tag:blogger.com,1999:blog-65277276672434635172024-03-24T00:09:48.066-07:00Fun with num3ersBenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.comBlogger689125tag:blogger.com,1999:blog-6527727667243463517.post-18843217017373069762011-06-17T16:46:00.000-07:002011-06-17T16:49:58.084-07:00Here's Why Our Long-Term Growth Expectations Are Absurd<span style="font-style:italic;">Investor Jeremy Grantham of GMO illustrates this in his recent apocalyptic commodity analysis with an anecdote about ancient Egypt, which was one of the most successful civilizations in human history.<br /><br />Grantham asked a group of mathematicians how big Ancient Eqypt would have gotten if its economy had growth 4.5% a year for the 3,000 years the civilization lasted.<br />The mathematicians were directionally correct: Very big.<br /><br />But not one of them came even remotely close to the actual number (which is mind-boggling).</span><div><span style="font-style:italic;"><br /></span></div><div><span style="font-style:italic;"><a href="http://www.businessinsider.com/heres-why-our-long-term-growth-expectations-are-absurd-2011-6">http://www.businessinsider.com/heres-why-our-long-term-growth-expectations-are-absurd-2011-6</a></span></div><div><br /></div><div><span class="Apple-style-span" style="color: rgb(51, 51, 51); font-family: 'Lucida Grande', sans-serif; font-size: 14px; line-height: 15px; ">I posted on @<a class="tweet-url username" href="http://twitter.com/Quora" rel="nofollow" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-decoration: none; color: rgb(0, 132, 180); ">Quora</a>: Here's Why Our Long-Term Growth Expectations Are Absurd <a href="http://qr.ae/7Xjhm" class="tweet-url web" rel="nofollow" target="_blank" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-decoration: underline; color: rgb(0, 132, 180); ">http://qr.ae/7Xjhm</a> <a href="http://twitter.com/search?q=%23Math" title="#Math" class="tweet-url hashtag" rel="nofollow" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-decoration: none; color: rgb(0, 132, 180); ">#Math</a> <a href="http://twitter.com/search?q=%23Business" title="#Business" class="tweet-url hashtag" rel="nofollow" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-decoration: none; color: rgb(0, 132, 180); ">#Business</a></span></div><div><br /></div><div><br /></div>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com1tag:blogger.com,1999:blog-6527727667243463517.post-12195397230931182172011-06-17T16:43:00.000-07:002011-06-17T16:45:57.141-07:00What is the difference between causation and correlation?<a href="http://stats.org/in_depth/faq/causation_correlation.htm">http://stats.org/in_depth/faq/causation_correlation.htm</a><div><br /></div><div><span class="Apple-style-span" style="color: rgb(51, 51, 51); font-family: Georgia, 'Bitstream Charter', serif; line-height: 24px; "><h1 class="entry-title" style="background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: transparent; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; vertical-align: baseline; clear: both; font-weight: bold; font-family: 'Helvetica Neue', Arial, Helvetica, 'Nimbus Sans L', sans-serif; color: rgb(0, 0, 0); line-height: 1.3em; font-size: 21px; background-position: initial initial; background-repeat: initial initial; ">Selection of Best Beer Commercials</h1><div><a href="http://benvitalis.wordpress.com/2011/06/17/selection-of-best-beer-commercials/">http://benvitalis.wordpress.com/2011/06/17/selection-of-best-beer-commercials/</a></div><div><br /></div><div><br /></div></span></div>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com1tag:blogger.com,1999:blog-6527727667243463517.post-15576897955061117682011-06-03T21:44:00.000-07:002011-06-03T21:45:48.152-07:00Chess Puzzle<span style="font-weight:bold;">A game begins with 1.e4 and ends in the fifth move with knight takes rook mate.</span><div><span style="font-weight:bold;"><br /></span></div><div><span style="font-weight:bold;"><a href="http://www.chessbase.com/puzzle/puzz05d.htm">http://www.chessbase.com/puzzle/puzz05d.htm</a></span></div><div><br /></div><div><br /></div><div><br /></div>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com1tag:blogger.com,1999:blog-6527727667243463517.post-39342535423500742562011-05-25T19:34:00.000-07:002011-05-25T19:36:06.539-07:00Ratio of triangle ABC to BCE ?