<?xml version='1.0' encoding='UTF-8'?><?xml-stylesheet href="http://www.blogger.com/styles/atom.css" type="text/css"?><feed xmlns='http://www.w3.org/2005/Atom' xmlns:openSearch='http://a9.com/-/spec/opensearchrss/1.0/' xmlns:georss='http://www.georss.org/georss' xmlns:gd='http://schemas.google.com/g/2005' xmlns:thr='http://purl.org/syndication/thread/1.0'><id>tag:blogger.com,1999:blog-6058061522806863813</id><updated>2011-07-31T10:01:48.330+08:00</updated><title type='text'>OMGWTFBBQPWNT!</title><subtitle type='html'></subtitle><link rel='http://schemas.google.com/g/2005#feed' type='application/atom+xml' href='http://nuubcake.blogspot.com/feeds/posts/default'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/6058061522806863813/posts/default?max-results=100'/><link rel='alternate' type='text/html' href='http://nuubcake.blogspot.com/'/><link rel='hub' href='http://pubsubhubbub.appspot.com/'/><author><name>Memories of Nobody</name><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='http://img2.blogblog.com/img/b16-rounded.gif'/></author><generator version='7.00' uri='http://www.blogger.com'>Blogger</generator><openSearch:totalResults>7</openSearch:totalResults><openSearch:startIndex>1</openSearch:startIndex><openSearch:itemsPerPage>100</openSearch:itemsPerPage><entry><id>tag:blogger.com,1999:blog-6058061522806863813.post-1446688042122459694</id><published>2009-06-12T04:18:00.007+08:00</published><updated>2009-06-12T05:02:11.811+08:00</updated><title type='text'>Top 10 Hybrid Animals</title><content type='html'>&lt;span style="font-family:georgia;"&gt;Source: &lt;/span&gt;&lt;a href="http://www.hemmy.net/2006/06/19/top-10-hybrid-animals/"&gt;&lt;span style="font-family:georgia;"&gt;http://www.hemmy.net/2006/06/19/top-10-hybrid-animals/&lt;/span&gt;&lt;/a&gt;&lt;br /&gt;&lt;br /&gt;&lt;span style="font-family:georgia;"&gt;Hybrid animals are cross-breds between animals of similar genetics. They mostly exist in captivity and are the result of human intervention. &lt;/span&gt;&lt;br /&gt;&lt;br /&gt;&lt;span style="font-family:georgia;"&gt;&lt;strong&gt;10. Liger/Tigon&lt;/strong&gt;&lt;br /&gt;Though they are fascinating animals, they get the last of the list because they are the most popular known among the hybrids. Ligers are crossbreeds between a male lion while Tigons are crossbreds between a male tiger and a female lion. Ligers are the world’s largest cats. Tigons on the other hand, are prone towards dwarfism and are usually smaller than either of their parents. Male Ligers/Tigons are sterile while the females are often fertile. Below shows a liger and its trainer Dr. Bhagavan Antle at a Renaissance Festival in Massachusetts, USA, October 2005.&lt;/span&gt;&lt;br /&gt;&lt;br /&gt;&lt;p&gt;&lt;a href="http://www.hemmy.net/images/animals/liger02.jpg"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 500px; CURSOR: hand; HEIGHT: 375px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/liger02.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;/p&gt;&lt;p&gt;&lt;a href="http://www.hemmy.net/images/animals/liger01.jpg"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 500px; CURSOR: hand; HEIGHT: 413px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/liger01.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;span style="font-family:georgia;"&gt; copyright, Andy Carvin &lt;/span&gt;&lt;br /&gt;&lt;/p&gt;&lt;p&gt;&lt;strong&gt;&lt;span style="font-family:georgia;"&gt;9. Wolf Dog&lt;/span&gt;&lt;/strong&gt;&lt;/p&gt;&lt;p&gt;&lt;span style="font-family:georgia;"&gt;Dogs and wolves tend to crossbreed rather freely. The wolf is a shy animal depending on nuances in body language, facial expression and on hunting skills to survive. Their jaws are much stronger than those of a dog and are often used to exert dominance. For a dog wolf hybrid, it is not known when it will display a wolf behaviour or dog behaviour or something in between. Obedience training is a must in order to tame the animal. &lt;/span&gt;&lt;/p&gt;&lt;p&gt;&lt;a href="http://www.hemmy.net/images/animals/wolfdog01.jpg"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 400px; CURSOR: hand; HEIGHT: 382px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/wolfdog01.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;span style="font-family:georgia;"&gt;&lt;br /&gt;&lt;br /&gt;&lt;strong&gt;8. Iron Age Pig&lt;/strong&gt;&lt;br /&gt;Domestic Tamworth pigs are crossbred with wild boar to create ‘Iron Age Pigs’. The hybrids are tamer than wild boar but less tractable than domestic swine and generally become specialist pork sausages. Most of them are bred for the specialist meat trade.&lt;/span&gt;&lt;/p&gt;&lt;p&gt;&lt;a href="http://www.hemmy.net/images/animals/ironagepig.jpg"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 500px; CURSOR: hand; HEIGHT: 351px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/ironagepig.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;span style="font-family:georgia;"&gt;&lt;br /&gt;&lt;br /&gt;&lt;/span&gt;&lt;span style="font-family:georgia;"&gt;&lt;strong&gt;7. Zebroid&lt;/strong&gt;&lt;/span&gt;&lt;/p&gt;&lt;p&gt;&lt;span style="font-family:georgia;"&gt;A zorse is the result of crossbreeding a horse and a zebra. A zonkey is the result of crossbreeding a donkey with a zebra. The Zony is the result of crossbreeding a pony to a zebra. All these three are called zebroids - defined as a cross between a zebra and any other equid. Zebroids are preferred over zebra for practical uses such as riding because of its body shape. However it is more inclined to be temperamental and can prove to be difficult to handle. &lt;/span&gt;&lt;/p&gt;&lt;p&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 275px; CURSOR: hand; HEIGHT: 196px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/zony01.jpg" border="0" /&gt;&lt;/span&gt;&lt;/p&gt;&lt;p align="center"&gt;&lt;span style="font-family:georgia;"&gt;A Zony&lt;br /&gt;&lt;/span&gt;&lt;/p&gt;&lt;br /&gt;&lt;br /&gt;&lt;div align="center"&gt;&lt;a href="http://www.hemmy.net/images/animals/zonkey01.jpg"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 211px; CURSOR: hand; HEIGHT: 227px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/zonkey01.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;span style="font-family:georgia;"&gt; A Zonkey&lt;br /&gt;&lt;br /&gt;&lt;/span&gt;&lt;a href="http://www.hemmy.net/images/animals/zebroid01.jpg"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 286px; CURSOR: hand; HEIGHT: 250px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/zebroid01.