https://mathpuzzlewiki.com/index.php?title=Consecutive_hot_dogs&feed=atom&action=historyConsecutive hot dogs - Revision history2024-03-28T11:02:48ZRevision history for this page on the wikiMediaWiki 1.29.0https://mathpuzzlewiki.com/index.php?title=Consecutive_hot_dogs&diff=1377&oldid=prevOlevin-test: /* Help */2017-07-25T12:49:00Z<p><span dir="auto"><span class="autocomment">Help</span></span></p>
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 12:49, 25 July 2017</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Solution | Our task is equivalent to finding all the ways in which 253 can be expressed as the sum of consecutive positive integers. In fact, there are three solutions: 126 + 127, but this is not the answer because we know there are at least three contestants.  253 can be expressed as 1 + 2 +...+ 22, but that is not the answer, since Alvin ate at least 2 hot dogs.  The only other solution is 17 + 18 + ... + 29, and this give 11 contestants.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Solution | Our task is equivalent to finding all the ways in which 253 can be expressed as the sum of consecutive positive integers. In fact, there are three solutions: 126 + 127, but this is not the answer because we know there are at least three contestants.  253 can be expressed as 1 + 2 +...+ 22, but that is not the answer, since Alvin ate at least 2 hot dogs.  The only other solution is 17 + 18 + ... + 29, and this give 11 contestants.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>In general, with <m>c</m> contestants there will be a solution with <m>h</m> total hot dogs if and only if <m>c<del class="diffchange diffchange-inline">|</del>2h</m> and <m>\frac{c}{2} < \frac{h}{c}</m>. To see why this is, and as such to answer the extra credit parts, is left as an exercise for the reader.}}</div></td><td class='diff-marker'>+</td><td style="color:black; 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>In general, with <m>c</m> contestants there will be a solution with <m>h</m> total hot dogs if and only if <m>c<ins class="diffchange diffchange-inline">\mid </ins>2h</m> and <m>\frac{c}{2} < \frac{h}{c}</m>. To see why this is, and as such to answer the extra credit parts, is left as an exercise for the reader.}}</div></td></tr>
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</table>Olevin-testhttps://mathpuzzlewiki.com/index.php?title=Consecutive_hot_dogs&diff=1056&oldid=prevOscarlevin at 21:11, 16 March 20132013-03-16T21:11:11Z<p></p>
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</table>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Consecutive_hot_dogs&diff=993&oldid=prevOscarlevin: /* Help */2012-01-16T15:43:04Z<p><span dir="auto"><span class="autocomment">Help</span></span></p>
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 15:43, 16 January 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l21" >Line 21:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Solution | Our task is equivalent to finding all the ways in which 253 can be expressed as the sum of consecutive positive integers. In fact, there are three solutions: 126 + 127, but this is not the answer because we know there are at least three contestants.  253 can be expressed as 1 + 2 +...+ 22, but that is not the answer, since Alvin ate at least 2 hot dogs.  The only other solution is 17 + 18 + ... + 29, and this give 11 contestants.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Solution | Our task is equivalent to finding all the ways in which 253 can be expressed as the sum of consecutive positive integers. In fact, there are three solutions: 126 + 127, but this is not the answer because we know there are at least three contestants.  253 can be expressed as 1 + 2 +...+ 22, but that is not the answer, since Alvin ate at least 2 hot dogs.  The only other solution is 17 + 18 + ... + 29, and this give 11 contestants.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>In general, with <m>c</m> contestants will be a solution with <m>h</m> total hot dogs<del class="diffchange diffchange-inline">, </del>if and only if <m>c|2h</m> and <m>\frac{c}{2} < \frac{h}{c}</m>. To see why this is, and as such to answer the extra credit parts, is left as an exercise for the reader.}}</div></td><td class='diff-marker'>+</td><td style="color:black; 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>In general, with <m>c</m> contestants <ins class="diffchange diffchange-inline">there </ins>will be a solution with <m>h</m> total hot dogs if and only if <m>c|2h</m> and <m>\frac{c}{2} < \frac{h}{c}</m>. To see why this is, and as such to answer the extra credit parts, is left as an exercise for the reader.}}</div></td></tr>
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</table>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Consecutive_hot_dogs&diff=992&oldid=prevOscarlevin at 15:10, 12 January 20122012-01-12T15:10:00Z<p></p>
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 15:10, 12 January 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19" >Line 19:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Answer | There were 11 contestants}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Answer | There were 11 contestants}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>{{Solution | Our task is equivalent to finding all the ways in which 253 can be expressed as the sum of consecutive positive integers. <del class="diffchange diffchange-inline">One of my friends recommended me to order [http://www.essaysprofessors.com custom writing]on EssaysProfessors.Com. To tell you the truth, I have never regretted my decision. The writers are real professionals and know how to write impressive work full of knowledgeable information.  </del>In fact, there are three solutions: 126 + 127, but this is not the answer because we know there are at least three contestants.  253 can be expressed as 1 + 2 +...+ 22, but that is not the answer, since Alvin ate at least 2 hot dogs.  