<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://i196.photobucket.com/albums/aa270/SkateGG/Untitled-37.jpg"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 575px; height: 411px;" src="http://i196.photobucket.com/albums/aa270/SkateGG/Untitled-37.jpg" border="0" alt="" /></a>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com1tag:blogger.com,1999:blog-6527727667243463517.post-79497037699615541082011-05-25T19:16:00.000-07:002011-05-25T19:20:04.958-07:00What's the area of a five-pointed star?<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://upload.wikimedia.org/wikipedia/commons/1/18/Five-pointed_star.svg"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 382px; height: 360px;" src="http://upload.wikimedia.org/wikipedia/commons/1/18/Five-pointed_star.svg" border="0" alt="" /></a>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com2tag:blogger.com,1999:blog-6527727667243463517.post-34702048155495304432011-05-24T22:07:00.000-07:002011-05-24T22:08:25.899-07:00Buffon's Needle Problem<a href="http://www.gaussianmath.com/misc/buffons/buffon.html">http://www.gaussianmath.com/misc/buffons/buffon.html</a><div><br /></div><br /><iframe width="480" height="390" src="http://www.youtube.com/embed/Vws1jvMbs64" frameborder="0" allowfullscreen></iframe>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-57518874918373666222011-05-24T21:49:00.000-07:002011-05-24T22:05:11.019-07:00MOBIUS FUNCTION<a href="http://www.gaussianmath.com/misc/mobiusfunction/mobius.html">http://www.gaussianmath.com/misc/mobiusfunction/mobius.html</a><div><br /></div><div><br /></div><span style="font-weight:bold;">Introduction to the Möbius Function</span><div><b><a href="http://www.gaussianmath.com/misc/mobiusfunction/intro/intromobius.html">http://www.gaussianmath.com/misc/mobiusfunction/intro/intromobius.html</a></b></div><div><span></span><b><br /></b><iframe width="480" height="390" src="http://www.youtube.com/embed/zlRm1Lnz6fg" frameborder="0" allowfullscreen=""></iframe></div><br /><br /><span style="font-weight:bold;">Hat 0 </span><div><span style="font-weight:bold;"><a href="http://www.gaussianmath.com/misc/mobiusfunction/hat0/hat0.html">http://www.gaussianmath.com/misc/mobiusfunction/hat0/hat0.html</a></span></div><div><br /></div><iframe width="480" height="390" src="http://www.youtube.com/embed/ldIk2qZtgRQ" frameborder="0" allowfullscreen=""></iframe><br /><br /><span style="font-weight:bold;">Fundamental Theorem of Arithmetic</span><br /><div><span style="font-weight:bold;"><a href="http://www.gaussianmath.com/misc/mobiusfunction/fundtheoremarith/fundtheoremarith.html">http://www.gaussianmath.com/misc/mobiusfunction/fundtheoremarith/fundtheoremarith.html</a></span></div><iframe width="480" height="390" src="http://www.youtube.com/embed/_3XTNFTlWdc" frameborder="0" allowfullscreen=""></iframe><br /><span style="font-weight:bold;"><br /></span><div><span style="font-weight:bold;">Uniqueness of Factorization</span></div><div><span style="font-weight:bold;"><a href="http://www.gaussianmath.com/misc/mobiusfunction/uniquefactor/uniquefactor.html">http://www.gaussianmath.com/misc/mobiusfunction/uniquefactor/uniquefactor.html</a></span></div><iframe width="480" height="390" src="http://www.youtube.com/embed/Oavn5ISWTlU" frameborder="0" allowfullscreen=""></iframe><br /><br /><span style="font-weight:bold;">Mertens Function</span><div><span style="font-weight:bold;"><a href="http://www.gaussianmath.com/misc/mobiusfunction/mertens/mertens.html">http://www.gaussianmath.com/misc/mobiusfunction/mertens/mertens.html</a></span></div><div><br /></div><iframe width="480" height="390" src="http://www.youtube.com/embed/yiyuu9HiXUI" frameborder="0" allowfullscreen=""></iframe><br /><br /><span style="font-weight:bold;">Mertens Conjecture</span><div><span style="font-weight:bold;"><a href="http://www.gaussianmath.com/misc/mobiusfunction/mertenconj/mertenconj.html">http://www.gaussianmath.com/misc/mobiusfunction/mertenconj/mertenconj.html</a></span></div><br /><iframe width="480" height="390" src="http://www.youtube.com/embed/jjuNYAkPVTY" frameborder="0" allowfullscreen=""></iframe><br /><br /><span style="font-weight:bold;">Identities and Mobidromes </span><div><span style="font-weight:bold;"><a href="http://www.gaussianmath.com/misc/mobiusfunction/identities/identities.html">http://www.gaussianmath.com/misc/mobiusfunction/identities/identities.html</a></span></div><div><br /></div><iframe width="480" height="390" src="http://www.youtube.com/embed/KUv6emqNAkg" frameborder="0" allowfullscreen></iframe>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-38566569235696174572011-05-24T21:38:00.000-07:002011-05-24T21:43:35.784-07:00Mathematical Wormhole 1 & 2Time travel is possible in mathematics!<br /><br />Check out ... <a href="http://www.gaussianmath.com/">http://www.gaussianmath.com/</a><br />for a more indepth explanation and other interesting topics in mathematics<br /><br /><iframe width="480" height="390" src="http://www.youtube.com/embed/l3ZUW0LYUD0" frameborder="0" allowfullscreen=""></iframe><br /><iframe width="480" height="390" src="http://www.youtube.com/embed/nvW6XtnMeP4" frameborder="0" allowfullscreen=""></iframe>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-68181781359076401502011-05-24T14:50:00.000-07:002011-05-24T15:00:34.568-07:00The M&M game<span style="font-style:italic;">It has been said that, "Life is like a box of chocolates—you never know what you're going to get." (Forrest Gump in Forrest Gump, 1994.) In this experiment you can test the "Forrest Gump Chaos Theory" by using M&M's, which are much cheaper than a box of chocolates. What if life is more like a bag of M&M's?</span><br /><br /><span style="font-weight:bold;">We start with two bags of M&Ms, one with 19 and one with 20. </span><div><span style="font-weight:bold;">In her turn, a player has to eat all the M&Ms in one of the bags, and then split the M&M from the other bag between the two, leaving at least one on each bag, and not necessarily evenly. The she gives the two bags to the other player. The player who receives two bags with one M&M each loses as he can no longer move.</span></div><div><span style="font-weight:bold;"><br /></span></div><div><span style="font-weight:bold;"><a href="http://www.mathcamp.org/gettoknowmathcamp/academics/archives/Alfonso-CGT-handout.pdf">http://www.mathcamp.org/gettoknowmathcamp/academics/archives/Alfonso-CGT-handout.pdf</a></span></div><div><br /></div><div><br /></div>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-22656938871713981032011-05-24T13:47:00.001-07:002011-05-24T13:47:55.428-07:00Farey Sums and Dedekind Sums By Kurt Girstmair<a href="http://www.jstor.org/pss/10.4169/amermathmont.117.1.72">http://www.jstor.org/pss/10.4169/amermathmont.117.1.72</a><div><br /></div><div><br /></div>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-8008076615339721612011-05-24T11:25:00.001-07:002011-05-24T11:32:19.289-07:00Mathematical Platonism<a href="http://www.lehman.edu/deanhum/philosophy/platofootnote/PlatoFootnote.org/Outreach_files/Mathematical%20Platonism.pdf">http://www.lehman.edu/deanhum/philosophy/platofootnote/PlatoFootnote.org/Outreach_files/Mathematical%20Platonism.pdf</a><br /><br />Read:<br /><span style="font-weight:bold;">James Robert Brown’s Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures (Routledge, 2008).</span><br /><br /><br /><span style="font-style:italic;">There is a difference between general Platonism and the mathematical flavor. For Plato, each apple, say, is but an imperfect example of the absolute (and perfect) Idea of an apple. But as Aristotle quickly realized, Plato has it exactly backwards: we arrive at the general idea of ‘apple’ by mentally abstracting a set of characteristics we think common to all actual apples. It is we who conjure the ‘perfect’ idea from the world, not the world copying the concept.