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;span style="font-family:georgia;"&gt; A Zorse&lt;/span&gt;&lt;br /&gt;&lt;/div&gt;&lt;div align="center"&gt;&lt;br /&gt;&lt;br /&gt;&lt;div align="left"&gt;&lt;span style="font-family:georgia;"&gt;&lt;strong&gt;6. Cama&lt;/strong&gt;&lt;br /&gt;A Cama is a hybrid between a camel and a llama. They are born via artificial insemination due to the huge difference in sizes of the animals which disallow natural breeding. A Cama usually has the short ears and long tails of a camel but the cloven hooves of a llama. Also most noticeably is the absence of the hump.&lt;/span&gt;&lt;/div&gt;&lt;br /&gt;&lt;div align="center"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 442px; CURSOR: hand; HEIGHT: 299px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/cama01.jpg" border="0" /&gt;Rama’s parents shown behind, a camel and llama.&lt;br /&gt;&lt;br /&gt;&lt;/span&gt;&lt;div align="center"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 350px; CURSOR: hand; HEIGHT: 377px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/cama02.jpg" border="0" /&gt;&lt;/span&gt;&lt;span style="font-family:georgia;"&gt; This is Rama the Cama at two days old.&lt;/span&gt;&lt;/div&gt;&lt;div align="center"&gt;&lt;/div&gt;&lt;div align="center"&gt;&lt;span style="font-family:georgia;"&gt;&lt;br /&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 450px; CURSOR: hand; HEIGHT: 394px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/cama03.jpg" border="0" /&gt; Rama at two years of age as a young adult.&lt;/span&gt;&lt;/div&gt;&lt;/div&gt;&lt;br /&gt;&lt;br /&gt;&lt;p align="left"&gt;&lt;span style="font-family:georgia;"&gt;&lt;strong&gt;5. Grolar, Pizzly&lt;/strong&gt;&lt;br /&gt;A grolar/pizzly hybrid is the product of a grizzly bear and a polar bear. Although the two bears are genetically similar, they tend to avoid each other in the wild. During 16 April 2006, a hybrid bear was shot dead by Jim Martell,a hunter from the United States, in Canada. It was the first time a hybrid was found in the wild where previous records of grolars or pizzlies have only been found in zoos.&lt;br /&gt;&lt;br /&gt;&lt;/span&gt;&lt;a href="http://www.hemmy.net/images/animals/grolar01.jpg"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 307px; CURSOR: hand; HEIGHT: 354px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/grolar01.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;/p&gt;&lt;a href="http://www.hemmy.net/images/animals/grolar02.jpg"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 194px; CURSOR: hand; HEIGHT: 358px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/grolar02.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;span style="font-family:georgia;"&gt; &lt;/span&gt;&lt;span style="font-family:georgia;"&gt;A grolar, pizzly displayed at the Rothschild Museum, Tring, copyright Sarah Hartwell&lt;/span&gt;&lt;br /&gt;&lt;br /&gt;&lt;br /&gt;&lt;p align="left"&gt;&lt;strong&gt;&lt;span style="font-family:georgia;"&gt;4. Leopon&lt;/span&gt;&lt;/strong&gt;&lt;/p&gt;&lt;p align="left"&gt;&lt;span style="font-family:georgia;"&gt;A Leopon is the result of breeding a male leopard and a female lion. The head of the animal is similar to that of a lion while the rest of the bodies carries similarities to leopards. The most successful breeding programme was at the Koshien Hanshin Park in Nishinomiya City, Japan. Leopons are larger than leopards and likes to climb and enjoy water.&lt;/span&gt;&lt;/p&gt;&lt;p align="left"&gt;&lt;a href="http://www.hemmy.net/images/animals/leopon01.jpg"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 250px; CURSOR: hand; HEIGHT: 196px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/leopon01.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;/p&gt;&lt;a href="http://www.hemmy.net/images/animals/leopon02.jpg"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 243px; CURSOR: hand; HEIGHT: 196px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/leopon02.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;span style="font-family:georgia;"&gt;A leopon at a zoo.&lt;br /&gt;&lt;br /&gt;&lt;/span&gt;&lt;br /&gt;&lt;span style="font-family:georgia;"&gt;&lt;br /&gt;&lt;/span&gt;&lt;p align="left"&gt;&lt;span style="font-family:georgia;"&gt;&lt;strong&gt;3. Hybrid Pheasant&lt;/strong&gt;&lt;br /&gt;The Golden Phesant has commonly been crossed with the similar Lady Amherst’s Pheasant. The result is a hybrid with distinguished colors from its parents.&lt;/span&gt;&lt;/p&gt;&lt;div&gt;&lt;a href="http://www.hemmy.net/images/animals/goldenpheasant.jpg"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 500px; CURSOR: hand; HEIGHT: 338px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/goldenpheasant.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;span style="font-family:georgia;"&gt; A Golden Pheasant&lt;br /&gt;&lt;br /&gt;&lt;br /&gt;&lt;/span&gt;&lt;div&gt;&lt;a href="http://www.hemmy.net/images/animals/amherstpheasant.jpg"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 500px; CURSOR: hand; HEIGHT: 395px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/amherstpheasant.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;span style="font-family:georgia;"&gt;A Lady Amherst Pheasant&lt;br /&gt;&lt;br /&gt;&lt;br /&gt;&lt;/span&gt;&lt;div&gt;&lt;a href="http://www.hemmy.net/images/animals/hybridpheasant.jpg"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 430px; CURSOR: hand; HEIGHT: 344px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/hybridpheasant.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;span style="font-family:georgia;"&gt; Hybrid Pheasant displayed at Rothschild Museum Copyright Sarah Hartwell&lt;br /&gt;&lt;/span&gt;&lt;span style="font-family:georgia;"&gt;&lt;br /&gt;&lt;/span&gt;&lt;/div&gt;&lt;div&gt;&lt;p align="left"&gt;&lt;span style="font-family:georgia;"&gt;&lt;strong&gt;2. Wolphin&lt;/strong&gt;&lt;br /&gt;A wolphin is a rare hybrid formed from a cross between a bottlenose dolphin and a false killer whale. There are currently only two in captivity at the Sea Life Park in Hawaii. A wolphin’s size, colour and shape are intermediate between the parent species. The first captive wolphin was Kekaimalu, which shows mixed heritage even in its teeth: bottlenose dolphins have 88, false killer whales have 44 and Kekaimalu has 66!