none;">and the number of white chameleons increases by 2.  Thus &lt;m&gt;D&lt;/m&gt; decreases by 3. </ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:Case 3. A white and a blue </ins>chameleon <ins style="font-weight: bold; text-decoration: none;">meet: both change to red</ins>.  <ins style="font-weight: bold; text-decoration: none;">The number of red chameleons increases by 2, and </ins>the <ins style="font-weight: bold; text-decoration: none;">number of white </ins>chameleons <ins style="font-weight: bold; text-decoration: none;">decreases by 1.  Thus &lt;m&gt;D&lt;/m&gt; increases by 3.</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Notice that in each case, the remainder of &lt;m&gt;D&lt;/m&gt; when divided by 3 remains the same.  Since &lt;m&gt;D&lt;/m&gt; was 1 or 2 modulo 3 to begin with, it will always </ins>be <ins style="font-weight: bold; text-decoration: none;">such.  Therefore it is impossible for </ins>all the <ins style="font-weight: bold; text-decoration: none;">chameleons to be simultaneously blue, as that would make &lt;m&gt;D = 0&lt;/m&gt;.}}</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Number theory]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Number theory]]</div></td></tr> </table>

Oscarlevin http://mathpuzzlewiki.com/index.php?title=Patriotic_chameleons&diff=741&oldid=prev 2010-11-30T20:25:27Z

<p>Created page with &quot;This is a problem I saw on someone&#039;s Putnam Exam review sheet. ==Puzzle== For last year&#039;s 4th of July party, you bought 2010 chameleons: an equal number colored red, white, and...&quot;</p> <p><b>New page</b></p><div>This is a problem I saw on someone&#039;s Putnam Exam review sheet.<br /> <br /> ==Puzzle==<br /> <br /> For last year&#039;s 4th of July party, you bought 2010 chameleons: an equal number colored red, white, and blue. These chameleons have the strange property that whenever two of different colors meet, they both change into the third color. After the party, you worry that while in storage for next year&#039;s party, the chameleons will all change into a single color, thus making it impossible to recreate the patriotic mix of colors. To remedy this problem, you go ahead an buy one more chameleon. Will the 2011 chameleons ever be all the same color? <br /> <br /> [[Category: Number theory]]</div>

Oscarlevin