<br /><br /></span><div><span style="font-style:italic;">But now contrast the idea of an apple with the idea of a circle. Here Aristotle’s approach becomes more problematic, as we don’t find any true circles in nature. No natural object has the precise geometric characteristics of a circle, and in a very strong sense we can also say that the circles we draw are but imperfect representations of the perfect idea of a circle. Ah – but whence does such a perfect idea come from?</span><div><span style="font-style:italic;"><br />Consider another way to put the problem. One major difference between science and technology is that science discovers things, while technology is about human inventions. We discover the law of gravity; but we invent airplanes to allow heavier-than-air flight despite the law of gravity. But where do mathematical objects, like circles and numbers, or mathematical theorems like the Pythagorean one, or Fermat’s Last one, come from? Are they inventions of the human mind, or are they discoveries?</span></div></div>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-56122749930989004342011-05-23T23:58:00.000-07:002011-05-24T00:00:25.190-07:00Which of the following statements are true?<span style="font-weight:bold;">1. Precisely one of these statements is untrue.<br /><br />2. Precisely two of these statements are untrue.<br /><br />3. Precisely three of these statements are untrue.<br /><br />4. Precisely four of these statements are untrue.<br /><br />5. Precisely five of these statements are untrue.<br /><br />6. Precisely six of these statements are untrue.<br /><br />7. Precisely seven of these statements are untrue.<br /><br />8. Precisely eight of these statements are untrue.<br /><br />9. Precisely nine of these statements are untrue.<br /><br />10. Precisely ten of these statements are untrue.</span>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-42568378713931656252011-05-23T12:27:00.000-07:002011-05-23T12:31:57.286-07:00Visual.ly<span style="font-weight:bold;">visual.ly is launching soon!</span> <div><span style="font-weight:bold;"><br /></span></div><div><span style="font-weight:bold;"><a href="http://visual.ly/ihoq1">http://visual.ly/ihoq1</a></span></div><div><br /></div><div><br /></div><div><br /></div><iframe width="640" height="390" src="http://www.youtube.com/embed/AiVKfNeRbPQ" frameborder="0" allowfullscreen></iframe>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-36139687351230764482011-05-23T12:26:00.001-07:002011-05-23T12:26:35.188-07:00Tweepi<a href="http://tweepi.com/?2">http://tweepi.com/?2</a><div><br /></div><div><br /></div>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-28688440101652990132011-05-23T12:24:00.000-07:002011-05-23T12:25:07.771-07:00Principles of Uncertainty<a href="http://uncertainty.stat.cmu.edu/">http://uncertainty.stat.cmu.edu/</a><div><br /></div><div><br /></div><div><br /></div>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-62290395812537449002011-05-23T12:22:00.000-07:002011-05-23T12:23:10.348-07:00Math Problems, Episode 8: Corn Mo<iframe width="540" height="390" src="http://www.youtube.com/embed/mJXkGAE3WeM" frameborder="0" allowfullscreen></iframe>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-5560012762231311962011-05-22T17:01:00.000-07:002011-05-22T17:23:10.046-07:00Sum of two dice2 .................. 1+1 .................................................1/36<div>3 ...................1+2, 2+1 ..........................................2/36<br />4 ...................1+3, 2+2, 3+1 ..................................3/36<br />5 ...................1+4, 2+3, 3+2, 4+1 .........................4/36<br />6 ...................1+5, 2+4, 3+3, 4+2, 5+1 .................5/36<br />7 ...................1+6, 2+5, 3+4, 4+3, 5+2, 6+1 ..........6/36<br />8 ...................2+6, 3+5, 4+4, 5+3, 6+2 .................5/36<br />9 ...................3+6, 4+5, 5+4, 6+3 ........................4/36<br />10 ..................4+6, 5+5, 6+4 ................................3/36<br />11 ..................5+6, 6+5 .........................................2/36<br />12 ..................6+6 ..................................................1/36<br /><br /><span style="font-weight:bold;">Is it possible to load the dice in such a way that these eleven scores are equally probable?