The only other solution is 17 + 18 + ... + 29, and this give 11 contestants.</div></td><td class='diff-marker'>+</td><td style="color:black; 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>{{Solution | Our task is equivalent to finding all the ways in which 253 can be expressed as the sum of consecutive positive integers. In fact, there are three solutions: 126 + 127, but this is not the answer because we know there are at least three contestants.  253 can be expressed as 1 + 2 +...+ 22, but that is not the answer, since Alvin ate at least 2 hot dogs.  The only other solution is 17 + 18 + ... + 29, and this give 11 contestants.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>In general, with <m>c</m> contestants will be a solution with <m>h</m> total hot dogs, if and only if <m>c|2h</m> and <m>\frac{c}{2} < \frac{h}{c}</m>. To see why this is, and as such to answer the extra credit parts, is left as an exercise for the reader.}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>In general, with <m>c</m> contestants will be a solution with <m>h</m> total hot dogs, if and only if <m>c|2h</m> and <m>\frac{c}{2} < \frac{h}{c}</m>. To see why this is, and as such to answer the extra credit parts, is left as an exercise for the reader.}}</div></td></tr>
</table>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Consecutive_hot_dogs&diff=991&oldid=prevStacieGoodwin at 14:20, 12 January 20122012-01-12T14:20:13Z<p></p>
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 14:20, 12 January 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19" >Line 19:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Answer | There were 11 contestants}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Answer | There were 11 contestants}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>{{Solution | Our task is equivalent to finding all the ways in which 253 can be expressed as the sum of consecutive positive integers.  In fact, there are three solutions: 126 + 127, but this is not the answer because we know there are at least three contestants.  253 can be expressed as 1 + 2 +...+ 22, but that is not the answer, since Alvin ate at least 2 hot dogs.  The only other solution is 17 + 18 + ... + 29, and this give 11 contestants.</div></td><td class='diff-marker'>+</td><td style="color:black; 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>{{Solution | Our task is equivalent to finding all the ways in which 253 can be expressed as the sum of consecutive positive integers<ins class="diffchange diffchange-inline">. One of my friends recommended me to order [http://www.essaysprofessors.com custom writing]on EssaysProfessors.Com. To tell you the truth, I have never regretted my decision. The writers are real professionals and know how to write impressive work full of knowledgeable information</ins>.  In fact, there are three solutions: 126 + 127, but this is not the answer because we know there are at least three contestants.  253 can be expressed as 1 + 2 +...+ 22, but that is not the answer, since Alvin ate at least 2 hot dogs.  The only other solution is 17 + 18 + ... + 29, and this give 11 contestants.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>In general, with <m>c</m> contestants will be a solution with <m>h</m> total hot dogs, if and only if <m>c|2h</m> and <m>\frac{c}{2} < \frac{h}{c}</m>. To see why this is, and as such to answer the extra credit parts, is left as an exercise for the reader.}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>In general, with <m>c</m> contestants will be a solution with <m>h</m> total hot dogs, if and only if <m>c|2h</m> and <m>\frac{c}{2} < \frac{h}{c}</m>. To see why this is, and as such to answer the extra credit parts, is left as an exercise for the reader.}}</div></td></tr>
</table>StacieGoodwinhttps://mathpuzzlewiki.com/index.php?title=Consecutive_hot_dogs&diff=779&oldid=prevOscarlevin at 16:20, 29 January 20112011-01-29T16:20:57Z<p></p>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; 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;">[[Category: New]]</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>__NOTOC__</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>__NOTOC__</div></td></tr>
</table>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Consecutive_hot_dogs&diff=771&oldid=prevOscarlevin at 22:59, 17 January 20112011-01-17T22:59:40Z<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 22:59, 17 January 2011</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11" >Line 11:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Extra extra credit==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Extra extra credit==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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 which total numbers of hot dogs is it possible to determine the number of contestants participating in such an event? Assume there must be at least three contestants<del class="diffchange diffchange-inline">?</del></div></td><td class='diff-marker'>+</td><td style="color:black; 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 which total numbers of hot dogs is it possible to determine the number of contestants participating in such an event? Assume there must be at least three contestants <ins class="diffchange diffchange-inline">and that everyone ate more than 1 hot dog.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td></tr>
</table>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Consecutive_hot_dogs&diff=770&oldid=prevOscarlevin at 18:59, 13 January 20112011-01-13T18:59:58Z<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 18:59, 13 January 2011</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l20" >Line 20:</td>
