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Shlomi Fish committed a02d6a6

Remove most trailing space from the t2 directory.

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t2/DeCSS/JavaScript/decss_js.html

 <h2>Congratulations!</h2>
 
 <p>
-You have violated the DMCA and have a working DeCSS code inside your 
+You have violated the DMCA and have a working DeCSS code inside your
 computer’s memory. To run it (and <b>really</b> violate the DMCA) follow the
 following instructions:
 </p>
     var the_string_extracted = the_string.substring(0,2048);
     var the_string_decoded = decode_a_string_with_escape_sequences(the_string_extracted);
     var the_string_scrambled = CSSdescramble(the_string_decoded, key_decoded);
-    
-    
-    document.decss_js_form.myoutput.value = 
+
+
+    document.decss_js_form.myoutput.value =
         encode_string_with_escape_sequences(
             the_string_scrambled,
             false
 Another nail in the DMCA coffin.
 </li>
 <li>
-A hack that allows you to induce innocent web-surfers into violating the 
+A hack that allows you to induce innocent web-surfers into violating the
 DMCA on their home computer.
 </li>
 </ol>
 <h2>About Us</h2>
 
 <p>
-<b>Shlomi Fish</b> - Shlomi Fish who did most of the conversion from C to 
-JavaScript, considers himself a professional programmer and a hobbyist 
-mathematician, writer and philosopher. He was born in Israel in 1977, and 
+<b>Shlomi Fish</b> - Shlomi Fish who did most of the conversion from C to
+JavaScript, considers himself a professional programmer and a hobbyist
+mathematician, writer and philosopher. He was born in Israel in 1977, and
 has lived there for the majority of his life.
 </p>
 
 <p>
-At the moment, he is a student at the Electrical Engineering Department of 
-<a href="http://www.technion.ac.il/">the Technion</a>. 
+At the moment, he is a student at the Electrical Engineering Department of
+<a href="http://www.technion.ac.il/">the Technion</a>.
 </p>
 
 <p>
 <b>Chen Shapira</b> - a renowed Israeli hackeress and the founder of
-Hackers-IL. She came up with the idea for a JavaScript implementation of 
+Hackers-IL. She came up with the idea for a JavaScript implementation of
 DeCSS and gave some useful advice about JavaScript.
 
 </p>
 
 <p>
 Hackers-IL is a mailing-list dedicated to the discussion of various topics
-that are related to computers. (with a very broad definition of “related”). 
-The majority of the members of Hackers-IL are professional IT workers and 
+that are related to computers. (with a very broad definition of “related”).
+The majority of the members of Hackers-IL are professional IT workers and
 wannabe philosophers.
 </p>
 
 <p>
-Most of the Hackers-IL are either Israelis, or people who are affiliated with 
+Most of the Hackers-IL are either Israelis, or people who are affiliated with
 Israel in some way. You are welcome to join the mailing list if you enjoy
-hacking, and are tolerable of some various Israelist idioms. 
+hacking, and are tolerable of some various Israelist idioms.
 </p>
 
 <p>
-Even those who are not members may check out the 
+Even those who are not members may check out the
 <a href="http://groups.yahoo.com/group/hackers-il/">
 homepage of the mailing-list
-</a> at <a href="http://groups.yahoo.com/">Yahoo! Groups</a> and enjoy the 
+</a> at <a href="http://groups.yahoo.com/">Yahoo! Groups</a> and enjoy the
 nonsense that we discuss.
 </p>
 
 </p>
 
 <p>
-DeCSS was the topic of discussion in some threads before, and in 
+DeCSS was the topic of discussion in some threads before, and in
 <a href="http://groups.yahoo.com/group/hackers-il/message/910">one message</a>,
-it was Chen’s suggestion that it should be written in JavaScript. 
-It was mostly ignored, until Shlomi Fish realized it could actually be 
+it was Chen’s suggestion that it should be written in JavaScript.
+It was mostly ignored, until Shlomi Fish realized it could actually be
 a good idea™ to do so, and he started working on this hack.
 </p>
 

t2/DeCSS/JavaScript/decss_js.js

 //
 // Fake DeCSS for JavaScript - A hack of Hackers-IL
 // version 0.2.2
-// 
+//
 // This code descrypts according to Julius Caesar's method of adding 3
 // to the character code.
-// 
+//
 
 
 //
     var a;
     var ret = "";
     var c_code
-    
+
     for (a = 0; a < sec.length ; a++)
     {
     	c_code = sec.charCodeAt(a);
     	ret += String.fromCharCode((c_code-3)%256);
     }
-    
+
     return ret;
 }
 
     var len = str.length;
     var ret = "";
     var c, c_code;
-   
+
     for( a = 0; a < len ; a++ )
     {
         c = str.charAt(a);
                 ret += c;
             }
         }
-        
+
         if (! on_one_line)
         {
             // Add a newline every 20 characters
 
         if (c != "\\")
         {
-            ret += c;        
+            ret += c;
         }
         else
         {

t2/DeCSS/JavaScript/index.html

 <h1>DeCSS for JavaScript Embedding HTML</h1>
 
 <p>
-On this page: <a href="decss_js.html"><tt>decss_js.html</tt></a>, you can find 
+On this page: <a href="decss_js.html"><tt>decss_js.html</tt></a>, you can find
 an HTML file that powers the DeCSS for JavaScript source code. Note that its
 decryption backend is _not_ DeCSS.
 </p>
 </p>
 
 <p>
-To activate the HTML with the real DeCSS backend place both it and the 
+To activate the HTML with the real DeCSS backend place both it and the
 <b>real</b> <tt>decss_js.js</tt> file in the same directory, assuming you know
 what you are doing. I am not responsible for the results.
 </p>

t2/DeCSS/index.html.wml

 <p>
 I removed the offending files from here. The reason for that is that
 I don’t want to drag the Technion’s undergraduate students’ server into the
-MPAA witch-hunt. 
+MPAA witch-hunt.
 </p>
 
 <p>
 What I did prepare is a port of DeCSS to JavaScript. You can’t find it here,
-but it can be found in the 
+but it can be found in the
 <a href="http://www.cs.cmu.edu/~dst/DeCSS/Gallery/">
 DeCSS Gallery</a>.
 </p>
 <p>
 Well, to show my empathy with the DeCSS side of the MPAA vs. DeCSS hackers
 fiasco that is going on, I decided to set up this page which will explain my
-take on this subject.  No, I am not an anarchist or a nihilist or anything like 
-that. I am a  <a href="http://www.neo-tech.com/">Neo-Tech</a> Objectivist. 
+take on this subject.  No, I am not an anarchist or a nihilist or anything like
+that. I am a  <a href="http://www.neo-tech.com/">Neo-Tech</a> Objectivist.
 Still, I believe that objective ethics is not on the side of the MPAA in this
 regard. And following are my reasons:
 </p>
 
-<p>1. <b>Code is Speech</b> - computer source code can do many things that 
+<p>1. <b>Code is Speech</b> - computer source code can do many things that
 speech alone previously could not do, but it still speech and is protected
 by the First Amendment of the American Constitution and objective ethics in
 general. And the “Digital Millennium Copyright Act” or any other law that
 jumps from the head of the American Government cannot change that fact.</p>
 
-<p>2. <b>No code is illegitimate</b> - whether it was acquired by reverse 
+<p>2. <b>No code is illegitimate</b> - whether it was acquired by reverse
 engineering or not, code cannot be made illegitimate or illegal.</p>
 
