Problems B: Octagons

<p></p> <h2>Problems B: Octagons</h2> <p>Below is a picture of an infinite hyperbolic tessellation of octagons. If we think of this as a graph of vertices (of degree three), then there exists an isomorphism of the graph which maps any vertex <i>x</i> onto any other vertex <i>y</i>. Every edge is given a label from the set <i>{a,b,c}</i> in such a way that every vertex has all three types of edges incident on it, and the labels alternate around each octagon. Part of this labeling is illustrated in the diagram.</p> <p><img src="http://tk.hustoj.com/admin/../upload/pimg28250_1.png" alt="" /></p> <p>So a path in this graph (starting from any vertex) can be specified by a sequence of edge labels. Your job is to write a program which, given a squence of labels such as &quot;abcbcbcabcaccabb&quot;, returns &quot;closed&quot; if the path ends on the same vertex where it starts, and returns &quot;open&quot; otherwise.</p>

<p>The input will begin with a number <i>Z</i> &le; 200 on a line by itself. This is followed by <i>Z</i> lines, each of which is a squence of length at least 1 and at most 40 of 'a's 'b's and 'c's.</p>

<p>For each input instance, the output will be the words &quot;closed&quot; or &quot;open&quot;, each on a single line.</p>