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    <div id="titlebar">Graph Theory: Industrial Drilling</div>
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			Home &gt; Graph Theory: Industrial Drilling
					&gt; The Drilling Problem</div>
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        <a class="toc_link" href="page1.xml">The Drilling Problem</a>
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        <a class="toc_link" href="page4.xml">Objectives</a>
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        <a class="toc_link" href="">Foundations</a>
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        <a class="toc_link" href="page5.xml">What is a Graph?</a>
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        <a class="toc_link" href="page10.xml">A First Theorem</a>
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        <a class="toc_link" href="">Solving the Drilling Problem</a>
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    <div id="content"><h1>The Drilling Problem</h1>
			<h2>The History of The Solution</h2>
			<p>
			Originally, the task was given to a <b>junior engineer</b>, who designed a tool 
			that accomplished the task using 33 passes. The result was considered to be less than satisfactory.
			</p>
			<p>
			Afterward, a <b>more experienced engineer</b> came up with a better result: the new design 
			allowed the job to be done in 31 passes.
			</p>
			<p>
			Then, the <b>chief engineer</b> of the plant challenged all the engineers to find a better solution. 
			A <b>team of experienced engineers</b> won a prize offered by the chief engineer by finding a solution 
			requiring only 29 passes.
			</p>
			<h2>Graph Theory to the Rescue!</h2>
			<p>
			One of the engineers from the plant consulted a friend who was (and is) 
			a mathematician specializing in Graph Theory.
			The engineer showed the problem to his mathematician friend, 
			who immediately recognized it as a graph theory problem: 
			a special case of the famous <i>Vertex Coloring Problem</i>.
			</p>
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				<b><i>The mathematician did two things:</i></b>
				
				<ol><li>
					He developed a scheme which did the job in only 25 passes.
					</li><li>
					He proved that any scheme requires at least 25 passes.
					</li></ol>
				
			</div>
			<p>
			In addition, it turned out that the secondary goal also was achieved: 
			the number of different gang drills used for the proposed scheme was 3 -
			fewer than required by any previous design.
			</p>
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