 <p>3. <b>Links are speech</b> - the fact that links are convenient does not
 categorize them as weapons or anything like that. They are speech because
-they are made of ASCII characters and that all there is to it. Assuming 
+they are made of ASCII characters and that all there is to it. Assuming
 the DMCA, now my homepage as well as everything in it, becomes something
 you are not allowed to link to. So if you are scared of the “Big Brother”,
 don’t. ;-)</p>
 In a couple of years, DVD copiers will become commonplace, so people will be
 able to copy them without opening the encryption.</p>
 
-<p>Perhaps the point of the fiasco is to charge a fee for those players 
+<p>Perhaps the point of the fiasco is to charge a fee for those players
 which can play it. But that has nothing to do with protecting intellectual
 property rights. (!)</p>
 
 
 <p>
 I believe it is perfectly legitimate to film such motion pictures. However,
-they should understand that if they call for the government’s help to censor 
+they should understand that if they call for the government’s help to censor
 code, web-sites and links, they in a fact giving it the power and legitimacy
 to censor them. IMO, they above all should hold the liberty of speech and
 it’s a shame they don’t.

t2/Kfar-Saba-Site-Email-Exchange.txt

 
 > אנו מודים לך על פנייתך,
 >
-> באתר כתוב:  האתר מומלץ לצפייה באמצעות דפדפן Internet Explorer 5.5 
+> באתר כתוב:  האתר מומלץ לצפייה באמצעות דפדפן Internet Explorer 5.5
 > <http://www.microsoft.com/downloads/details.aspx?FamilyID=1e1550cb-5e5d-48f
->5-b02b-20b602228de6&DisplayLang=he>   עם קישור להורדה/שידרוג  התוכנה (חינם) 
+>5-b02b-20b602228de6&DisplayLang=he>   עם קישור להורדה/שידרוג  התוכנה (חינם)
 > - למי שאין.
 >
 
-ראשית אני חייב לציין שכאשר אני מסתכל בדף הראשי בעזרת דפדפן מוזילה (או דפדפן 
-Konqueror(, איני רואה הודעה זאת. בנוסף, גם אילו הייתי רואה אותה, היא לא הייתה 
-עוזרת לי. אינני יכול להריץ את הדפדפן על מערכת ההפעלה לינוקס איתה אני עובד, 
-משום שהוא נגיש רק למערכות Win32. גם אילו הייתי משתמש ב-Windows, אני מעדיף שלא 
-להשתמש בו, משום שהוא מלא בחורי אבטחה, שאינם מתוקנים מהר, ולכן מהווה סכנה 
+ראשית אני חייב לציין שכאשר אני מסתכל בדף הראשי בעזרת דפדפן מוזילה (או דפדפן
+Konqueror(, איני רואה הודעה זאת. בנוסף, גם אילו הייתי רואה אותה, היא לא הייתה
+עוזרת לי. אינני יכול להריץ את הדפדפן על מערכת ההפעלה לינוקס איתה אני עובד,
+משום שהוא נגיש רק למערכות Win32. גם אילו הייתי משתמש ב-Windows, אני מעדיף שלא
+להשתמש בו, משום שהוא מלא בחורי אבטחה, שאינם מתוקנים מהר, ולכן מהווה סכנה
 למערכת:
 
 http://shlomif.il.eu.org/no-ie/
 > דפדפן מוזילה אפילו לא מוזכר ואולי כלול ב- "אחרים" (0.16%)
 >
 
-עשויות להיות לכך מספר סיבות. ייתכן שגולשים המגיעים לדף הראשי בעזרת מוזילה, 
-מתייאשים משום שהם רואים שהאתר לא מתפקד, ואז עוברים לגלוש באקספלורר בשאר הדפים 
-ובפעמים הבאות (מה שמגדיל את התפוסה). בכל מקרה, חוסר תמיכת האתר במוזילה לא 
-עוזר לשפר סטטיסטיקה זאת. בנוסף, חוסר התמיכה של האתר מפלה נגד חלק גדול 
-מהאוכלוסיה שאינה רוצה או יכולה להשתמש במערכת חלונות או שמודעת לבעיות האבטחה 
+עשויות להיות לכך מספר סיבות. ייתכן שגולשים המגיעים לדף הראשי בעזרת מוזילה,
+מתייאשים משום שהם רואים שהאתר לא מתפקד, ואז עוברים לגלוש באקספלורר בשאר הדפים
+ובפעמים הבאות (מה שמגדיל את התפוסה). בכל מקרה, חוסר תמיכת האתר במוזילה לא
+עוזר לשפר סטטיסטיקה זאת. בנוסף, חוסר התמיכה של האתר מפלה נגד חלק גדול
+מהאוכלוסיה שאינה רוצה או יכולה להשתמש במערכת חלונות או שמודעת לבעיות האבטחה
 של הדפדפן. יהיה האחוז קטן ככל שיהיה, זאת עדיין אפליה.
 
 >
 > לכתובות כאלה.
 >
 
-ניתן לסנן את דואר הזבל באמצעות תוכנות מסננות כמו SpamAssassin או 
-SpamBayes. )אצלי זה עובד מצויין). בכל מקרה זו לא סיבה להקשות על הגולשים שפשוט 
-רוצים לבוא אליכם במגע, ואינם יכולים לגשת לטופס באתר. (אני הייתי יכול רק 
+ניתן לסנן את דואר הזבל באמצעות תוכנות מסננות כמו SpamAssassin או
+SpamBayes. )אצלי זה עובד מצויין). בכל מקרה זו לא סיבה להקשות על הגולשים שפשוט
+רוצים לבוא אליכם במגע, ואינם יכולים לגשת לטופס באתר. (אני הייתי יכול רק
 כשהתשמשתי בדפדפן אקספלורר).
 
 > בכניסה לאתר, בסרגל העליון, יש כפתור בשם "צור קשר"  אשר מחובר לטופס פנייה
 > אנא חזור שוב להתעדכן באתר.
 >
 
-אין לי כוונה לעשות זאת, עד שהאתר יתוקן להיות בנוי לפי התקנים ותומך בכל 
+אין לי כוונה לעשות זאת, עד שהאתר יתוקן להיות בנוי לפי התקנים ותומך בכל
 הדפדפנים המודרנים.
 
 בכבוד רב,
 >
 > מין: זכר
 
--- 
+--
 
 ---------------------------------------------------------------------
 Shlomi Fish      shlomif@iglu.org.il

t2/MathVentures/3d-outof-4d-mathml.xhtml.wml

 #include "prelude.wml"
 <define-tag latemp_html_doctype>
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0//EN" 
-"http://www.w3.org/Math/DTD/mathml2/xhtml-math11-f.dtd" > 
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0//EN"
+"http://www.w3.org/Math/DTD/mathml2/xhtml-math11-f.dtd" >
 </define-tag>
 #include "driver.wml"
 