&lt;/span&gt;&lt;a href="http://www.hemmy.net/images/animals/wolphin.jpg"&gt;&lt;/p&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 400px; CURSOR: hand; HEIGHT: 290px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/wolphin.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;span style="font-family:georgia;"&gt; Kekaimalu, The Wolphin&lt;br /&gt;&lt;br /&gt;&lt;br /&gt;&lt;br /&gt;&lt;/span&gt;&lt;p align="left"&gt;&lt;strong&gt;&lt;span style="font-family:georgia;"&gt;1. Ti-Liger, Ti-Tigon, Li-Tigon, Li-Liger&lt;/span&gt;&lt;/strong&gt;&lt;/p&gt;&lt;p align="left"&gt;&lt;span style="font-family:georgia;"&gt;The top spot goes to ti-ligers/ti-tigon/li-tigons/li-ligers because it is a hybrid among the hybrids. It is a cross breed between a male tiger and a female liger/tigon or a male lion with a female tigon/liger. Do note that female ligers or tigons are fertile. They are extremely rare and are in mostly private ownership within a behavioural studies programme. In the case of ti-ligers, they have unusual striping where it breaks up and display a blotchy appearance. Since they are 3/4 tiger, their characteristics inhibit more of those of a tiger than a lion. &lt;/span&gt;&lt;/p&gt;&lt;div&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 364px; CURSOR: hand; HEIGHT: 227px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/tiliger01.jpg" border="0" /&gt;&lt;br /&gt;&lt;/span&gt;&lt;div&gt;&lt;a href="http://www.hemmy.net/images/animals/tiliger02.jpg"&gt;&lt;span style="font-family:georgia;"&gt;&lt;img style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 269px; CURSOR: hand; HEIGHT: 284px; TEXT-ALIGN: center" alt="" src="http://www.hemmy.net/images/animals/tiliger02.jpg" border="0" /&gt;&lt;/span&gt;&lt;/a&gt;&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;&lt;/div&gt;&lt;div class="blogger-post-footer"&gt;&lt;img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/6058061522806863813-1446688042122459694?l=nuubcake.blogspot.com' alt='' /&gt;&lt;/div&gt;</content><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/6058061522806863813/posts/default/1446688042122459694'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/6058061522806863813/posts/default/1446688042122459694'/><link rel='alternate' type='text/html' href='http://nuubcake.blogspot.com/2009/06/top-10-hybrid-animals.html' title='Top 10 Hybrid Animals'/><author><name>Memories of Nobody</name><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='http://img2.blogblog.com/img/b16-rounded.gif'/></author></entry><entry><id>tag:blogger.com,1999:blog-6058061522806863813.post-2557021917812674095</id><published>2008-10-14T14:31:00.002+08:00</published><updated>2008-10-14T15:09:23.088+08:00</updated><title type='text'>Palindromes</title><content type='html'>A palindrome is a word or phrase that is the same backwards and forwards. The name was coined in the 1600s by English writer Ben Jonson from the Greek words Palin (back) and Dromos (way or direction), although the first known palindrome was a Latin phrase discovered in 79AD: "Sator Arepo Tenet Opera Rotas". This is known as the Sator square, since it can be written out as a five x five square and is the same whether read across the squares or down.&lt;br /&gt;&lt;br /&gt;S A T O R&lt;br /&gt;A R E P O&lt;br /&gt;T E N E T&lt;br /&gt;O P E R A&lt;br /&gt;R O T A S&lt;br /&gt;&lt;br /&gt;Other examples of palindromic phrases or sentences include: "Drab as a fool, as aloof as a bard." "Egad, an adage!" "Here so long? No loser, eh?" "Live not on evil deed, live not on evil." " Madam, in Eden I'm Adam."&lt;br /&gt;&lt;br /&gt;Most people who live in Adelaide, Australia know what palindromes are because there is a suburb whose name is a palindrome: Glenelg (named after a village in Scotland). Other geographic palindromes include Kanakanak (Alaska), Saxet, Texas, Ellemelle (Belgium), Qaanaaq (Greenland), Kinikinik (Alberta), and Anahanahana (Madagascar).&lt;br /&gt;&lt;br /&gt;Palindromic company and brand names include Oxo, Omo, Axa, Aviva, Civic, Rotavator, and M&amp;amp;M, but interestingly enough, there are more companies using Palindrome in their names, than there are palindromic brand names. Palindromic names include Anna, Otto, Elle, Bob, Eve, Hannah. And there's also Mom, Mam, Mum, Dad, Pop, and of course, Madam.&lt;br /&gt;&lt;br /&gt;Numbers can also be palindromes, such as 345543 or 9096909, and perfect squares such as 121, 484 and 10201. A date can be a palindrome, such as 6-06-06, as can a time: 21:12, or a year: 2002 or 1991 (and we won't be around for the next one: 2112).&lt;br /&gt;&lt;br /&gt;The longest palindrome sentence in English is over 17,000 characters long, but it's completely meaningless. You can read it at: &lt;a href="http://www.norvig.com/pal1txt.html"&gt;http://www.norvig.com/pal1txt.html&lt;/a&gt;, but don't expect it to make any sense because it's computer generated gibberish. The longer palindromes are, the less sense they make. The following is probably about the limit: "Are we not drawn onward, we few, drawn onward to a new era."&lt;br /&gt;&lt;br /&gt;The longest English palindrome word is "tattarrattat", invented by James Joyce in "Ulysses" to mean a knock on the door. One of the second longest is detartrated, which means having tartrates (a chemical) removed. Another, also with 11 letters is aibohphobia, an invented word, meaning, appropriately enough: fear of palindromes!