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t2/MathVentures/disco_circle.html.wml

`-<HTML>`
`-<TITLE>Math-Ventures: A Solidarian Disco Circle</TITLE>`
`+#include '../template.wml'`
` `
`-<BODY BGCOLOR="#FFFFFF">`
`+<subject "A Solidarian Disco Circle" />`
` `
`-<CENTER><H2>A Solidarian Disco Circle</H2></CENTER>`
`-`
`+<p>`
` Are you familiar with disco circles? I participated in quite a lot of them`
` when in the parties I attended. The concept is that a group of boys and`
` girls are standing in a circle with pop music playing in the background. A`
` after a while the boy walks-dances out of the center, while the girl invites`
` a different boy to dance with her. After a while longer, the girl leaves,`
` allowing the new boy to invite a new female partner. Again, the boy leaves,`
`-and so on.<P>`
`+and so on.`
`+</p>`
` `
`-I eventually came to wonder about this question regarding disco circles:<P>`
`+<p>`
`+I eventually came to wonder about this question regarding disco circles:`
`+</p>`
` `
`-<TABLE BORDER=1>`
`-<TR><TD>`
`-<B>1.</B> When can a group of dancers form a disco circle, such that every`
`+<table border="1">`
`+<tr><td>`
`+<ol>`
`+<li>When can a group of dancers form a disco circle, such that every`
` girl dances with every boy once and only once?`
`-<P>`
`-`
`-<B>2.</B> In case this feat cannot be achieved, what possible times can`
`+</li>`
`+<li>`
`+In case this feat cannot be achieved, what possible times can`
` every pair dance, so that all boy-girl combinations will dance the same`
` number of times together?`
`+</li>`
`+</ol>`
`+</td></tr>`
`+</table>`
` `
`-</TR></TD>`
`-</TABLE>`
`+<p>`
`+The solution can be found some space below:`
`+</p>`
` `
`-The solution can be found some space below:`
`-<BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR>`
`-<BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR>`
`-<BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR>`
`-<BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR><BR>`
`+<longblank />`
` `
`-<H3>Solution:</H3>`
`+<h2>Solution:</h2>`
` `
`+<p>`
` The secret to solving this problem lies in representing it in a proper`
` model. Let's take a piece of paper (or an empty pad in a drawing program)`
` and draw one circle for every boy and one for every girl. We can mark the`
` boys' circles with "B1", "B2", "B3" and so on and the girls' with "G1",`
`-"G2".<P>`
`+"G2".`
`+</p>`
` `
`+<p>`
` Now, let's suppose a girl invited a boy to dance with her in the middle of`
` the disco circle. After a short while, the girl would leave and the boy`
` would be able to invite a girl to join him in the center. If we keep tracing`
` who is the person in the middle who can (now or in a short while) invite a`
` member of the opposite sex to join him, we can see that we move from a boy`
`-to a girl, and then to another boy, another girl, and so on.<P>`
`+to a girl, and then to another boy, another girl, and so on.`
`+</p>`
` `
`+<p>`
` Imagine we put the pencil inside the circle of the kid that is in the`
` middle , and when he is replaced by the one he invited, we move the`
` pencil towards the circle of the new kid, thus drawing a line between`
` the two circles. Assuming that a dancing routine in which every pair`
` danced once can be achieved, we'll end up with a drawing in which every`
`-boy is connected to every girl (and vice-versa).<P>`
`+boy is connected to every girl (and vice-versa).`
`+</p>`
` `
`+<p>`
` If we can traverse this drawing, so that every link is drawn once,`
` and the pencil is not lifted off the paper, then we found a dancing`
` order that satistfies the request. In mathematical terminology a`
` drawing with lines and circles is commonly called a <B>"Graph"</B>, and`
` the field of mathemtics that deals with graphs is called <B>Graph`
`-Theory</B>.<P>`
`+Theory</B>.`
`+</p>`
` `
`+<p>`
` A very famous theorem in Graph Theory, called Euler's Circle Law,`
` proves that a graph can be drawn in one draw if and only if the`
`-following condition is fullfilled:<BR>`
`+following condition is fullfilled:<br />`
` Every circle is connected by an even number of links or there are only`
`-two circles that are connected by an uneven number of links.<P>`
`+two circles that are connected by an uneven number of links.`
`+</p>`
` `
`+<p>`
` In our case, it's obvious that the number of links for every girl is`
` equal to the number of boys and vice versa. Thus, to have an even`
` number of links that emerge from every circle, we will need to have an`
` even number of boys <B>and</B> an even number of girls. Plus, if there`
` are two boys and an uneven number of girls, we'll still get a graph`
` with only two circles having an uneven number of links. (and likewise`
`-for two girls and an uneven number of boys).<P>`
`+for two girls and an uneven number of boys).`
`+</p>`
` `
`+<p>`
` Yet another option is the trivial case in which there is one boy and one`
` girl. The reason that it is not included in the other sub-cases, is because`
`-that way the two nodes are all the existing nodes in the graph.<p>`
`+that way the two nodes are all the existing nodes in the graph.`
`+</p>`
` `
`+<p>`
` To sum up, a disco circled in which every pair of boy and girl dances`
`-together once can be formed on one of the following conditions:<P>`
`+together once can be formed on one of the following conditions:`
`+</p>`
` `
`-1. There are 2 boys present.<BR>`
`-2. There are 2 girls present.<BR>`
`-3. There is an even number of boys and an even number of girls who wish to`
`-pariticipate in the circle.<BR>`
`-4. There is exactly one boy and exactly one girl.<br>`
`+<ol>`
`+<li>There are 2 boys present.</li>`
`+<li>There are 2 girls present.</li>`
`+<li>There is an even number of boys and an even number of girls who wish to`
`+pariticipate in the circle.</li>`
`+<li>There is exactly one boy and exactly one girl.</li>`
`+</ol>`
` `
`-<P>`
`+<hr />`
` `
`-`
`-<HR>`
`-`
`-<A HREF="./"><H3>Back to the Math-Ventures Web page</H3></A><P>`
`-<A HREF="../"><H3>Back to Shlomi Fish' Homepage</H3></A>`
`+<h3><a href="./">Back to the Math-Ventures web page</a></h3>`
` `
` `
` </BODY>`

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t2/old-news.html

`-<html>`
`-<head>`
`-<title>Shlomi Fish' Homepage Old News Items</title>`
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`-`
`-<h1>Shlomi Fish' Homepage Old News Items</h1>`
`-`
`-<h3>15-May-2002</h3>`
`-`
`-<p>`
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t2/old-news.html.wml

`+#include '../template.wml'`
`+`
`+<subject "Shlomi Fish' Homepage Old News Items" />`
`+`
`+<h3>15-May-2002</h3>`
`+`
`+<p>`
`+I set up a `
`+<a href="http://groups.yahoo.com/group/shlomif-stories/">mailing-list</a> for `
`+discussing my stories on Yahoo-Groups. Feel free to join it or post messages`
`+there. (one does not need to be a subscriber to post).`
`+</p>`
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