 <!--l. 15--><p class="noindent">Like I said, it did not come to me right away and I had to think about it for a while. I pondered various
 methods, and then came to think about the question this way:
 </p><!--l. 19--><p class="noindent">In each throw the numbers 1..6 can be substracted. I &#xFB01;rst realized that I could immediately substract 1 from all
-<!--l. 20--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><msup><mrow 
-><mn>6</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
+<!--l. 20--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><msup><mrow
+><mn>6</mn></mrow><mrow
+><mn>4</mn></mrow></msup
 ></math> of the
 possible throws, because one always substracts <b>at least</b> 1. Furthermore, when is another 1 point substracted from
 the &#xFB01;nal sum? Obviously this is the case when 2 or more points are substracted. When does it happen? It happens
 when all the dice are bigger than 1. (or else 1 would have been substracted.) Since all the dice are in the range 2-6
-there are <!--l. 25--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><msup><mrow 
-><mn>5</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
+there are <!--l. 25--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><msup><mrow
+><mn>5</mn></mrow><mrow
+><mn>4</mn></mrow></msup
 ></math>
 di&#xFB00;erent possibilities for this case.
 </p><!--l. 28--><p class="noindent">The next point is substracted when all the dice are in the range 3-6, hence
-<!--l. 29--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><msup><mrow 
-><mn>4</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
+<!--l. 29--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><msup><mrow
+><mn>4</mn></mrow><mrow
+><mn>4</mn></mrow></msup
 ></math>
 possibilities and so on. Now, to calculate the total, let&#x2019;s start from the sum of all
 possible throws of 4 dice, without the minimal die removed. This sum is equal to
-<!--l. 31--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><msup><mrow 
-><mn>6</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
-> <mo 
-class="MathClass-punc">&#x22C5;</mo> <mn>1</mn><mn>4</mn> <mo 
+<!--l. 31--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><msup><mrow
+><mn>6</mn></mrow><mrow
+><mn>4</mn></mrow></msup
+> <mo
+class="MathClass-punc">&#x22C5;</mo> <mn>1</mn><mn>4</mn> <mo
 class="MathClass-rel">=</mo> <mn>1</mn><mn>8</mn><mn>1</mn><mn>4</mn><mn>4</mn></math>. Then, let&#x2019;s remove
-<!--l. 31--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><msup><mrow 
-><mn>6</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
-> <mo 
+<!--l. 31--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><msup><mrow
+><mn>6</mn></mrow><mrow
+><mn>4</mn></mrow></msup
+> <mo
 class="MathClass-rel">=</mo> <mn>1</mn><mn>2</mn><mn>9</mn><mn>6</mn></math> because of the &#xFB01;rst
-point removed, <!--l. 32--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><msup><mrow 
-><mn>5</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
+point removed, <!--l. 32--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><msup><mrow
+><mn>5</mn></mrow><mrow
+><mn>4</mn></mrow></msup
 ></math>
 because of the second point etc. Eventually we get:
-</p><!--l. 35--><p class="noindent"><!--l. 35--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><msup><mrow 
-><mn>6</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
-> <mo 
-class="MathClass-bin">&#x2217;</mo> <mn>1</mn><mn>4</mn> <mo 
-class="MathClass-bin">&#x2212;</mo> <msup><mrow 
-><mn>6</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
-> <mo 
-class="MathClass-bin">&#x2212;</mo> <msup><mrow 
-><mn>5</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
-> <mo 
-class="MathClass-bin">&#x2212;</mo> <msup><mrow 
-><mn>4</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
-> <mo 
-class="MathClass-bin">&#x2212;</mo> <msup><mrow 
-><mn>3</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
-> <mo 
-class="MathClass-bin">&#x2212;</mo> <msup><mrow 
-><mn>2</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
-> <mo 
-class="MathClass-bin">&#x2212;</mo> <msup><mrow 
-><mn>1</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
-> <mo 
+</p><!--l. 35--><p class="noindent"><!--l. 35--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><msup><mrow
+><mn>6</mn></mrow><mrow
+><mn>4</mn></mrow></msup
+> <mo
+class="MathClass-bin">&#x2217;</mo> <mn>1</mn><mn>4</mn> <mo
+class="MathClass-bin">&#x2212;</mo> <msup><mrow
+><mn>6</mn></mrow><mrow
+><mn>4</mn></mrow></msup
+> <mo
+class="MathClass-bin">&#x2212;</mo> <msup><mrow
+><mn>5</mn></mrow><mrow
+><mn>4</mn></mrow></msup
+> <mo
+class="MathClass-bin">&#x2212;</mo> <msup><mrow
+><mn>4</mn></mrow><mrow
+><mn>4</mn></mrow></msup
+> <mo
+class="MathClass-bin">&#x2212;</mo> <msup><mrow
+><mn>3</mn></mrow><mrow
+><mn>4</mn></mrow></msup
+> <mo
+class="MathClass-bin">&#x2212;</mo> <msup><mrow
+><mn>2</mn></mrow><mrow
+><mn>4</mn></mrow></msup
+> <mo
+class="MathClass-bin">&#x2212;</mo> <msup><mrow
+><mn>1</mn></mrow><mrow
+><mn>4</mn></mrow></msup
+> <mo
 class="MathClass-rel">=</mo> <mn>1</mn><mn>5</mn><mn>8</mn><mn>6</mn><mn>9</mn></math>
 </p><!--l. 37--><p class="noindent">To &#xFB01;nd the average, all one has to do is divide it by
-<!--l. 37--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><msup><mrow 
-><mn>6</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
+<!--l. 37--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><msup><mrow
+><mn>6</mn></mrow><mrow
+><mn>4</mn></mrow></msup
 ></math>, which is the number
-of individual throws. <!--l. 38--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><mfrac><mrow 
+of individual throws. <!--l. 38--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><mfrac><mrow
 ><mn>1</mn><mn>5</mn><mn>8</mn><mn>6</mn><mn>9</mn></mrow>
-  <mrow 
-><msup><mrow 
-><mn>6</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
-></mrow></mfrac>    <mo 
-class="MathClass-rel">=</mo> <mn>1</mn><mn>2</mn><mo 
+  <mrow
+><msup><mrow
+><mn>6</mn></mrow><mrow
+><mn>4</mn></mrow></msup
+></mrow></mfrac>    <mo
+class="MathClass-rel">=</mo> <mn>1</mn><mn>2</mn><mo
 class="MathClass-punc">.</mo><mn>2</mn><mn>4</mn></math>,
 which is also the number that my friend&#x2019;s program returned.
 </p><!--l. 41--><p class="noindent">To generalise it for n dice each having the numbers 1..m on its side (AD&#x0026;D) also involves
 using dice with 4, 8, 10,12, 20 and sometimes 100 sides. :-)) all one has to do is replace
-<!--l. 43--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
+<!--l. 43--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
 display="inline" ><mn>6</mn></math> with
-<!--l. 43--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><mi 
+<!--l. 43--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><mi
 >n</mi></math> and
-<!--l. 43--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
+<!--l. 43--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
 display="inline" ><mn>4</mn></math> with
-<!--l. 43--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><mi 
+<!--l. 43--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><mi
 >m</mi></math>
 where appropriate. One should remember that the average throw for an individual die of 1..m is
-<!--l. 45--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><mfrac><mrow><mn>1</mn> <mo 
-class="MathClass-bin">+</mo> <mi 
+<!--l. 45--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><mfrac><mrow><mn>1</mn> <mo
+class="MathClass-bin">+</mo> <mi
 >m</mi></mrow>
-  <mrow><mi 
+  <mrow><mi
 >m</mi></mrow></mfrac>  </math> and for
-n such dice <!--l. 46--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><mfrac><mrow><mi 
->n</mi> <mfenced separators="" 
-open="("  close=")" ><mrow><mn>1</mn> <mo 
-class="MathClass-bin">+</mo> <mi 
+n such dice <!--l. 46--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><mfrac><mrow><mi
+>n</mi> <mfenced separators=""
+open="("  close=")" ><mrow><mn>1</mn> <mo
+class="MathClass-bin">+</mo> <mi
 >m</mi></mrow></mfenced></mrow>
-   <mrow><mi 
+   <mrow><mi
 >m</mi></mrow></mfrac>    </math>.
 We eventually get to:
-</p><!--l. 48--><p class="noindent"><!--l. 48--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><mfrac><mrow 
-><mfenced separators="" 
-open="("  close=")" ><mrow><mfrac><mrow 
-><msup><mrow 
-><mi 
->m</mi></mrow><mrow 
-><mi 
->n</mi></mrow></msup 
-><mo 
-class="MathClass-punc">&#x22C5;</mo><mi 
->n</mi><mo 
-class="MathClass-punc">&#x22C5;</mo><mfenced separators="" 
-open="("  close=")" ><mrow><mn>1</mn><mo 
-class="MathClass-bin">+</mo><mi 
+</p><!--l. 48--><p class="noindent"><!--l. 48--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><mfrac><mrow
+><mfenced separators=""
+open="("  close=")" ><mrow><mfrac><mrow
+><msup><mrow
+><mi
+>m</mi></mrow><mrow
+><mi
+>n</mi></mrow></msup
+><mo
+class="MathClass-punc">&#x22C5;</mo><mi
+>n</mi><mo
+class="MathClass-punc">&#x22C5;</mo><mfenced separators=""
+open="("  close=")" ><mrow><mn>1</mn><mo
+class="MathClass-bin">+</mo><mi
 >m</mi></mrow></mfenced></mrow>
-     <mrow 
-><mi 
->m</mi></mrow></mfrac>   </mrow></mfenced><mo 
-class="MathClass-bin">&#x2212;</mo><msup><mrow 
-><mi 
->m</mi></mrow><mrow 
-><mi 
->n</mi></mrow></msup 
-><mo 
-class="MathClass-bin">&#x2212;</mo><msup><mrow 
-><mfenced separators="" 
-open="("  close=")" ><mrow><mi 
->m</mi><mo 
-class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></mfenced></mrow><mrow 
-><mi 
->n</mi></mrow></msup 
-><mo 
-class="MathClass-bin">&#x2212;</mo><msup><mrow 
-><mfenced separators="" 
-open="("  close=")" ><mrow><mi 
->m</mi><mo 
-class="MathClass-bin">&#x2212;</mo><mn>2</mn></mrow></mfenced></mrow><mrow 
-><mi 
->n</mi></mrow></msup 
-><mo 
-class="MathClass-bin">&#x2212;</mo><msup><mrow 
-><mfenced separators="" 
-open="("  close=")" ><mrow><mi 
->m</mi><mo 
-class="MathClass-bin">&#x2212;</mo><mn>3</mn></mrow></mfenced></mrow><mrow 
-><mi 
->n</mi></mrow></msup 
-><mo 
-class="MathClass-punc">.</mo><mo 
-class="MathClass-punc">.</mo><mo 
-class="MathClass-punc">.</mo><mo 
-class="MathClass-bin">&#x2212;</mo><msup><mrow 
-><mn>1</mn></mrow><mrow 
-><mi 
->n</mi></mrow></msup 
+     <mrow
+><mi
+>m</mi></mrow></mfrac>   </mrow></mfenced><mo
+class="MathClass-bin">&#x2212;</mo><msup><mrow
+><mi
+>m</mi></mrow><mrow
+><mi
+>n</mi></mrow></msup
+><mo
+class="MathClass-bin">&#x2212;</mo><msup><mrow
+><mfenced separators=""
+open="("  close=")" ><mrow><mi
+>m</mi><mo
+class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></mfenced></mrow><mrow
+><mi
+>n</mi></mrow></msup
+><mo
+class="MathClass-bin">&#x2212;</mo><msup><mrow
+><mfenced separators=""
+open="("  close=")" ><mrow><mi
+>m</mi><mo
+class="MathClass-bin">&#x2212;</mo><mn>2</mn></mrow></mfenced></mrow><mrow
+><mi
+>n</mi></mrow></msup
+><mo
+class="MathClass-bin">&#x2212;</mo><msup><mrow
+><mfenced separators=""
+open="("  close=")" ><mrow><mi
+>m</mi><mo
+class="MathClass-bin">&#x2212;</mo><mn>3</mn></mrow></mfenced></mrow><mrow
+><mi
+>n</mi></mrow></msup
+><mo
+class="MathClass-punc">.</mo><mo
+class="MathClass-punc">.</mo><mo
+class="MathClass-punc">.</mo><mo
+class="MathClass-bin">&#x2212;</mo><msup><mrow
+><mn>1</mn></mrow><mrow
+><mi
+>n</mi></mrow></msup
 ></mrow>
 