</span></div>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-34492869706214123742011-05-22T16:31:00.000-07:002011-05-22T16:33:47.998-07:00A clock puzzle<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://www.toothpastefordinner.com/010508/clocks-with-no-hands.gif"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 388px; height: 636px;" src="http://www.toothpastefordinner.com/010508/clocks-with-no-hands.gif" border="0" alt="" /></a><br /><br /><span style="font-weight:bold;">Imagine a clock that does not have any hands or numbers on it, but it has a chimer.<br /><br />If the time is 1 o'clock, it chimes once. If the time is 2 o'clock, it chimes twice, and so forth. The time gap between any two chimes is 3 seconds.<br /><br />How many seconds would it take you to know the time, after the first chime is heard, if it is 3 o'clock?</span>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-17052528693202473662011-05-22T16:08:00.000-07:002011-05-22T16:18:16.199-07:00Game show: 5 fortune tellers<span style="font-weight:bold;">On a game show there are 5 fortune tellers (A, B, C, D and E)<br /><br />A has 81% chance of being correct<br />B ..... 65% .........................................<br />C ..... 43% .........................................<br />D...... 35% .........................................<br />E ...... 8% ............................................<br /><br />If you were trying to get the best prediction of your future, which fortune teller would you go to?</span>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-25837195262367821192011-05-22T16:00:00.000-07:002011-05-22T16:05:53.287-07:00n times the sum of its digits<span style="font-weight:bold;">Which number is 5 times the sum of its digits?<br /><br />In general, to find 2-digit and 3-digit numbers that are (2, 3, 4, 5, ...) times the sum of their digits<br /><br />And any d-digits (d > 3)?</span>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-75145148387656097832011-05-21T10:26:00.000-07:002011-05-21T10:32:31.457-07:00n circles with radius r ...<span style="font-weight:bold;">... are inscribed in a circle with radius 1</span><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnh8PPT0jqnjLPI326OHVE4afthQaT0ri8O4xaMHHDC3wn-IpoKOHY4e7ZHaBMbw_itNxagnaX2ezaBw74EYskHbUibBGCwTLwUPC_eyhCC-wxukSNVbtMjJ6c1HsQDO5YlamarKMdwG0/s1600/img5.png"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 238px; height: 258px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnh8PPT0jqnjLPI326OHVE4afthQaT0ri8O4xaMHHDC3wn-IpoKOHY4e7ZHaBMbw_itNxagnaX2ezaBw74EYskHbUibBGCwTLwUPC_eyhCC-wxukSNVbtMjJ6c1HsQDO5YlamarKMdwG0/s400/img5.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5609222423581161410" /></a><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAh0IjAiEVN4-ta4WBeezeetZUaKyHcTTmmHiogXOsmTVYgyVh_HLmcee1lnO9tkqlMvUk226zlR_x6_Q3d5-18v6bUIK07amsDRckGPhjYK2W-WPORgXn0uW0i64dAKgdmKkpduMU7_M/s1600/img4.png"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 238px; height: 258px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAh0IjAiEVN4-ta4WBeezeetZUaKyHcTTmmHiogXOsmTVYgyVh_HLmcee1lnO9tkqlMvUk226zlR_x6_Q3d5-18v6bUIK07amsDRckGPhjYK2W-WPORgXn0uW0i64dAKgdmKkpduMU7_M/s400/img4.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5609222220632666562" /></a><br /><br /><span style="font-weight:bold;">Determine the radius r and the area A of the shaded region between the circles for n = 4 and n = 3</span><br /><br /><span style="font-weight:bold;">Hint:</span> <span style="font-style:italic;"> In an equilateral triangle the medians divide each other in the ratio of 2:1</span>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com4tag:blogger.com,1999:blog-6527727667243463517.post-52814045786480533742011-05-15T15:35:00.000-07:002011-05-15T15:56:27.442-07:00Smallest positive integer that includes among its factors perfect squares<span><b>1^2 = 1 ..... 2^2 = 4 ..... 3^2 = 9</b><br /><br /><b>4^2 = 16 ... 5^2 = 25 ... 6^2 = 36 ... 7^2 = 49 ... 8^2 = 64</b><br /><br /><div style="font-weight: bold; ">10^2 = 100 ..... 11^2 = 121 ..... 12^2 = 144 ..... 13^2 = 169 ..... 