&lt;div class="blogger-post-footer"&gt;&lt;img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/6058061522806863813-2557021917812674095?l=nuubcake.blogspot.com' alt='' /&gt;&lt;/div&gt;</content><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/6058061522806863813/posts/default/2557021917812674095'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/6058061522806863813/posts/default/2557021917812674095'/><link rel='alternate' type='text/html' href='http://nuubcake.blogspot.com/2008/10/palindromes.html' title='Palindromes'/><author><name>Memories of Nobody</name><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='http://img2.blogblog.com/img/b16-rounded.gif'/></author></entry><entry><id>tag:blogger.com,1999:blog-6058061522806863813.post-7305534024099058298</id><published>2008-09-20T21:42:00.007+08:00</published><updated>2008-09-22T16:48:03.891+08:00</updated><title type='text'>Paradox</title><content type='html'>"&lt;strong&gt;paradox&lt;/strong&gt;: something which is contrary to received conventional opinion; something which is apparently absurd but is or may be really true; a self-contradictory statement" (The Chambers Dictionary 2006)&lt;br /&gt;&lt;br /&gt;&lt;u&gt;&lt;strong&gt;A paradoxical notice &lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;&lt;br /&gt;&lt;table&gt;&lt;tr&gt;&lt;td bgcolor="pink"&gt;PLEASE IGNORE THIS NOTICE&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;&lt;br /&gt;&lt;u&gt;&lt;strong&gt;Happiness or a ham sandwich? &lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;&lt;br /&gt;Which is better, eternal happiness or a ham sandwich? It would appear that eternal happiness is better, but this is really not so! After all, nothing is better than eternal happiness, and a ham sandwich is certainly better than nothing. Therefore a ham sandwich is better than eternal happiness. &lt;br /&gt;&lt;br /&gt;&lt;u&gt;&lt;strong&gt;The liar paradox &lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;&lt;br /&gt;The following version is the version which we will refer to as the liar paradox. Consider the statement in the following box: &lt;br /&gt;&lt;br /&gt;&lt;table&gt;&lt;tr&gt;&lt;td bgcolor="pink"&gt;THIS SENTENCE IS FALSE  &lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;&lt;br /&gt;Is that sentence true or false? If it is false then it is true, and if it is true then it is false... &lt;br /&gt;&lt;br /&gt;&lt;u&gt;&lt;strong&gt;Hanging or beheading &lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;&lt;br /&gt;Poaching on the hunting preserves of a powerful prince was punishable by death, but the prince further decreed that anyone caught poaching was to be given the privilege of deciding whether he should be hanged or beheaded. The culprit was permitted to make a statement - if it were false, he was to be hanged; if it were true, he was to be beheaded. One logical rogue availed himself of this dubious prerogative - to be hanged if he didn't and to be beheaded if he did - by stating: "I shall be hanged." Here was a dilemma not anticipated. For, as the poacher put it, "If you now hang me, you break the laws made by the prince, for my statement is true, and I ought to be beheaded, but if you behead me, you are also breaking the laws, for then what I said was false and I should therefore be hanged." &lt;br /&gt;&lt;br /&gt;&lt;u&gt;&lt;strong&gt;Knights and knaves &lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;&lt;br /&gt;There is a wide variety of puzzles about an island in which certain inhabitants called "knights" always tell the truth, and others called "knaves" always lie. It is assumed that every inhabitant of the island is either a knight or a knave... &lt;br /&gt;&lt;br /&gt;Suppose A says, "Either I am a knave or else two plus two equals five." What would you conclude.? &lt;br /&gt;&lt;br /&gt;If A is a knight, then two plus two equals five, which is not true. If A is a knave, then he is speaking the truth, which is not possible. &lt;br /&gt;&lt;br /&gt;Smullyan comments: &lt;br /&gt;&lt;br /&gt;The only valid conclusion is that the author of this problem is not a knight. The fact is that neither a knight nor a knave could make such a statement. &lt;br /&gt;&lt;br /&gt;&lt;u&gt;&lt;strong&gt;The visiting logician &lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;&lt;br /&gt;We are back on the Island of Knights and Knaves, where the following three propositions hold: (1) knights make only true statements; (2) knaves make only false ones; (3) every inhabitant is either a knght or a knave. These three propositions will be collectively referred to as the "rules of the island." &lt;br /&gt;&lt;br /&gt;We recall that no inhabitant can claim that he is not a knight, since no knight would make the false statement that he isn't a knight and no knave would make the true statement that he isn't a knight. &lt;br /&gt;&lt;br /&gt;Now suppose a logician visits the island and meets a native who makes the following statement to him: " You will never know that I am a knight. " &lt;br /&gt;&lt;br /&gt;Do we get a paradox? Let us see. The logician starts reasoning as follows: "Suppose he is a knave. Then his statement is false, which means that at some time I will know that he is a knight, but I can't know that he is a knight unless he really is one. So, if he is a knave, it follows that he must be a knight, which is a contradiction. Therefore he can't be a knave; he must be a knight." &lt;br /&gt;&lt;br /&gt;So far so good - there is as yet no contradiction. But then he continues reasoning: "Now I know that he is a knight, although he said that I never would. Hence his statement was false, which means that he must be a knave. Paradox!" &lt;br /&gt;&lt;br /&gt;&lt;u&gt;&lt;strong&gt;Hilbert's hotel paradox &lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;&lt;br /&gt;Imagine a hotel with a finite number of rooms, and assume that all the rooms are occupied. A new guest arrives and asks for a room. "Sorry" - says the proprietor - "but all the rooms are occupied." Now let us imagine a hotel with an infinite number of rooms, and all the rooms are occupied. To this hotel, too, comes a new guest and asks for a room. "But of course!" - exclaims the proprietor, and he moves the person previously occupying room N1 into room N2, the person from room N2 into room N3, the person from room N3 into room N4, and so on... And the new customer receives room N1, which becomes free as a result of these transpositions. &lt;br /&gt;&lt;br /&gt;Let us imagine now a hotel with an infinite number of rooms, all taken up, and an infinite number of new guests who come in, and ask for rooms. &lt;br /&gt;&lt;br /&gt;"Certainly, gentlemen," says the proprietor, "just wait a minute." He moves the occupant of N1 into N2, the occupant of N2 into N4, the occupant of N3 into N6, and so on, and so on... &lt;br /&gt;&lt;br /&gt;Now all odd numbered rooms become free and the infinity of new guests can easily be accommodated in them. &lt;br /&gt;&lt;br /&gt;The proprietor's "just wait a minute" seems optimistic; it would surely take him an infinite time to shift the guests around&lt;br /&gt;&lt;br /&gt;&lt;u&gt;&lt;strong&gt;The unexpected hanging &lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;&lt;br /&gt;[A man condemned to be hanged] was sentenced on Saturday. "The hanging will take place at noon," said the judge to the prisoner, "on one of the seven days of next week. But you will not know which day it is until you are so informed on the morning of the day of the hanging." &lt;br /&gt;&lt;br /&gt;The judge was known to be a man who always kept his word. The prisoner, accompanied by his lawyer, went back to his cell. As soon as the two men were alone, the lawyer broke into a grin. "Don't you see?" he exclaimed. "The judge's sentence cannot possibly be carried out." &lt;br /&gt;&lt;br /&gt;"I don't see," said the prisoner. &lt;br /&gt;&lt;br /&gt;"Let me explain They obviously can't hang you next Saturday. Saturday is the last day of the week. On Friday afternoon you would still be alive and you would know with absolute certainty that the hanging would be on Saturday. You would know this before you were told so on Saturday morning. That would violate the judge's decree." &lt;br /&gt;&lt;br /&gt;"True," said the prisoner. &lt;br /&gt;&lt;br /&gt;"Saturday, then is positively ruled out," continued the lawyer. "This leaves Friday as the last day they can hang you. But they can't hang you on Friday because by Thursday only two days would remain: Friday and Saturday. Since Saturday is not a possible day, the hanging would have to be on Friday. Your knowledge of that fact would violate the judge's decree again. So Friday is out. This leaves Thursday as the last possible day. But Thursday is out because if you're alive Wednesday afternoon, you'll know that Thursday is to be the day." &lt;br /&gt;&lt;br /&gt;"I get it," said the prisoner, who was beginning to feel much better. "In exactly the same way I can rule out Wednesday, Tuesday and Monday. That leaves only tomorrow. But they can't hang me tomorrow because I know it today!" &lt;br /&gt;&lt;br /&gt;... He is convinced, by what appears to be unimpeachable logic, that he cannot be hanged without contradicting the conditions specified in his sentence. Then on Thursday morning, to his great surprise, the hangman arrives. Clearly he did not expect him. What is more surprising, the judge's decree is now seen to be perfectly correctly. The sentence can be carried out exactly as stated. &lt;br /&gt;&lt;br /&gt;&lt;u&gt;&lt;strong&gt;Prediction paradox (Newcomb's problem)&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;&lt;br /&gt;Two closed boxes, Bl and B2, are on a table. Bl contains $1,000. B2 contains either nothing or $1 million. You do not know which. You have an irrevocable choice between two actions:&lt;br /&gt;&lt;br /&gt;1. Take what is in both boxes. &lt;br /&gt;2. Take only what is in B2. &lt;br /&gt;&lt;br /&gt;At some time before the test a superior Being has made a prediction about what you will decide. It is not necessary to assume determinism. You only need be persuaded that the Being's predictions are "almost certainly" correct. If you like, you can think of the Being as God, but the paradox is just as strong if you regard the Being as a superior intelligence from another planet or a supercomputer capable of probing your brain and making highly accurate predictions about your decisions. If the Being expects you to choose both boxes, he has left B2 empty. If he expects you to take only B2, he has put $1 million in it. (If he expects you to randomize your choice by, say, flipping a coin, he has left B2 empty.) In all cases Bl contains $1,000. You understand the situation fully, the Being knows you understand, you know that he knows, and so on. &lt;br /&gt;&lt;br /&gt;What should you do? Clearly it is not to your advantage to flip a coin, so that you must decide on your own. The paradox lies in the disturbing fact that a strong argument can be made for either decision. Both arguments cannot be right. The problem is to explain why one is wrong.&lt;br /&gt;&lt;br /&gt;Let us look first at the argument for taking only B2. You believe the Being is an excellent predictor. If you take both boxes, the Being almost certainly will have anticipated your action and have left B2 empty. You will get only the $1,000 in Bl. On the other hand, if you take only B2, the Being, expecting that, almost certainly will have placed $1 million in it. Clearly it is to your advantage to take only B2. &lt;br /&gt;&lt;br /&gt;Convincing? Yes, but the Being made his prediction, say a week ago and then left. Either he put the $1 million in B2, or he did not. "If the money is already there, it will stay there whatever you choose. It is not going to disappear. If it is not already there, it is not going to suddenly appear if you choose only what is in the second box." It is assumed that no "backward causality" is operating; that is, your present actions cannot influence what the Being did last week. So why not take both boxes and get everything that is there? If B2 is filled, you get $1,001,000. If it is empty, you get at least $1,000. If you are so foolish as to take only B2, you know you cannot get more than $1 million, and there is even a slight possibility of getting nothing. Clearly it is to your advantage to take both boxes!&lt;br /&gt;&lt;br /&gt;&lt;u&gt;&lt;strong&gt;"Interesting" and "uninteresting" numbers &lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;&lt;br /&gt;The question arises: Are there any uninteresting numbers? We can prove that there are none by the following simple steps. If there are dull numbers, then we can divide all numbers into two sets - interesting and dull. In the set of dull numbers there will be only one number that is the smallest. Since it is the smallest uninteresting number it becomes, ipso facto , an interesting number. We must therefore remove it from the dull set and place it in the other. But now there will be another smallest uninteresting number. Repeating this process will make any dull number interesting.&lt;br /&gt;&lt;br /&gt;&lt;u&gt;&lt;strong&gt;Grelling's paradox - autological and heterological&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;&lt;br /&gt;Divide the adjectives in English which are self-descriptive, such as "pentasyllabic", "awkwardnessful", and "recherché", and those which are not, such as "edible", "incomplete", and "bisyllabic". Now if we admit "non-self-descriptive" as an adjective, to which class does it belong? If it seems questionable to include hyphenated words, we can use two terms invented specially for this paradox: autological (= "self-descriptive"), and heterological (= "non-self-descriptive"). The question then becomes: "Is 'heterological' heterological?" Try it!