-                           <mrow 
-><msup><mrow 
-><mi 
->m</mi></mrow><mrow 
-><mi 
->n</mi></mrow></msup 
+                           <mrow
+><msup><mrow
+><mi
+>m</mi></mrow><mrow
+><mi
+>n</mi></mrow></msup
 ></mrow></mfrac>                </math>
 </p><!--l. 53--><p class="noindent">Eliminating the next smallest die, and the other dice in order, is a bit more tricky. As a matter of fact, it took
 me two more years to &#xFB01;nally come up with a solution. (not that I spent all my time thinking about
 it)
 </p><!--l. 57--><p class="noindent">The basic idea is this: regarding the &#xFB01;rst point of the second least die, it&#x2019;s obvious that we still have to remove
-<!--l. 58--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><msup><mrow 
-><mn>6</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
+<!--l. 58--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><msup><mrow
+><mn>6</mn></mrow><mrow
+><mn>4</mn></mrow></msup
 ></math> from
 the total. About the second point: it is removed only if all the dice, except maybe one has a face value of 2 or
 more. Phrased in a di&#xFB00;erent way it is removed:
      </p><ol  class="enumerate1" >
      <li class="enumerate"
-><a 
+><a
  id="x1-3x1"></a>If there are 4 dice in the range 2..6.<b>Or</b>:
      </li>
      <li class="enumerate"
-><a 
+><a
  id="x1-5x2"></a>If there are 3 dice in the range 2..6 and one die whose value is 1.</li></ol>
 <!--l. 67--><p class="noindent">The conditions for the other 4 cases are similiar: if the additional point is the Kth than there could be either 4
 dice in the range K..6 or 3 dice in the range K..6 and one die in the range 1..(K-1). To evaluate it
 mathematically, I&#x2019;ll use the standard formulas and eventually get to:
 