14^2 = 196</div><div><b>15^2 = 225 ..... 16^2 = 256 ..... 17^2 = 289 ..... 18^2 = 324 ..... 19^2 = 361</b></div><div><b>20^2 = 400 ..... 21^2 = 441 ..... 22^2 = 484 ..... </b><span class="Apple-style-span" style="font-weight: bold; ">23^2 = 529 ..... 24^2 = 576 </span></div><div><b>25^2 = 625 ..... 26^2 = 676 ..... 27^2 = 729 ..... 28^2 = 784 ..... 29^2 = 841</b></div><div><b>30^2 = 900 ..... 31^2 = 961 </b></div></span><br /><br /><br />The factors of <span style="font-weight:bold;">2304</span> are:<br /><br />1 2 3 4 6 8 9 12 16 18 24 32 36 48 64 72 96 128 144 192 256 288 384 576 768 1152 2304<br /><br />The prime factors are: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 or 2^8 * 3^2<br /><br /><span style="font-weight:bold;">2304</span> is the smallest positive integer that includes among its factors at least 10 perfect squares.BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-58903897780810480412011-05-15T13:24:00.001-07:002011-05-15T13:26:45.322-07:00(sqrt(n)^sqrt(n))^sqrt(n) an integer?<a href="http://www.wolframalpha.com/input/?i=%28sqrt%28n%29^sqrt%28n%29%29^Sqrt%28n%29">http://www.wolframalpha.com/input/?i=%28sqrt%28n%29^sqrt%28n%29%29^Sqrt%28n%29</a><div><br /></div><div><br /></div><span style="font-weight:bold;">For how many integral values of n in the range 1 to 1000 (inclusive) is the following statement true?</span>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-65314243241479120582011-05-15T12:46:00.000-07:002011-05-15T18:56:04.556-07:00Calendar: July : 5 Fridays, 5 Saturdays and 5 Sundays<div style="text-align: center;"><span style="font-weight:bold;">How often do we have this combination?</span></div><div style="text-align: left;"><span style="font-weight:bold;"><br /></span></div><div><br /></div><div style="text-align: center;"><span style="font-weight:bold;">July</span></div><div>2005</div><div><a href="http://www.palestinehistory.com/issues/images/05cal.jpg">http://www.palestinehistory.com/issues/images/05cal.jpg</a></div><div><br /></div><div>2011</div><div><a href="http://www.thfire.com/wp-content/uploads/2010/12/2011_calendar.jpg">http://www.thfire.com/wp-content/uploads/2010/12/2011_calendar.jpg</a></div><div><br /></div><div>2016</div><div><a href="http://www.printfree.com/Calendar_files/YearlyDecorativeCalendars/PlainWhite/2016.gif">http://www.printfree.com/Calendar_files/YearlyDecorativeCalendars/PlainWhite/2016.gif</a></div><div><br /></div><div>2022</div><div><a href="http://www.printfree.com/calendar_files/yearlydecorativecalendars/plainwhite/2022.gif">http://www.printfree.com/calendar_files/yearlydecorativecalendars/plainwhite/2022.gif</a></div><div><br /></div><div>2033</div><div><a href="http://www.printfree.com/Calendar_files/YearlyDecorativeCalendars/PlainWhite/2033plain.gif">http://www.printfree.com/Calendar_files/YearlyDecorativeCalendars/PlainWhite/2033plain.gif</a></div><div><br /></div><div><br /></div><div><br /></div><div>Other months, for example,</div><div><br /></div><div style="text-align: center;"><span style="font-weight:bold;">Octobre</span></div><div>2004</div><div><a href="http://www.bygsoftware.com/images/BygCalendar2004.gif">http://www.bygsoftware.com/images/BygCalendar2004.gif</a></div><div><a href="http://www.bygsoftware.com/images/BygCalendar2004.gif"></a>2010</div><div><div><a href="http://www.generalblue.com/Portals/0/Image_2010_Calendar_Template_BW.PNG">http://www.generalblue.com/Portals/0/Image_2010_Calendar_Template_BW.PNG</a></div></div><div><br /></div><div><br /></div>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0tag:blogger.com,1999:blog-6527727667243463517.post-67303285312375575182011-05-15T11:24:00.000-07:002011-05-15T11:26:34.922-07:00Prime factoization<span style="font-weight:bold;">Find the smallest positive integer with exactly n di fferent factors</span><div><span style="font-weight:bold;"><br /></span></div><div><span style="font-weight:bold;"><a href="http://math.berkeley.edu/~williams/55/hw3sol.pdf">http://math.berkeley.edu/~williams/55/hw3sol.pdf</a></span></div><div><br /></div><div><br /></div>BenVitalehttp://www.blogger.com/profile/08252218231924085041noreply@blogger.com0