&lt;br /&gt;&lt;br /&gt;&lt;u&gt;&lt;strong&gt;Protagoras's pupil&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;&lt;br /&gt;Another paradox which has its foundation - real or legendary - in antiquity concerns the sophist Protagoras, who lived and taught in the fifth century BC. It is said that Protagoras made an arrangement with one of his pupils whereby the pupil was to pay for his instruction after he had won his first case. The young man completed his course, hung up the traditional shingle, and waited for clients. None appeared. Protagoras grew impatient and decided to sue his former pupil for the amount owed him.&lt;br /&gt;&lt;br /&gt;'For,' argued Protagoras, 'either I win this suit, or you win it. If I win, you pay me according to the judgement of the court. If you win, you pay me according to our agreement. In either case I am bound to be paid.'&lt;br /&gt;&lt;br /&gt;'Not so,' replied the young man. 'If I win, then by the judgement of the court I need not pay you. If you win, then by our agreement I need not pay you. In either case I am bound not to have to pay you.'&lt;br /&gt;&lt;br /&gt;Whose argument was right? Who knows?&lt;br /&gt;&lt;br /&gt;&lt;u&gt;&lt;strong&gt;Sorites paradox (paradox of the heap)&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;&lt;br /&gt;Suppose you have a heap of sand. If you take away one grain of sand, what remains is still a heap: removing a single grain cannot turn a heap into something that is not a heap. If two collections of grains of sand differ in number by just one grain, then both or neither are heaps. This apparently obvious and uncontroversial supposition appears to lead to the paradoxical conclusion that all collections of grains of sand, even one-membered collections, are heaps.&lt;br /&gt;&lt;br /&gt;&lt;strong&gt;Drawing Hands&lt;/strong&gt;&lt;br /&gt;&lt;a href="http://4.bp.blogspot.com/_jtWgreWo0yc/SNUTk16RvKI/AAAAAAAAAIk/Fd_ZNUlpP0w/s1600-h/smallescher-hands%5B1%5D.jpg"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://4.bp.blogspot.com/_jtWgreWo0yc/SNUTk16RvKI/AAAAAAAAAIk/Fd_ZNUlpP0w/s400/smallescher-hands%5B1%5D.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5248122464437910690" /&gt;&lt;/a&gt;&lt;br /&gt;&lt;br /&gt;&lt;strong&gt;Belvedere&lt;/strong&gt;&lt;br /&gt;&lt;a href="http://2.bp.blogspot.com/_jtWgreWo0yc/SNUTk3mXQnI/AAAAAAAAAIs/17dJT8WIfDY/s1600-h/smallbelvedere%5B2%5D.jpg"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://2.bp.blogspot.com/_jtWgreWo0yc/SNUTk3mXQnI/AAAAAAAAAIs/17dJT8WIfDY/s400/smallbelvedere%5B2%5D.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5248122464891257458" /&gt;&lt;/a&gt;&lt;br /&gt;&lt;br /&gt;&lt;strong&gt;Waterfall&lt;/strong&gt;&lt;br /&gt;&lt;a href="http://2.bp.blogspot.com/_jtWgreWo0yc/SNUTlLM2TJI/AAAAAAAAAI0/JFBxL2L2Sog/s1600-h/smallwaterfall%5B2%5D.jpg"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://2.bp.blogspot.com/_jtWgreWo0yc/SNUTlLM2TJI/AAAAAAAAAI0/JFBxL2L2Sog/s400/smallwaterfall%5B2%5D.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5248122470152948882" /&gt;&lt;/a&gt;&lt;br /&gt;&lt;br /&gt;&lt;strong&gt;Mobius Band&lt;/strong&gt;&lt;br /&gt;&lt;a href="http://1.bp.blogspot.com/_jtWgreWo0yc/SNUTlAUFjHI/AAAAAAAAAI8/L7g7kwAPcKo/s1600-h/mobius2%5B1%5D.jpg"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://1.bp.blogspot.com/_jtWgreWo0yc/SNUTlAUFjHI/AAAAAAAAAI8/L7g7kwAPcKo/s400/mobius2%5B1%5D.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5248122467230518386" /&gt;&lt;/a&gt;&lt;br /&gt;A Mobius band has just one side and just one edge. &lt;br /&gt;&lt;br /&gt;&lt;strong&gt;Penrose Triangle&lt;/strong&gt;&lt;br /&gt;&lt;a href="http://2.bp.blogspot.com/_jtWgreWo0yc/SNUTla3hm1I/AAAAAAAAAJE/cHa3AWIMXes/s1600-h/penrose%5B1%5D.jpg"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://2.bp.blogspot.com/_jtWgreWo0yc/SNUTla3hm1I/AAAAAAAAAJE/cHa3AWIMXes/s400/penrose%5B1%5D.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5248122474358479698" /&gt;&lt;/a&gt;&lt;br /&gt;&lt;br /&gt;Necker's Cube (Freemish crate)&lt;br /&gt;&lt;a href="http://4.bp.blogspot.com/_jtWgreWo0yc/SNUUx79lBwI/AAAAAAAAAJM/FqcgdL5fU9s/s1600-h/necker%5B1%5D.jpg"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://4.bp.blogspot.com/_jtWgreWo0yc/SNUUx79lBwI/AAAAAAAAAJM/FqcgdL5fU9s/s400/necker%5B1%5D.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5248123788912297730" /&gt;&lt;/a&gt;&lt;br /&gt;&lt;a href="http://2.bp.blogspot.com/_jtWgreWo0yc/SNUUx1G3OuI/AAAAAAAAAJU/tW5Oraev_PY/s1600-h/crate%5B1%5D.jpg"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://2.bp.blogspot.com/_jtWgreWo0yc/SNUUx1G3OuI/AAAAAAAAAJU/tW5Oraev_PY/s400/crate%5B1%5D.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5248123787072191202" /&gt;&lt;/a&gt;&lt;br /&gt;&lt;br /&gt;Missing Square Paradox&lt;br /&gt;&lt;a href="http://2.bp.blogspot.com/_jtWgreWo0yc/SNUUx__7FjI/AAAAAAAAAJc/ddDATLNwbwI/s1600-h/missing_square%5B1%5D.png"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://2.bp.blogspot.com/_jtWgreWo0yc/SNUUx__7FjI/AAAAAAAAAJc/ddDATLNwbwI/s400/missing_square%5B1%5D.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5248123789995873842" /&gt;&lt;/a&gt;&lt;br /&gt;&lt;a href="http://1.bp.blogspot.com/_jtWgreWo0yc/SNUUyEkNG2I/AAAAAAAAAJk/NTaFb2z3nwU/s1600-h/missing_square_animation%5B1%5D.gif"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://1.bp.blogspot.com/_jtWgreWo0yc/SNUUyEkNG2I/AAAAAAAAAJk/NTaFb2z3nwU/s400/missing_square_animation%5B1%5D.gif" border="0" alt=""id="BLOGGER_PHOTO_ID_5248123791221791586" /&gt;&lt;/a&gt;&lt;br /&gt;&lt;br /&gt;&lt;u&gt;&lt;strong&gt;Proving that 2 = 1&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;&lt;br /&gt;Here is the version offered by Augustus De Morgan: Let x = 1. Then x² = x. So x² - 1 = x -1. Dividing both sides by x -1, we conclude that x + 1 = 1; that is, since x = 1, 2 = 1.&lt;br /&gt;&lt;br /&gt;Assume that&lt;br /&gt;&lt;br /&gt;a = b.    (1)&lt;br /&gt;&lt;br /&gt;Multiplying both sides by a,&lt;br /&gt;&lt;br /&gt;a² = ab.     (2)&lt;br /&gt;&lt;br /&gt;Subtracting b² from both sides,&lt;br /&gt;&lt;br /&gt;a² - b² = ab - b² .    (3)&lt;br /&gt;&lt;br /&gt;Factorizing both sides,&lt;br /&gt;&lt;br /&gt;(a + b)(a - b) = b(a - b).    (4)&lt;br /&gt;&lt;br /&gt;Dividing both sides by (a - b),&lt;br /&gt;&lt;br /&gt;a + b = b.    (5)&lt;br /&gt;&lt;br /&gt;If now we take a = b = 1, we conclude that 2 = 1. Or we can subtract b from both sides and conclude that a, which can be taken as any number, must be equal to zero. Or we can substitute b for a and conclude that any number is double itself. Our result can thus be interpreted in a number of ways, all equally ridiculous.&lt;br /&gt;&lt;br /&gt;The paradox arises from a disguised breach of the arithmetical prohibition on division by zero, occurring at (5): since a = b, dividing both sides by (a - b) is dividing by zero, which renders the equation meaningless. As Northrop goes on to show, the same trick can be used to prove, e.g., that any two unequal numbers are equal, or that all positive whole numbers are equal.&lt;br /&gt;&lt;br /&gt;Here is another example:&lt;br /&gt;&lt;br /&gt;Proving that 3 + 2 = 0&lt;br /&gt;&lt;br /&gt;Assume A + B = C, and assume A = 3 and B = 2.&lt;br /&gt;&lt;br /&gt;Multiply both sides of the equation A + B = C by (A + B).&lt;br /&gt;&lt;br /&gt;We obtain A² + 2AB + B² = C(A + B)&lt;br /&gt;&lt;br /&gt;Rearranging the terms we have&lt;br /&gt;&lt;br /&gt;A² + AB - AC = - AB - B² + BC&lt;br /&gt;&lt;br /&gt;Factoring out (A + B - C), we have&lt;br /&gt;&lt;br /&gt;A(A + B - C) = - B(A + B - C)&lt;br /&gt;&lt;br /&gt;Dividing both sides by (A + B - C), that is, dividing by zero, we get A = - B, or A + B = 0, which is evidently absurd.&lt;br /&gt;&lt;br /&gt;Proving that n = n + 1&lt;br /&gt;&lt;br /&gt;(a)   (n + 1)² = n² + 2n + 1&lt;br /&gt;&lt;br /&gt;(b)   (n + 1)² - (2n + 1) = n² &lt;br /&gt;&lt;br /&gt;(c)   Subtracting n(2n + 1) from both sides and factoring, we have&lt;br /&gt;&lt;br /&gt;(d)   (n + 1)² - (n + 1)(2n + 1) = n² - n(2n +1)&lt;br /&gt;&lt;br /&gt;(e)   Adding ¼(2n + 1)² to both sides of (d) yields&lt;br /&gt;&lt;br /&gt;(n + 1)² - (n + 1)(2n + 1) + ¼(2n + 1)² = n² - n(2n + 1) + ¼(2n + 1)² &lt;br /&gt;&lt;br /&gt;This may be written:&lt;br /&gt;&lt;br /&gt;(f)   [(n + 1) - ½(2n + 1)]² = [(n - ½(2n + 1)]² &lt;br /&gt;&lt;br /&gt;Taking square roots of both sides,&lt;br /&gt;&lt;br /&gt;(g)   n + 1 - ½(2n + 1) = n - ½(2n + 1)&lt;br /&gt;&lt;br /&gt;and, therefore,&lt;br /&gt;&lt;br /&gt;(h)   n = n + 1&lt;br /&gt;&lt;br /&gt;The trick here is to ignore the fact that there are two square roots for any positive number, one positive and one negative: the square roots of 4 are 2 and -2, which can be written as ±2. So (g) should properly read:&lt;br /&gt;&lt;br /&gt;±(n + 1 - ½(2n + 1)) = ±(n - ½(2n + 1))&lt;br /&gt;&lt;br /&gt;&lt;br /&gt;congratz youve juz wasted your life awayy........&lt;br /&gt;&lt;u&gt;&lt;strong&gt;Reference:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;Wilkins, G. (2006). &lt;em&gt;Some paradoxes - an anthology.&lt;/em&gt; Retrieved Setember 20, 2008, from &lt;a href="http://www.paradoxes.co.uk/"&gt;http://www.paradoxes.co.uk/&lt;/a&gt;&lt;div class="blogger-post-footer"&gt;&lt;img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/6058061522806863813-7305534024099058298?l=nuubcake.blogspot.com' alt='' /&gt;&lt;/div&gt;</content><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/6058061522806863813/posts/default/7305534024099058298'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/6058061522806863813/posts/default/7305534024099058298'/><link rel='alternate' type='text/html' href='http://nuubcake.blogspot.com/2008/09/paradox.html' title='Paradox'/><author><name>Memories of Nobody</name><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='http://img2.blogblog.com/img/b16-rounded.gif'/></author><media:thumbnail xmlns:media='http://search.yahoo.com/mrss/' url='http://4.bp.blogspot.com/_jtWgreWo0yc/SNUTk16RvKI/AAAAAAAAAIk/Fd_ZNUlpP0w/s72-c/smallescher-hands%5B1%5D.jpg' height='72' width='72'/></entry><entry><id>tag:blogger.com,1999:blog-6058061522806863813.post-3391627880765691975</id><published>2008-09-05T11:27:00.001+08:00</published><updated>2008-09-05T11:30:52.753+08:00</updated><title type='text'></title><content type='html'>&lt;a href="http://3.bp.blogspot.com/_jtWgreWo0yc/SMCnxZ2v2tI/AAAAAAAAAGs/B3bzY47kPTI/s1600-h/Emo.jpg"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://3.bp.blogspot.com/_jtWgreWo0yc/SMCnxZ2v2tI/AAAAAAAAAGs/B3bzY47kPTI/s400/Emo.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5242374433455200978" /&gt;&lt;/a&gt;&lt;br /&gt;&lt;a href="http://4.bp.blogspot.com/_jtWgreWo0yc/SMCnx8smr6I/AAAAAAAAAG0/P2JKrTqxFXU/s1600-h/emocat1.bmp"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://4.bp.blogspot.com/_jtWgreWo0yc/SMCnx8smr6I/AAAAAAAAAG0/P2JKrTqxFXU/s400/emocat1.bmp" border="0" alt=""id="BLOGGER_PHOTO_ID_5242374442807898018" /&gt;&lt;/a&gt;&lt;br /&gt;&lt;a href="http://4.bp.blogspot.com/_jtWgreWo0yc/SMCnyJttowI/AAAAAAAAAG8/jqsWItT_MW4/s1600-h/emocat.bmp"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://4.bp.blogspot.com/_jtWgreWo0yc/SMCnyJttowI/AAAAAAAAAG8/jqsWItT_MW4/s400/emocat.bmp" border="0" alt=""id="BLOGGER_PHOTO_ID_5242374446302208770" /&gt;&lt;/a&gt;&lt;br /&gt;&lt;a href="http://3.bp.blogspot.com/_jtWgreWo0yc/SMCnyRBHwhI/AAAAAAAAAHE/DXNVu0uoD6M/s1600-h/emocat.jpg"&gt;&lt;img style="display:block; 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margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 400px;" src="http://www.yinyangyo.ws/assets/images/yin-yang-yo-037.jpg" border="0" alt="" /&gt;&lt;/a&gt;&lt;br /&gt;&lt;a href="http://www.jungfreudlich.de/wp-content/uploads/spongebob.jpg"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 400px;" src="http://www.jungfreudlich.de/wp-content/uploads/spongebob.jpg" border="0" alt="" /&gt;&lt;/a&gt;&lt;br /&gt;&lt;a href="http://2.bp.blogspot.com/_jtWgreWo0yc/SLxXQKKHLII/AAAAAAAAAE0/4hfJuM0_tNs/s1600-h/dota.JPG"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://2.bp.blogspot.com/_jtWgreWo0yc/SLxXQKKHLII/AAAAAAAAAE0/4hfJuM0_tNs/s400/dota.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5241160001468705922" /&gt;&lt;/a&gt;&lt;div class="blogger-post-footer"&gt;&lt;img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/6058061522806863813-1953564958374585585?l=nuubcake.blogspot.com' alt='' /&gt;&lt;/div&gt;</content><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/6058061522806863813/posts/default/1953564958374585585'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/6058061522806863813/posts/default/1953564958374585585'/><link rel='alternate' type='text/html' href='http://nuubcake.blogspot.com/2008/09/randomcake.html' title='RANDOMCAKE!'