-</p><!--l. 72--><p class="noindent"><!--l. 72--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><msup><mrow 
-> <mfenced separators="" 
-open="("  close=")" ><mrow><mn>6</mn> <mo 
-class="MathClass-bin">&#x2212;</mo> <mi 
->K</mi> <mo 
-class="MathClass-bin">+</mo> <mn>1</mn></mrow></mfenced></mrow><mrow 
-><mn>4</mn></mrow></msup 
-><mo 
-class="MathClass-bin">+</mo> <mfenced separators="" 
-open="("  close=")" ><mrow><msup><mrow 
-><mfenced separators="" 
-open="("  close=")" ><mrow><mn>6</mn> <mo 
-class="MathClass-bin">&#x2212;</mo> <mi 
->K</mi> <mo 
-class="MathClass-bin">+</mo> <mn>1</mn></mrow></mfenced></mrow><mrow 
-><mn>3</mn></mrow></msup 
-> <mo 
-class="MathClass-punc">&#x22C5;</mo><mfenced separators="" 
-open="("  close=")" ><mrow><mi 
->K</mi> <mo 
-class="MathClass-bin">&#x2212;</mo> <mn>1</mn></mrow></mfenced></mrow></mfenced> <mo 
+</p><!--l. 72--><p class="noindent"><!--l. 72--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><msup><mrow
+> <mfenced separators=""
+open="("  close=")" ><mrow><mn>6</mn> <mo
+class="MathClass-bin">&#x2212;</mo> <mi
+>K</mi> <mo
+class="MathClass-bin">+</mo> <mn>1</mn></mrow></mfenced></mrow><mrow
+><mn>4</mn></mrow></msup
+><mo
+class="MathClass-bin">+</mo> <mfenced separators=""
+open="("  close=")" ><mrow><msup><mrow
+><mfenced separators=""
+open="("  close=")" ><mrow><mn>6</mn> <mo
+class="MathClass-bin">&#x2212;</mo> <mi
+>K</mi> <mo
+class="MathClass-bin">+</mo> <mn>1</mn></mrow></mfenced></mrow><mrow
+><mn>3</mn></mrow></msup
+> <mo
+class="MathClass-punc">&#x22C5;</mo><mfenced separators=""
+open="("  close=")" ><mrow><mi
+>K</mi> <mo
+class="MathClass-bin">&#x2212;</mo> <mn>1</mn></mrow></mfenced></mrow></mfenced> <mo
 class="MathClass-punc">&#x22C5;</mo> <mn>4</mn></math>
-</p><!--l. 75--><p class="noindent">If we substract <!--l. 75--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><msup><mrow 
-><mn>6</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
+</p><!--l. 75--><p class="noindent">If we substract <!--l. 75--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><msup><mrow
+><mn>6</mn></mrow><mrow
+><mn>4</mn></mrow></msup
 ></math>
 and those 5 sums from the total sum that was acquired in the previous stage, we&#x2019;d get to the
 new total with the two least dice removed on each throw. Divide it by the number of throws -
-<!--l. 77--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><msup><mrow 
-><mn>6</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
+<!--l. 77--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><msup><mrow
+><mn>6</mn></mrow><mrow
+><mn>4</mn></mrow></msup
 ></math> - and
 we get the new average. The grand formula, then, is:
-</p><!--l. 80--><p class="noindent"><!--l. 80--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><mfrac><mrow 
-><msup><mrow 
-><mn>6</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
-><mo 
-class="MathClass-punc">&#x22C5;</mo><mn>1</mn><mn>4</mn><mo 
-class="MathClass-bin">&#x2212;</mo><munderover accentunder="false" accent="false"><mrow  
-><mo 
-class="MathClass-op">&#x2211;</mo></mrow><mrow 
+</p><!--l. 80--><p class="noindent"><!--l. 80--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><mfrac><mrow
+><msup><mrow
+><mn>6</mn></mrow><mrow
+><mn>4</mn></mrow></msup
+><mo
+class="MathClass-punc">&#x22C5;</mo><mn>1</mn><mn>4</mn><mo
+class="MathClass-bin">&#x2212;</mo><munderover accentunder="false" accent="false"><mrow
+><mo
+class="MathClass-op">&#x2211;</mo></mrow><mrow
 >
-<mi 
->i</mi><mo 
-class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
-><mn>6</mn></mrow></munderover 
-> <mfenced separators="" 
-open=""  close="" ><mrow><msup><mrow 
-><mi 
->i</mi></mrow><mrow 
-><mn>4</mn></mrow></msup 
-> </mrow></mfenced><mo 
-class="MathClass-bin">&#x2212;</mo><munderover accentunder="false" accent="false"><mrow  
-><mo 
-class="MathClass-op">&#x2211;</mo></mrow><mrow 
+<mi
+>i</mi><mo
+class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow
+><mn>6</mn></mrow></munderover
+> <mfenced separators=""
+open=""  close="" ><mrow><msup><mrow
+><mi
+>i</mi></mrow><mrow
+><mn>4</mn></mrow></msup
+> </mrow></mfenced><mo
+class="MathClass-bin">&#x2212;</mo><munderover accentunder="false" accent="false"><mrow
+><mo
+class="MathClass-op">&#x2211;</mo></mrow><mrow
 >
-<mi 
->i</mi><mo 
-class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
-><mn>6</mn></mrow></munderover 
-> <mfenced separators="" 
-open="["  close="]" ><mrow><msup><mrow 
-> <mfenced separators="" 
-open="("  close=")" ><mrow><mn>6</mn><mo 
-class="MathClass-bin">&#x2212;</mo><mi 
->i</mi><mo 
-class="MathClass-bin">+</mo><mn>1</mn></mrow></mfenced></mrow><mrow 
-><mn>4</mn></mrow></msup 
-><mo 
-class="MathClass-bin">+</mo><mfenced separators="" 
-open="("  close=")" ><mrow><msup><mrow 
-><mfenced separators="" 
-open="("  close=")" ><mrow><mn>6</mn><mo 
-class="MathClass-bin">&#x2212;</mo><mi 
->i</mi><mo 
-class="MathClass-bin">+</mo><mn>1</mn></mrow></mfenced></mrow><mrow 
-><mn>3</mn></mrow></msup 
-><mo 
-class="MathClass-punc">&#x22C5;</mo><mfenced separators="" 
-open="("  close=")" ><mrow><mi 
->i</mi><mo 
-class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mo 
+<mi
+>i</mi><mo
+class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow
+><mn>6</mn></mrow></munderover
+> <mfenced separators=""
+open="["  close="]" ><mrow><msup><mrow
+> <mfenced separators=""
+open="("  close=")" ><mrow><mn>6</mn><mo
+class="MathClass-bin">&#x2212;</mo><mi
+>i</mi><mo
+class="MathClass-bin">+</mo><mn>1</mn></mrow></mfenced></mrow><mrow
+><mn>4</mn></mrow></msup
+><mo
+class="MathClass-bin">+</mo><mfenced separators=""
+open="("  close=")" ><mrow><msup><mrow
+><mfenced separators=""
+open="("  close=")" ><mrow><mn>6</mn><mo
+class="MathClass-bin">&#x2212;</mo><mi
+>i</mi><mo
+class="MathClass-bin">+</mo><mn>1</mn></mrow></mfenced></mrow><mrow
+><mn>3</mn></mrow></msup
+><mo
+class="MathClass-punc">&#x22C5;</mo><mfenced separators=""
+open="("  close=")" ><mrow><mi
+>i</mi><mo
+class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mo
 class="MathClass-punc">&#x22C5;</mo><mn>4</mn></mrow></mfenced></mrow>
 
-                                 <mrow 
-><msup><mrow 
-><mn>6</mn></mrow><mrow 
-><mn>4</mn></mrow></msup 
+                                 <mrow
+><msup><mrow
+><mn>6</mn></mrow><mrow
+><mn>4</mn></mrow></msup
 ></mrow></mfrac>                 </math>
 </p><!--l. 86--><p class="noindent">From the 3rd least dice onward, this method gets more and more complicated because we have to consider 3
 di&#xFB00;erent cases for every point we substract, 4 cases for the fourth die, and so on. If anyone has an idea on how
 removed by order, please let me know and I&#x2019;ll post it here.
 </p><!--l. 93--><p class="noindent">I, for my part, am no longer interested in this problem, and do not deal with it, at least not until I extend my
 knowldege in Combinatorics and the Theory of Probability.
-</p><!--l. 97--><p class="noindent">A final note: you can find a <a href="calc_dice_average.c">C program</a>, not unlike the one my friend used, that 
-calculates the average of such throws . It can do that for any number of dice 
-with any number of sides, and while eliminating any number of minimal 
+</p><!--l. 97--><p class="noindent">A final note: you can find a <a href="calc_dice_average.c">C program</a>, not unlike the one my friend used, that
+calculates the average of such throws . It can do that for any number of dice
+with any number of sides, and while eliminating any number of minimal
 dice.
 </p><!--l. 104--><p class="noindent">The program is quite stupid and merely iterates over all the di&#xFB00;erent throws, and adds the sum of
 every throw to the total. It becomes less e&#xFB03;cient as the number of sides and/or the number of dice
 increases.
-</p><!--l. 109--><p class="noindent">Writing a program that implements my method <span 
+</p><!--l. 109--><p class="noindent">Writing a program that implements my method <span
 class="cmbx-10">for any number of removed dice </span>is a bit complicated
 because the formula itself gets more and more complicated. Maybe I&#x2019;ll get to it one day.
 </p>

t2/MathVentures/3d_outof_4d.html.wml

 </pre>
 
 <p>
-If we substract 6^4 and those 5 sums from the total sum that was acquired in 
+If we substract 6^4 and those 5 sums from the total sum that was acquired in
 the previous stage, we’d get to the new total with the two least dice
 removed on each throw. Divide it by the number of throws - 6^4 - and we get
 the new average. The grand formula, then, is:
 
