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473 lines
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473 lines
12 KiB
HTML
---
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title: Birthday Puzzle
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author: Correl Roush
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---
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<p>
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This logic puzzle has been floating around the internet lately. When I
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caught wind of it, I thought it would be a great exercise to tackle
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using Prolog. I'm not especially good with the language yet, so it
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added to the challenge a bit, but it was a pretty worthwhile
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undertaking. When I got stumped, I discovered that mapping out the
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birthdays into a grid helped me visualize the problem and ultimately
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solve it, so I've included that with my prolog code so you can see how
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I arrived at the answer.
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</p>
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<div id="outline-container-sec-1" class="outline-2">
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<h2 id="sec-1">The Puzzle</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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Albert and Bernard have just met Cheryl. “When is your birthday?”
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Albert asked Cheryl. Cheryl thought for a moment and said, “I won’t
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tell you, but I’ll give you some clues”. She wrote down a list of
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ten dates:
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</p>
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<ul class="org-ul">
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<li>May 15, May 16, May 19
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</li>
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<li>June 17, June 18
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</li>
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<li>July 14, July 16
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</li>
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<li>August 14, August 15, August 17
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</li>
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</ul>
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<p>
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“One of these is my birthday,” she said.
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</p>
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<p>
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Cheryl whispered in Albert’s ear the month, and only the month, of
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her birthday. To Bernard, she whispered the day, and only the
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day. “Can you figure it out now?” she asked Albert.
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</p>
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<p>
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Albert: “I don’t know when your birthday is, but I know Bernard
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doesn’t know, either.”
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</p>
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<p>
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Bernard: “I didn’t know originally, but now I do.”
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</p>
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<p>
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Albert: “Well, now I know, too!”
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</p>
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<p>
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<i>When is Cheryl’s birthday?</i>
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</p>
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</div>
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</div>
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<div id="outline-container-sec-2" class="outline-2">
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<h2 id="sec-2">The Solution</h2>
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<div class="outline-text-2" id="text-2">
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</div><div id="outline-container-sec-2-1" class="outline-3">
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<h3 id="sec-2-1">The Dates</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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To start off, i entered each of the possible birthdays as facts:
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</p>
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<div class="org-src-container">
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<pre class="src src-prolog"><span style="color: #268bd2;">possible_birthday</span>(may, 15).
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<span style="color: #268bd2;">possible_birthday</span>(may, 16).
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<span style="color: #268bd2;">possible_birthday</span>(may, 19).
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<span style="color: #268bd2;">possible_birthday</span>(june, 17).
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<span style="color: #268bd2;">possible_birthday</span>(june, 18).
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<span style="color: #268bd2;">possible_birthday</span>(july, 14).
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<span style="color: #268bd2;">possible_birthday</span>(july, 16).
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<span style="color: #268bd2;">possible_birthday</span>(august, 14).
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<span style="color: #268bd2;">possible_birthday</span>(august, 15).
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<span style="color: #268bd2;">possible_birthday</span>(august, 17).
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</pre>
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</div>
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<p>
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And here they are, mapped out in a grid:
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</p>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<colgroup>
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<col class="right" />
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<col class="left" />
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<col class="left" />
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<col class="left" />
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<col class="left" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="right"> </th>
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<th scope="col" class="left">May</th>
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<th scope="col" class="left">June</th>
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<th scope="col" class="left">July</th>
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<th scope="col" class="left">August</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="right">14</td>
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<td class="left"> </td>
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<td class="left"> </td>
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<td class="left">X</td>
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<td class="left">X</td>
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</tr>
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<tr>
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<td class="right">15</td>
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<td class="left">X</td>
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<td class="left"> </td>
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<td class="left"> </td>
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<td class="left">X</td>
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</tr>
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<tr>
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<td class="right">16</td>
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<td class="left">X</td>
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<td class="left"> </td>
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<td class="left">X</td>
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<td class="left"> </td>
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</tr>
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<tr>
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<td class="right">17</td>
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<td class="left"> </td>
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<td class="left">X</td>
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<td class="left"> </td>
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<td class="left">X</td>
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</tr>
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<tr>
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<td class="right">18</td>
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<td class="left"> </td>
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<td class="left">X</td>
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<td class="left"> </td>
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<td class="left"> </td>
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</tr>
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<tr>
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<td class="right">19</td>
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<td class="left">X</td>
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<td class="left"> </td>
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<td class="left"> </td>
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<td class="left"> </td>
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</tr>
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</tbody>
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</table>
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</div>
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</div>
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<div id="outline-container-sec-2-2" class="outline-3">
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<h3 id="sec-2-2">Albert's Statement</h3>
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<div class="outline-text-3" id="text-2-2">
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<blockquote>
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<p>
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I don’t know when your birthday is,…
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</p>
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</blockquote>
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<p>
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Albert only knows the month, and the month isn't enough to uniquely
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identify Cheryl's birthday.