/><author><name>Memories of Nobody</name><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='http://img2.blogblog.com/img/b16-rounded.gif'/></author><media:thumbnail xmlns:media='http://search.yahoo.com/mrss/' url='http://2.bp.blogspot.com/_jtWgreWo0yc/SLxXQKKHLII/AAAAAAAAAE0/4hfJuM0_tNs/s72-c/dota.JPG' height='72' width='72'/></entry><entry><id>tag:blogger.com,1999:blog-6058061522806863813.post-9006891109596624235</id><published>2008-04-29T16:55:00.007+08:00</published><updated>2008-09-02T03:53:19.210+08:00</updated><title type='text'>How to fold a straw heart?</title><content type='html'>&lt;a href="http://1.bp.blogspot.com/_jtWgreWo0yc/SLxIIFiih-I/AAAAAAAAAEs/WoU4d4KZd2k/s1600-h/strawheart.JPG"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://1.bp.blogspot.com/_jtWgreWo0yc/SLxIIFiih-I/AAAAAAAAAEs/WoU4d4KZd2k/s400/strawheart.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5241143370115614690" /&gt;&lt;/a&gt;&lt;br /&gt;Step 1:&lt;br /&gt;&lt;a href="http://1.bp.blogspot.com/_jtWgreWo0yc/SBbi5jXC1bI/AAAAAAAAACk/wITqlsMGkPc/s1600-h/step1.JPG"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://1.bp.blogspot.com/_jtWgreWo0yc/SBbi5jXC1bI/AAAAAAAAACk/wITqlsMGkPc/s200/step1.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5194588698590762418" /&gt;&lt;/a&gt;&lt;br /&gt;Step 2:&lt;br /&gt;&lt;a href="http://2.bp.blogspot.com/_jtWgreWo0yc/SBbi5zXC1cI/AAAAAAAAACs/HyMUi9H5G7o/s1600-h/step2.JPG"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://2.bp.blogspot.com/_jtWgreWo0yc/SBbi5zXC1cI/AAAAAAAAACs/HyMUi9H5G7o/s200/step2.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5194588702885729730" /&gt;&lt;/a&gt;&lt;br /&gt;Step 3:&lt;br /&gt;&lt;a href="http://3.bp.blogspot.com/_jtWgreWo0yc/SBbi7DXC1dI/AAAAAAAAAC0/KLfq2ewz6d0/s1600-h/step3.JPG"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://3.bp.blogspot.com/_jtWgreWo0yc/SBbi7DXC1dI/AAAAAAAAAC0/KLfq2ewz6d0/s200/step3.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5194588724360566226" /&gt;&lt;/a&gt;&lt;br /&gt;Step 4:&lt;br /&gt;&lt;a href="http://3.bp.blogspot.com/_jtWgreWo0yc/SBbi7DXC1eI/AAAAAAAAAC8/E3IGeeYMHcU/s1600-h/step4.JPG"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://3.bp.blogspot.com/_jtWgreWo0yc/SBbi7DXC1eI/AAAAAAAAAC8/E3IGeeYMHcU/s200/step4.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5194588724360566242" /&gt;&lt;/a&gt;&lt;br /&gt;Step 5:&lt;br /&gt;&lt;a href="http://4.bp.blogspot.com/_jtWgreWo0yc/SBbi7TXC1fI/AAAAAAAAADE/amgDzBL1Quo/s1600-h/step5.JPG"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://4.bp.blogspot.com/_jtWgreWo0yc/SBbi7TXC1fI/AAAAAAAAADE/amgDzBL1Quo/s200/step5.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5194588728655533554" /&gt;&lt;/a&gt;&lt;br /&gt;Step 6:&lt;br /&gt;&lt;a href="http://2.bp.blogspot.com/_jtWgreWo0yc/SBbjPzXC1gI/AAAAAAAAADM/CIVMfKZddok/s1600-h/step6.JPG"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://2.bp.blogspot.com/_jtWgreWo0yc/SBbjPzXC1gI/AAAAAAAAADM/CIVMfKZddok/s320/step6.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5194589080842851842" /&gt;&lt;/a&gt;&lt;br /&gt;Step 7:&lt;br /&gt;&lt;a href="http://4.bp.blogspot.com/_jtWgreWo0yc/SBbjQTXC1hI/AAAAAAAAADU/Pn_cW-65A0o/s1600-h/step7.JPG"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://4.bp.blogspot.com/_jtWgreWo0yc/SBbjQTXC1hI/AAAAAAAAADU/Pn_cW-65A0o/s320/step7.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5194589089432786450" /&gt;&lt;/a&gt;&lt;br /&gt;Step 8:&lt;br /&gt;&lt;a href="http://4.bp.blogspot.com/_jtWgreWo0yc/SBbjQTXC1iI/AAAAAAAAADc/Z1ovCLKFJXI/s1600-h/step8.JPG"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://4.bp.blogspot.com/_jtWgreWo0yc/SBbjQTXC1iI/AAAAAAAAADc/Z1ovCLKFJXI/s320/step8.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5194589089432786466" /&gt;&lt;/a&gt;&lt;br /&gt;Step 9:&lt;br /&gt;&lt;a href="http://1.bp.blogspot.com/_jtWgreWo0yc/SBbjQjXC1jI/AAAAAAAAADk/eM8zNeCvgUY/s1600-h/step9.JPG"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://1.bp.blogspot.com/_jtWgreWo0yc/SBbjQjXC1jI/AAAAAAAAADk/eM8zNeCvgUY/s320/step9.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5194589093727753778" /&gt;&lt;/a&gt;&lt;br /&gt;Step 10:&lt;br /&gt;&lt;a href="http://1.bp.blogspot.com/_jtWgreWo0yc/SBbjQjXC1kI/AAAAAAAAADs/nB8znmbDT4Q/s1600-h/step10.JPG"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://1.bp.blogspot.com/_jtWgreWo0yc/SBbjQjXC1kI/AAAAAAAAADs/nB8znmbDT4Q/s320/step10.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5194589093727753794" /&gt;&lt;/a&gt;&lt;br /&gt;Step 11:&lt;br /&gt;&lt;a href="http://3.bp.blogspot.com/_jtWgreWo0yc/SBbjqDXC1lI/AAAAAAAAAD0/wR1gDg6-cok/s1600-h/step11.JPG"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://3.bp.blogspot.com/_jtWgreWo0yc/SBbjqDXC1lI/AAAAAAAAAD0/wR1gDg6-cok/s320/step11.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5194589531814418002" /&gt;&lt;/a&gt;&lt;br /&gt;&lt;a href="http://1.bp.blogspot.com/_jtWgreWo0yc/SLxIIFiih-I/AAAAAAAAAEs/WoU4d4KZd2k/s1600-h/strawheart.JPG"&gt;&lt;img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://1.bp.blogspot.com/_jtWgreWo0yc/SLxIIFiih-I/AAAAAAAAAEs/WoU4d4KZd2k/s400/strawheart.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5241143370115614690" /&gt;&lt;/a&gt;&lt;div class="blogger-post-footer"&gt;&lt;img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/6058061522806863813-9006891109596624235?l=nuubcake.blogspot.com' alt='' /&gt;&lt;/div&gt;</content><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/6058061522806863813/posts/default/9006891109596624235'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/6058061522806863813/posts/default/9006891109596624235'/><link rel='alternate' type='text/html' href='http://nuubcake.blogspot.com/2008/04/how-to-fold-straw-heart.html' title='How to fold a straw heart?'/><author><name>Memories of Nobody</name><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='http://img2.blogblog.com/img/b16-rounded.gif'/></author><media:thumbnail xmlns:media='http://search.yahoo.com/mrss/' url='http://1.bp.blogspot.com/_jtWgreWo0yc/SLxIIFiih-I/AAAAAAAAAEs/WoU4d4KZd2k/s72-c/strawheart.JPG' height='72' width='72'/></entry></feed>
