I really liked this problem that my friend Shamile gave during dinner:
Consider 50 people lined up in a random order. Show that there exists a subsequence of length 8 such that the height is non-increasing or non-decreasing.
The solution is quite elegant. Assign an ordered pair  to (7,7). There is only 49 total combinations, so by pigeonhole principle, there exists a pair of the same number. But this cannot be! The rest is left to the reader.</p>
</div><!-- .entry-content -->
<footer class="entry-footer"><span class="cat-tags-links"><span class="cat-links"><svg class="icon icon-folder-open" aria-hidden="true" role="img"> <use href="#icon-folder-open" xlink:href="#icon-folder-open"></use> </svg><span class="screen-reader-text">Categories</span><a href="https://marshalljiang.com/category/blargh/" rel="category tag">Blarghing</a></span></span></footer> <!-- .entry-footer -->
</article><!-- #post-1074 -->
<div id="comments" class="comments-area">
<div id="respond" class="comment-respond">
<h3 id="reply-title" class="comment-reply-title">Leave a Reply <small><a rel="nofollow" id="cancel-comment-reply-link" href="/subsequences-of-heights/#respond" style="display:none;">Cancel reply</a></small></h3><form action="https://marshalljiang.com/wp-comments-post.php" method="post" id="commentform" class="comment-form" novalidate><p class="comment-notes"><span id="email-notes">Your email address will not be published.</span> <span class="required-field-message">Required fields are marked <span class="required">*</span></span></p><p class="comment-form-comment"><label for="comment">Comment <span class="required">*</span></label> <textarea id="comment" name="comment" cols="45" rows="8" maxlength="65525" required></textarea></p><p class="comment-form-author"><label for="author">Name <span class="required">*</span></label> <input id="author" name="author" type="text" value="" size="30" maxlength="245" autocomplete="name" required /></p>
<p class="comment-form-email"><label for="email">Email <span class="required">*</span></label> <input id="email" name="email" type="email" value="" size="30" maxlength="100" aria-describedby="email-notes" autocomplete="email" required /></p>
<p class="comment-form-url"><label for="url">Website</label> <input id="url" name="url" type="url" value="" size="30" maxlength="200" autocomplete="url" /></p>
<p class="form-submit"><input name="submit" type="submit" id="submit" class="submit" value="Post Comment" /> <input type=)