 <pre>
           6            6
-6^4*14 - SIGMA(i^4) - SIGMA [(6-i+1)^4 + ( (6-i+1)^3 * (i-1)) * 4 ] 
+6^4*14 - SIGMA(i^4) - SIGMA [(6-i+1)^4 + ( (6-i+1)^3 * (i-1)) * 4 ]
          i=1          i=1
 -----------------------------------------------------------------------
-                    6^4              
+                    6^4
 </pre>
 
 
 substract, 4 cases for the fourth die, and so on. If anyone has an idea
 on how to improve this method, or can suggest an alternative method
 which is simpler as more and more dice are removed by order, please let
-me know and I’ll post it here. 
+me know and I’ll post it here.
 </p>
 
 <p>
 
 <p>
 A final note: you can find a C program, not unlike the one my friend used,
-that calculates the average of such throws 
+that calculates the average of such throws
 <a href="calc_dice_average.c">here</a>. It can do that for any number of
 dice with any number of sides, and while eliminating any number of minimal
-dice. 
+dice.
 </p>
 
 <p>

t2/MathVentures/bug_square.html.wml

 
 
 <pre>
-                                                       
-        p*a1             
-lim ---------------   =  
-p-&gt;0   ____________   
-    1-V 2*p^2-2*p+1      
+
+        p*a1
+lim ---------------   =
+p-&gt;0   ____________
+    1-V 2*p^2-2*p+1
 
 
                             _____________
 lim  ----------------  * -----------------   =
 p-&gt;0    ____________        _____________
      1-V 2*p^2-2*p+1     1+V 2*p^2-2*p+1
- 
 
- 
-                ____________     
-        a1*p*(1+V 2*p^2-2*p+1)    
-lim     ---------------------  =  
-p-&gt;0                        
+
+
+                ____________
+        a1*p*(1+V 2*p^2-2*p+1)
+lim     ---------------------  =
+p-&gt;0
          1 - 2*p^2 + 2*p - 1
 
 
                _____________
         a1 * (1+V 2*p^2-2*p+1 )
-lim     ----------------------  = 
+lim     ----------------------  =
 p-&gt;0       2 - 2*p
 
 
                  _____________
-        a1 * (1+V 2*0-2*0+1  )   
-lim     ---------------------  = 
-p-&gt;0      2 - 2*0           
+        a1 * (1+V 2*0-2*0+1  )
+lim     ---------------------  =
+p-&gt;0      2 - 2*0
 
 
 
-   a1 * (1+1)  
+   a1 * (1+1)
  ------------- = a1
-      2        
-          
+      2
+
 
 </pre>
 
 
 <p>
 Here’s <a href="bugs.scm">a script</a>
-for <a href="http://www.gimp.org/">Gimp</a> version 1.2 
-that generates a series of squares inside squares, and optionally marks the 
+for <a href="http://www.gimp.org/">Gimp</a> version 1.2
+that generates a series of squares inside squares, and optionally marks the
 path of a single bug. And here’s the <a href="bugs-2.2.scm">corresponding
 script for Gimp version 2.2</a>.
 </p>

t2/MathVentures/bugs-2.2.scm

 ;
 ; Copyright (C) 1999 Shlomi Fish
 ;
-; This file can be freely used, modified and distributed under the terms of 
+; This file can be freely used, modified and distributed under the terms of
 ; the MIT X11 license.
 ;
 ;
          (path-x 10)
          (path-y 10)
     )
-         
+
          (gimp-image-add-layer img bg-layer 1)
-     
+
          (gimp-context-set-background '(255 255 255))
          ;(gimp-edit-fill img bg-layer)
          (gimp-edit-fill bg-layer BG-IMAGE-FILL)
          (gimp-context-set-background old-bg)
-         
-        
+
+
          ; Set the ratio to a value between 0 and 1
          (if (< ratio 0)
              (set! ratio (* ratio -1))
          (if (or (= ratio 0) (= ratio 1))
              (set! ratio 0.1)
          )
-         
+
 
          (regular-brush)
 
             (*
                 (- (nth 3 coords) (nth 1 coords))
                 (- (nth 3 coords) (nth 1 coords))
-            )) 
+            ))
             3
          )
 
          )
 
          (gimp-context-set-brush old-brush)
-         
+
          (gimp-display-new img)
     )
 )
                     SF-TOGGLE "Mark Path?" TRUE
                     SF-VALUE "Ratio" "0.1"
                     )
-                    
+

t2/MathVentures/bugs-in-square-mathml.xhtml.wml

 #include "prelude.wml"
 <define-tag latemp_html_doctype>
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0//EN" 
-"http://www.w3.org/Math/DTD/mathml2/xhtml-math11-f.dtd" > 
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0//EN"
+"http://www.w3.org/Math/DTD/mathml2/xhtml-math11-f.dtd" >
 </define-tag>
 
 #include "driver.wml"
 
 <page_extra_head_elements>
-<link rel="stylesheet" type="text/css" href="bugs-in-square.css" /> 
+<link rel="stylesheet" type="text/css" href="bugs-in-square.css" />
 </page_extra_head_elements>
 