<td colspan="2" class="diff-lineno">Line 20:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Solution | Our task is equivalent to finding all the ways in which 253 can be expressed as the sum of consecutive positive integers.  In fact, there are three solutions: 126 + 127, but this is not the answer because we know there are at least three contestants.  253 can be expressed as 1 + 2 +...+ 22, but that is not the answer, since Alvin ate at least 2 hot dogs.  The only other solution is 17 + 18 + ... + 29, and this give 11 contestants.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Solution | Our task is equivalent to finding all the ways in which 253 can be expressed as the sum of consecutive positive integers.  In fact, there are three solutions: 126 + 127, but this is not the answer because we know there are at least three contestants.  253 can be expressed as 1 + 2 +...+ 22, but that is not the answer, since Alvin ate at least 2 hot dogs.  The only other solution is 17 + 18 + ... + 29, and this give 11 contestants.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>To see why <del class="diffchange diffchange-inline">these answers are unique</del>, and to answer the extra credit parts, is left as an exercise for the reader.}}</div></td><td class='diff-marker'>+</td><td style="color:black; 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"> </td><td class='diff-marker'>+</td><td style="color:black; 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 class="diffchange diffchange-inline">In general, with <m>c</m> contestants will be a solution with <m>h</m> total hot dogs, if and only if <m>c|2h</m> and <m>\frac{c}{2} < \frac{h}{c}</m>. </ins>To see why <ins class="diffchange diffchange-inline">this is</ins>, and <ins class="diffchange diffchange-inline">as such </ins>to answer the extra credit parts, is left as an exercise for the reader.}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Number theory]]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category: Number theory]]</div></td></tr>
</table>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Consecutive_hot_dogs&diff=769&oldid=prevOscarlevin at 18:53, 13 January 20112011-01-13T18:53:06Z<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 18:53, 13 January 2011</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19" >Line 19:</td>
<td colspan="2" class="diff-lineno">Line 19:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Answer | There were 11 contestants}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Answer | There were 11 contestants}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>{{Solution | Our task is equivalent to finding all the ways in which 253 can be expressed as the sum of consecutive positive integers.  In fact, there are three solutions: 126 + 127, but this is not the answer because we know there are at least three contestants.  <del class="diffchange diffchange-inline"><m></del>253 <del class="diffchange diffchange-inline">= </del>1 + 2 +...+ 22<del class="diffchange diffchange-inline"></m></del>, but that is not the answer, since Alvin ate at least 2 hot dogs.  The only other solution is 17 + 18 + ... + 29, and this give 11 contestants.</div></td><td class='diff-marker'>+</td><td style="color:black; 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>{{Solution | Our task is equivalent to finding all the ways in which 253 can be expressed as the sum of consecutive positive integers.  In fact, there are three solutions: 126 + 127, but this is not the answer because we know there are at least three contestants.  253 <ins class="diffchange diffchange-inline">can be expressed as </ins>1 + 2 +...+ 22, but that is not the answer, since Alvin ate at least 2 hot dogs.  The only other solution is 17 + 18 + ... + 29, and this give 11 contestants.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>To see why these answers are unique, and to answer the extra credit parts, is left as an exercise for the reader.}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>To see why these answers are unique, and to answer the extra credit parts, is left as an exercise for the reader.}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Consecutive_hot_dogs&diff=768&oldid=prevOscarlevin at 18:50, 13 January 20112011-01-13T18:50:57Z<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 18:50, 13 January 2011</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19" >Line 19:</td>
<td colspan="2" class="diff-lineno">Line 19:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Answer | There were 11 contestants}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Answer | There were 11 contestants}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>{{Solution | Our task is equivalent to finding all the ways in which 253 can be expressed as the sum of consecutive positive integers.  In fact, there are three solutions: 126 + 127, but this is not the answer because we know there are at least three contestants.  253 = 1 + 2 +...+ 22, but that is not the answer, since Alvin ate at least 2 hot dogs.  The only other solution is 17 + 18 + ... + 29, and this give 11 contestants.</div></td><td class='diff-marker'>+</td><td style="color:black; 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>{{Solution | Our task is equivalent to finding all the ways in which 253 can be expressed as the sum of consecutive positive integers.  In fact, there are three solutions: 126 + 127, but this is not the answer because we know there are at least three contestants.  <ins class="diffchange diffchange-inline"><m></ins>253 = 1 + 2 +...+ 22<ins class="diffchange diffchange-inline"></m></ins>, but that is not the answer, since Alvin ate at least 2 hot dogs.  The only other solution is 17 + 18 + ... + 29, and this give 11 contestants.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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> </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>To see why these answers are unique, and to answer the extra credit parts, is left as an exercise for the reader.}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>To see why these answers are unique, and to answer the extra credit parts, is left as an exercise for the reader.}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>Oscarlevin