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</p>
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<div class="org-src-container">
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<pre class="src src-prolog"><span style="color: #268bd2;">month_is_not_unique</span>(<span style="color: #268bd2;">M</span>) :-
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bagof(<span style="color: #268bd2;">D</span>, possible_birthday(<span style="color: #268bd2;">M</span>, <span style="color: #268bd2;">D</span>), <span style="color: #268bd2;">Days</span>),
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length(<span style="color: #268bd2;">Days</span>, <span style="color: #268bd2;">Len</span>),
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<span style="color: #268bd2;">Len</span> > 1.
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</pre>
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</div>
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<blockquote>
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<p>
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… but I know Bernard doesn’t know, either.
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</p>
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</blockquote>
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<p>
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Albert knows that Bernard doesn't know Cheryl's
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birthday. Therefore, the day alone isn't enough to know Cheryl's
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birthday, and we can infer that the month of Cheryl's birthday does
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not include any of the unique dates.
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</p>
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<div class="org-src-container">
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<pre class="src src-prolog"><span style="color: #268bd2;">day_is_not_unique</span>(<span style="color: #268bd2;">D</span>) :-
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bagof(<span style="color: #268bd2;">M</span>, possible_birthday(<span style="color: #268bd2;">M</span>, <span style="color: #268bd2;">D</span>), <span style="color: #268bd2;">Months</span>),
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length(<span style="color: #268bd2;">Months</span>, <span style="color: #268bd2;">Len</span>),
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<span style="color: #268bd2;">Len</span> > 1.
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<span style="color: #268bd2;">month_has_no_unique_days</span>(<span style="color: #268bd2;">M</span>) :-
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forall(possible_birthday(<span style="color: #268bd2;">M</span>,<span style="color: #268bd2;">D</span>),
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day_is_not_unique(<span style="color: #268bd2;">D</span>)).
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</pre>
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</div>
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<p>
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Based on what Albert knows at this point, let's see how we've
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reduced the possible dates:
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</p>
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<div class="org-src-container">
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<pre class="src src-prolog"><span style="color: #268bd2;">part_one</span>(<span style="color: #268bd2;">M</span>,<span style="color: #268bd2;">D</span>) :-
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possible_birthday(<span style="color: #268bd2;">M</span>,<span style="color: #268bd2;">D</span>),
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month_is_not_unique(<span style="color: #268bd2;">M</span>),
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month_has_no_unique_days(<span style="color: #268bd2;">M</span>),
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day_is_not_unique(<span style="color: #268bd2;">D</span>).
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</pre>
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</div>
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<pre class="example">
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Results = [ (july, 14), (july, 16), (august, 14), (august, 15), (august, 17)].
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</pre>
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<p>
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So the unique days (the 18th and 19th) are out, as are the months
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that contained them (May and June).
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</p>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<colgroup>
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<col class="right" />
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<col class="left" />
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<col class="left" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="right"> </th>
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<th scope="col" class="left">July</th>
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<th scope="col" class="left">August</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="right">14</td>
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<td class="left">X</td>
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<td class="left">X</td>
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</tr>
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<tr>
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<td class="right">15</td>
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<td class="left"> </td>
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<td class="left">X</td>
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</tr>
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<tr>
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<td class="right">16</td>
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<td class="left">X</td>
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<td class="left"> </td>
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</tr>
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<tr>
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<td class="right">17</td>
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<td class="left"> </td>
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<td class="left">X</td>
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</tr>
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</tbody>
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</table>
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</div>
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</div>
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<div id="outline-container-sec-2-3" class="outline-3">
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<h3 id="sec-2-3">Bernard's Statement</h3>
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<div class="outline-text-3" id="text-2-3">
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<blockquote>
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<p>
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I didn’t know originally, but now I do.
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</p>
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</blockquote>
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<p>
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For Bernard to know Cheryl's birthday, the day he knows must be
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unique within the constraints we have so far.