 <latemp_subject "Bugs in a Square (MathML Enabled Version)" />
 the initial distance between them. Then they move again, towards a position which is in proportion p to their
 distance, and so forth.
 </p><!--l. 22--><p class="noindent">Thus they form an in&#xFB01;nite series of squares inside each other. You can see an illustration of this scheme to the right of this text
-for the proportion <!--l. 24--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><mi 
->p</mi> <mo 
-class="MathClass-rel">=</mo> <mn>0</mn><mo 
+for the proportion <!--l. 24--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><mi
+>p</mi> <mo
+class="MathClass-rel">=</mo> <mn>0</mn><mo
 class="MathClass-punc">.</mo><mn>1</mn><mn>0</mn></math>.
-</p><!--l. 26--><p class="noindent">Now, if the length of a given square is <!--l. 26--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><mi 
+</p><!--l. 26--><p class="noindent">Now, if the length of a given square is <!--l. 26--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><mi
 >a</mi></math>,
 then the length of the square inside it is (according to pythgoras theorem):
 </p><!--l. 29--><p class="noindent">
-<!--tex4ht:inline--></p><!--l. 29--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
+<!--tex4ht:inline--></p><!--l. 29--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
 display="block" >
-<mtable 
+<mtable
 class="eqnarray-star" columnalign="right center left" >
-<mtr><mtd 
-class="eqnarray-1"> </mtd><mtd 
-class="eqnarray-2">   </mtd><mtd 
-class="eqnarray-3">   <msqrt><mrow><msup><mrow 
+<mtr><mtd
+class="eqnarray-1"> </mtd><mtd
+class="eqnarray-2">   </mtd><mtd
+class="eqnarray-3">   <msqrt><mrow><msup><mrow
 >
-<mfenced separators="" 
-open="("  close=")" ><mrow><mi 
->p</mi> <mo 
-class="MathClass-bin">&#x2217;</mo> <mi 
->a</mi></mrow></mfenced></mrow><mrow 
-><mn>2</mn></mrow></msup 
-> <mo 
-class="MathClass-bin">+</mo><msup><mrow 
-> <mfenced separators="" 
-open="("  close=")" ><mrow><mfenced separators="" 
-open="("  close=")" ><mrow><mn>1</mn> <mo 
-class="MathClass-bin">&#x2212;</mo> <mi 
->p</mi></mrow></mfenced> <mo 
-class="MathClass-bin">&#x2217;</mo> <mi 
->a</mi></mrow></mfenced></mrow><mrow 
-><mn>2</mn></mrow></msup 
-></mrow></msqrt> <mo 
-class="MathClass-rel">=</mo> <mi 
->a</mi><msqrt><mrow><mn>2</mn><msup><mrow 
-><mi 
->p</mi></mrow><mrow 
-><mn>2</mn> </mrow> </msup 
-> <mo 
-class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
->p</mi> <mo 
-class="MathClass-bin">+</mo> <mn>1</mn></mrow></msqrt></mtd><mtd 
+<mfenced separators=""
+open="("  close=")" ><mrow><mi
+>p</mi> <mo
+class="MathClass-bin">&#x2217;</mo> <mi
+>a</mi></mrow></mfenced></mrow><mrow
+><mn>2</mn></mrow></msup
+> <mo
+class="MathClass-bin">+</mo><msup><mrow
+> <mfenced separators=""
+open="("  close=")" ><mrow><mfenced separators=""
+open="("  close=")" ><mrow><mn>1</mn> <mo
+class="MathClass-bin">&#x2212;</mo> <mi
+>p</mi></mrow></mfenced> <mo
+class="MathClass-bin">&#x2217;</mo> <mi
+>a</mi></mrow></mfenced></mrow><mrow
+><mn>2</mn></mrow></msup
+></mrow></msqrt> <mo
+class="MathClass-rel">=</mo> <mi
+>a</mi><msqrt><mrow><mn>2</mn><msup><mrow
+><mi
+>p</mi></mrow><mrow
+><mn>2</mn> </mrow> </msup
+> <mo
+class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi
+>p</mi> <mo
+class="MathClass-bin">+</mo> <mn>1</mn></mrow></msqrt></mtd><mtd
 class="eqnarray-4"> <mtext class="eqnarray"></mtext></mtd>                                             </mtr></mtable>
 </math>
 <!--l. 32--><p class="nopar">
 The lengths of the squares form a decreasing geometrical series, with that proportion. Thus the length of the
 path a bug travel until they meet is:
 </p><!--l. 37--><p class="noindent">
-<!--tex4ht:inline--></p><!--l. 37--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
+<!--tex4ht:inline--></p><!--l. 37--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
 display="block" >
-<mtable 
+<mtable
 class="eqnarray-star" columnalign="right center left" >
-<mtr><mtd 
-class="eqnarray-1"> </mtd><mtd 
-class="eqnarray-2">   </mtd><mtd 
-class="eqnarray-3">        <mfrac><mrow 
-><mi 
->p</mi> <mo 
-class="MathClass-punc">&#x22C5;</mo> <msub><mrow 
-><mi 
->a</mi></mrow><mrow 
-><mn>1</mn></mrow></msub 
+<mtr><mtd
+class="eqnarray-1"> </mtd><mtd
+class="eqnarray-2">   </mtd><mtd
+class="eqnarray-3">        <mfrac><mrow
+><mi
+>p</mi> <mo
+class="MathClass-punc">&#x22C5;</mo> <msub><mrow
+><mi
+>a</mi></mrow><mrow
+><mn>1</mn></mrow></msub
 ></mrow>
-<mrow 
-><mn>1</mn> <mo 
-class="MathClass-bin">&#x2212;</mo><msqrt><mrow><mn>2</mn><msup><mrow 
-><mi 
->p</mi></mrow><mrow 
-><mn>2</mn> </mrow> </msup 
-> <mo 
-class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
->p</mi> <mo 
-class="MathClass-bin">+</mo> <mn>1</mn></mrow></msqrt></mrow></mfrac></mtd><mtd 
+<mrow
+><mn>1</mn> <mo
+class="MathClass-bin">&#x2212;</mo><msqrt><mrow><mn>2</mn><msup><mrow
+><mi
+>p</mi></mrow><mrow
+><mn>2</mn> </mrow> </msup
+> <mo
+class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi
+>p</mi> <mo
+class="MathClass-bin">+</mo> <mn>1</mn></mrow></msqrt></mrow></mfrac></mtd><mtd
 class="eqnarray-4"> <mtext class="eqnarray"></mtext></mtd>                                                                </mtr></mtable>
 </math>
 
