Subject: Re: The uncrossed hotwater tour<br/>
Date: Apr 23, 2002 @ 16:53<br/>
Author: granthutchison (&quot;granthutchison&quot; &lt;granthutchison@...&gt;)<br/>
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<div id="ygrps-yiv-1542752898">Michael:<br/>
<blockquote><span title="ireply"> &gt; & since this was probably the most difficult such maze ever found <br/>
&gt; anywhere in nature<br/>
&gt; you have practically proved that what i have been calling border <br/>
&gt; crosses are not necessarily border crosses at all<br/>
&gt; & tho you havent proved they dont cross<br/>
 </span></blockquote>Somewhere out there I imagine this simple situation of directed line <br/>
segments, quadripoints and an Euler tour has been addressed in the <br/>
maths literature, but I wouldn&#39;t know where to begin. Where is Martin <br/>
Gardner&#39;s or Ian Stewart&#39;s column in Scientific American when you <br/>
need it? Ian Stewart in particular might have picked this up and run <br/>
with it.<br/>
What we can say, though, is that a *fully crossed* hotwater tour is <br/>
impossible - the tiny dangling exclave attached to a larger clave by <br/>
two quadripoints prevents it, since the only fully crossed walk <br/>
around this would require two trips - one crosswise and one closed <br/>
loop. <br/>
A fully crossed tour was always going to be harder, since you only <br/>
have *one* way out of any given quadripoint (straight ahead), whereas <br/>
I&#39;m allowed two (left or right) and can therefore better avoid being <br/>
forced back to my starting point or creating detached, untravelled <br/>
loops (which are the only two rules of Fleury&#39;s algorithm).<br/>
 <br/>
<blockquote><span title="ireply"> &gt; so er what would you term these<br/>
 </span></blockquote>Are they not just bipartite quadripoints?<br/>
<br/>
Grant</div>