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</p>
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<div class="org-src-container">
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<pre class="src src-prolog"><span style="color: #268bd2;">day_is_unique</span>(<span style="color: #268bd2;">Month</span>, <span style="color: #268bd2;">Day</span>) :-
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findall(<span style="color: #268bd2;">M</span>, part_one(<span style="color: #268bd2;">M</span>, <span style="color: #268bd2;">Day</span>), <span style="color: #859900; font-weight: bold;">[</span><span style="color: #268bd2;">Month</span><span style="color: #859900; font-weight: bold;">]</span>).
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<span style="color: #268bd2;">part_two</span>(<span style="color: #268bd2;">Month</span>, <span style="color: #268bd2;">Day</span>) :-
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possible_birthday(<span style="color: #268bd2;">Month</span>, <span style="color: #268bd2;">Day</span>),
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day_is_unique(<span style="color: #268bd2;">Month</span>, <span style="color: #268bd2;">Day</span>).
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</pre>
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</div>
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<pre class="example">
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Results = [ (july, 16), (august, 15), (august, 17)].
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</pre>
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<p>
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Both July and August contain the 14th, so that row is out.
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</p>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<colgroup>
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<col class="right" />
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<col class="left" />
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<col class="left" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="right"> </th>
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<th scope="col" class="left">July</th>
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<th scope="col" class="left">August</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="right">15</td>
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<td class="left"> </td>
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<td class="left">X</td>
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</tr>
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<tr>
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<td class="right">16</td>
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<td class="left">X</td>
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<td class="left"> </td>
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</tr>
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<tr>
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<td class="right">17</td>
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<td class="left"> </td>
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<td class="left">X</td>
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</tr>
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</tbody>
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</table>
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</div>
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</div>
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<div id="outline-container-sec-2-4" class="outline-3">
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<h3 id="sec-2-4">Albert's Second Statement</h3>
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<div class="outline-text-3" id="text-2-4">
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<blockquote>
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<p>
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Well, now I know, too!
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</p>
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</blockquote>
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<p>
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Albert's month must be the remaining unique month:
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</p>
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<div class="org-src-container">
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<pre class="src src-prolog"><span style="color: #268bd2;">month_is_not_unique</span>(<span style="color: #268bd2;">Month</span>, <span style="color: #268bd2;">Day</span>) :-
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findall(<span style="color: #268bd2;">D</span>, part_two(<span style="color: #268bd2;">Month</span>, <span style="color: #268bd2;">D</span>), <span style="color: #859900; font-weight: bold;">[</span><span style="color: #268bd2;">Day</span><span style="color: #859900; font-weight: bold;">]</span>).
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<span style="color: #268bd2;">part_three</span>(<span style="color: #268bd2;">Month</span>, <span style="color: #268bd2;">Day</span>) :-
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possible_birthday(<span style="color: #268bd2;">Month</span>, <span style="color: #268bd2;">Day</span>),
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month_is_not_unique(<span style="color: #268bd2;">Month</span>, <span style="color: #268bd2;">Day</span>).
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</pre>
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</div>
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<pre class="example">
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Results = [ (july, 16)].
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</pre>
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<p>
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August had two possible days, so it's now clear that the only
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possible unique answer is July 16th.
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</p>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<colgroup>
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<col class="right" />
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<col class="left" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="right"> </th>
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<th scope="col" class="left">July</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="right">15</td>
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<td class="left"> </td>
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</tr>
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<tr>
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<td class="right">16</td>
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<td class="left">X</td>
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</tr>
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<tr>
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<td class="right">17</td>
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<td class="left"> </td>
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</tr>
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</tbody>
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</table>
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</div>
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</div>
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<div id="outline-container-sec-2-5" class="outline-3">
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<h3 id="sec-2-5">Cheryl's Birthday</h3>
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<div class="outline-text-3" id="text-2-5">
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<div class="org-src-container">
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<pre class="src src-prolog"><span style="color: #268bd2;">cheryls_birthday</span>(<span style="color: #268bd2;">Month</span>, <span style="color: #268bd2;">Day</span>) :-
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part_three(<span style="color: #268bd2;">Month</span>, <span style="color: #268bd2;">Day</span>).
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</pre>
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</div>
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<pre class="example">
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Month = july,
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Day = 16.
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</pre>
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<p>
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So, there we have it. Cheryl's birthday is July 16th!
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</p>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<colgroup>
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<col class="right" />
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<col class="left" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="right"> </th>
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<th scope="col" class="left">July</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="right">16</td>
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<td class="left">X</td>
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</tr>
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</tbody>
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</table>
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</div>
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</div>
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</div>
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