 <!--l. 39--><p class="nopar">
 </p><!--l. 41--><p class="noindent">This is according to the formula that the sum of an in&#xFB01;nite decresing geometric series is
-<!--l. 42--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><mfrac><mrow> <msub><mrow 
-><mi 
->a</mi></mrow><mrow 
-><mn>1</mn></mrow></msub 
+<!--l. 42--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><mfrac><mrow> <msub><mrow
+><mi
+>a</mi></mrow><mrow
+><mn>1</mn></mrow></msub
 ></mrow>
-<mrow><mn>1</mn> <mo 
-class="MathClass-bin">&#x2212;</mo> <mi 
+<mrow><mn>1</mn> <mo
+class="MathClass-bin">&#x2212;</mo> <mi
 >q</mi></mrow></mfrac></math> where
-<!--l. 42--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><msub><mrow 
-><mi 
->a</mi></mrow><mrow 
-><mn>1</mn></mrow></msub 
+<!--l. 42--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><msub><mrow
+><mi
+>a</mi></mrow><mrow
+><mn>1</mn></mrow></msub
 ></math> is the value of its
-&#xFB01;rst item and <!--l. 43--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><mi 
+&#xFB01;rst item and <!--l. 43--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><mi
 >q</mi></math>
 is the proportion between two consecutive items.
 </p><!--l. 45--><p class="noindent">Now, to &#xFB01;nd the length an in&#xFB01;nitesimal bug will travel, we just limit
-<!--l. 45--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
-display="inline" ><mi 
+<!--l. 45--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
+display="inline" ><mi
 >p</mi></math> to
 0:
 </p><!--l. 48--><p class="noindent">
-<!--tex4ht:inline--></p><!--l. 48--><math 
- xmlns="http://www.w3.org/1998/Math/MathML"  
+<!--tex4ht:inline--></p><!--l. 48--><math
+ xmlns="http://www.w3.org/1998/Math/MathML"
 display="block" >
-<mtable 
+<mtable
 class="eqnarray-star" columnalign="right center left" >
-<mtr><mtd 
-class="eqnarray-1"> </mtd><mtd 
-class="eqnarray-2">   </mtd><mtd 
-class="eqnarray-3">   <munder class="msub"><mrow 
-><mo class="qopname">lim</mo></mrow><mrow 
-><mi 
->p</mi><mo 
-class="MathClass-rel">&#x2192;</mo><mn>0</mn></mrow></munder 
->           <mfrac><mrow 
-><mi 
->p</mi> <mo 
-class="MathClass-punc">&#x22C5;</mo> <msub><mrow 
-><mi 
->a</mi></mrow><mrow 
-><mn>1</mn></mrow></msub 
-></mrow> 
-<mrow 
-><mn>1</mn> <mo 
-class="MathClass-bin">&#x2212;</mo><msqrt><mrow><mn>2</mn><msup><mrow 
-><mi 
->p</mi></mrow><mrow 
-><mn>2</mn> </mrow> </msup 
-> <mo 
-class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
->p</mi> <mo 
-class="MathClass-bin">+</mo> <mn>1</mn></mrow></msqrt></mrow></mfrac> <mo 
-class="MathClass-rel">=</mo>                  </mtd><mtd 
+<mtr><mtd
+class="eqnarray-1"> </mtd><mtd
+class="eqnarray-2">   </mtd><mtd
+class="eqnarray-3">   <munder class="msub"><mrow
+><mo class="qopname">lim</mo></mrow><mrow
+><mi
+>p</mi><mo
+class="MathClass-rel">&#x2192;</mo><mn>0</mn></mrow></munder
+>           <mfrac><mrow
+><mi
+>p</mi> <mo
+class="MathClass-punc">&#x22C5;</mo> <msub><mrow
+><mi
+>a</mi></mrow><mrow
+><mn>1</mn></mrow></msub
+></mrow>
+<mrow
+><mn>1</mn> <mo
+class="MathClass-bin">&#x2212;</mo><msqrt><mrow><mn>2</mn><msup><mrow
+><mi
+>p</mi></mrow><mrow
+><mn>2</mn> </mrow> </msup
+> <mo
+class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi
+>p</mi> <mo
+class="MathClass-bin">+</mo> <mn>1</mn></mrow></msqrt></mrow></mfrac> <mo
+class="MathClass-rel">=</mo>                  </mtd><mtd
 class="eqnarray-4"> <mtext class="eqnarray"></mtext></mtd>
-</mtr><mtr><mtd 
-class="eqnarray-1"> </mtd><mtd 
-class="eqnarray-2">   </mtd><mtd 
-class="eqnarray-3">   <munder class="msub"><mrow 
-><mo class="qopname">lim</mo></mrow><mrow 
-><mi 
->p</mi><mo 
-class="MathClass-rel">&#x2192;</mo><mn>0</mn></mrow></munder 
->           <mfrac><mrow 
-><mi 
->p</mi> <mo 
-class="MathClass-punc">&#x22C5;</mo> <msub><mrow 
-><mi 
->a</mi></mrow><mrow 
-><mn>1</mn></mrow></msub 
-></mrow> 
-<mrow 
-><mn>1</mn> <mo 
-class="MathClass-bin">&#x2212;</mo><msqrt><mrow><mn>2</mn><msup><mrow 
-><mi 
->p</mi></mrow><mrow 
-><mn>2</mn> </mrow> </msup 
-> <mo 
-class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
->p</mi> <mo 
-class="MathClass-bin">+</mo> <mn>1</mn></mrow></msqrt></mrow></mfrac> <mo 
-class="MathClass-punc">&#x22C5;</mo><mfrac><mrow 
-><mn>1</mn> <mo 
-class="MathClass-bin">+</mo> <msqrt><mrow><mn>2</mn><msup><mrow 
-><mi 
->p</mi></mrow><mrow 
-><mn>2</mn> </mrow> </msup 
-> <mo 
-class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
->p</mi> <mo 
-class="MathClass-bin">+</mo> <mn>1</mn></mrow></msqrt></mrow> 
-<mrow 
-><mn>1</mn> <mo 
-class="MathClass-bin">+</mo> <msqrt><mrow><mn>2</mn><msup><mrow 
-><mi 
->p</mi></mrow><mrow 
-><mn>2</mn> </mrow> </msup 
-> <mo 
-class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
->p</mi> <mo 
-class="MathClass-bin">+</mo> <mn>1</mn></mrow></msqrt></mrow></mfrac> <mo 
-class="MathClass-rel">=</mo> </mtd><mtd 
+</mtr><mtr><mtd
+class="eqnarray-1"> </mtd><mtd
+class="eqnarray-2">   </mtd><mtd
+class="eqnarray-3">   <munder class="msub"><mrow
+><mo class="qopname">lim</mo></mrow><mrow
+><mi
+>p</mi><mo
+class="MathClass-rel">&#x2192;</mo><mn>0</mn></mrow></munder
+>           <mfrac><mrow
+><mi
+>p</mi> <mo
+class="MathClass-punc">&#x22C5;</mo> <msub><mrow
+><mi
+>a</mi></mrow><mrow
+><mn>1</mn></mrow></msub
+></mrow>
+<mrow
+><mn>1</mn> <mo
+class="MathClass-bin">&#x2212;</mo><msqrt><mrow><mn>2</mn><msup><mrow
+><mi
+>p</mi></mrow><mrow
+><mn>2</mn> </mrow> </msup
+> <mo
+class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi
+>p</mi> <mo
+class="MathClass-bin">+</mo> <mn>1</mn></mrow></msqrt></mrow></mfrac> <mo
+class="MathClass-punc">&#x22C5;</mo><mfrac><mrow
+><mn>1</mn> <mo
+class="MathClass-bin">+</mo> <msqrt><mrow><mn>2</mn><msup><mrow
+><mi
+>p</mi></mrow><mrow
+><mn>2</mn> </mrow> </msup
+> <mo
+class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi
+>p</mi> <mo
+class="MathClass-bin">+</mo> <mn>1</mn></mrow></msqrt></mrow>
+<mrow
+><mn>1</mn> <mo
+class="MathClass-bin">+</mo> <msqrt><mrow><mn>2</mn><msup><mrow
+><mi
+>p</mi></mrow><mrow
+><mn>2</mn> </mrow> </msup
+> <mo
+class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi
+>p</mi> <mo
+class="MathClass-bin">+</mo> <mn>1</mn></mrow></msqrt></mrow></mfrac> <mo
+class="MathClass-rel">=</mo> </mtd><mtd
 class="eqnarray-4"> <mtext class="eqnarray"></mtext></mtd>
-</mtr><mtr><mtd 
-class="eqnarray-1"> </mtd><mtd 
-class="eqnarray-2">   </mtd><mtd 
-class="eqnarray-3">   <munder class="msub"><mrow 
-><mo class="qopname">lim</mo></mrow><mrow 
-><mi 
->p</mi><mo 
-class="MathClass-rel">&#x2192;</mo><mn>0</mn></mrow></munder 
-><mfrac><mrow 
-><msub><mrow 
-><mi 
->a</mi></mrow><mrow 
-><mn>1</mn></mrow></msub 
-> <mo 
-class="MathClass-punc">&#x22C5;</mo> <mi 
->p</mi> <mo 
-class="MathClass-punc">&#x22C5;</mo><mfenced separators="" 
-open="("  close=")" ><mrow><mn>1</mn> <mo 
-class="MathClass-bin">+</mo> <msqrt><mrow><mn>2</mn><msup><mrow 
-><mi 
->p</mi></mrow><mrow 
-><mn>2</mn> </mrow> </msup 
-> <mo 
-class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
->p</mi> <mo 
-class="MathClass-bin">+</mo> <mn>1</mn></mrow></msqrt></mrow></mfenced></mrow> 
-       <mrow 
-><mn>1</mn> <mo 
-class="MathClass-bin">&#x2212;</mo> <msup><mrow 
-><mi 
->p</mi></mrow><mrow 
-><mn>2</mn></mrow></msup 
-> <mo 
-class="MathClass-bin">+</mo> <mn>2</mn><mi 
->p</mi> <mo 
-class="MathClass-bin">&#x2212;</mo> <mn>1</mn></mrow></mfrac>        <mo 
-class="MathClass-rel">=</mo>           </mtd><mtd 
+</mtr><mtr><mtd
+class="eqnarray-1"> </mtd><mtd
+class="eqnarray-2">   </mtd><mtd
+class="eqnarray-3">   <munder class="msub"><mrow
+><mo class="qopname">lim</mo></mrow><mrow
+><mi
+>p</mi><mo
+class="MathClass-rel">&#x2192;</mo><mn>0</mn></mrow></munder
+><mfrac><mrow
+><msub><mrow
+><mi
+>a</mi></mrow><mrow
+><mn>1</mn></mrow></msub
+> <mo
+class="MathClass-punc">&am