Parabola  Issue 3
https://www.parabola.unsw.edu.au/20102019/volume482012/issue3
en

Volume 48 Issue 3 Header
https://www.parabola.unsw.edu.au/content/volume48issue3header
<section class="field fieldnamefieldnavpicvolumeissue fieldtypetaxonomytermreference fieldlabelabove viewmoderss"><h2 class="fieldlabel">Volume/Issue: </h2><ul class="fielditems"><li class="fielditem even"><a href="/20102019/volume482012/issue3">Issue 3</a></li></ul></section><section class="field fieldnamefieldnavpicimage fieldtypeimage fieldlabelabove viewmoderss"><h2 class="fieldlabel">Image: </h2><div class="fielditems"><figure class="clearfix fielditem even"><img class="imagestylevolumeissueheaderimage" src="https://www.parabola.unsw.edu.au/files/styles/volume_issue_header_image/public/promotional_images/Gateshead%20bridgeVol48_2.jpg?itok=4IqzGuY" width="640" height="250" alt="" /></figure></div></section><section class="field fieldnamefieldnavpicvolumenumber fieldtypetaxonomytermreference fieldlabelabove viewmoderss"><h2 class="fieldlabel">Volume Number: </h2><ul class="fielditems"><li class="fielditem even"><a href="/volume/volume48">Volume 48</a></li></ul></section>
Tue, 11 Feb 2014 04:41:06 +0000
z9803847
247 at https://www.parabola.unsw.edu.au

Solutions to Problems 1391  1400
https://www.parabola.unsw.edu.au/20102019/volume482012/issue3/article/solutionsproblems13911400
<div class="field fieldnamefieldarticleauthor fieldtypetext fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even">Various</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even"><div><strong>Q1391 </strong>Jack looked at the clock next to his front door as he left home one afternoon to visit Jill and watch a TV programme. Arriving exactly as the programme started, he set out for home again when it finished one hour later. As he did so he looked at her clock and noticed that it showed the same time as his had done when he left home. Puzzling over how Jill's clock could be so wrong, Jack travelled home at half the speed of his earlier journey. When he arrived home he saw from his clock that the whole expedition had taken two hours and fifteen minutes. He still hadn't worked out about Jill's clock and so he called her up on the phone. Jill explained that her clock was actually correct (as was Jack's), but it was an ``anticlockwise clock'' on which the hands travel in the opposite direction from usual. Jack had been in such a hurry to leave that he hadn't noticed the numbers on the clock face going the ``wrong'' way around the dial. At what time did Jack leave home?</div><div><span style="lineheight: 1.5;"> </span></div><div><strong>SOLUTION </strong>Jack's travel time home was twice that of his outward journey; the total travel time, plus the $60$ minutes' visit, adds up to $2$ hours $15$ minutes. So Jack's outward travel time was $25$ minutes, and the time when he looked at Jill's clock was $1$ hour $25$ minutes after looking at his own. Since it then appeared to show the time at which Jack left, double this time plus the $1$ hour $25$ minute stay must add up to $12$ hours (if you have trouble seeing why, draw pictures of the two clocks). So the time at which Jack left was 5:17:30.</div><div> </div><div><strong>Comment</strong>. The times involved could have added up to $24$ hours instead of $12$, in which case Jack's departure time would have been 11:17:30. However this does not fit in with the information that he ``left home one afternoon''.</div></div></div></div><div class="field fieldnamefieldarticlearticlepdf fieldtypefile fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even"><a href="https://www.parabola.unsw.edu.au/files/articles/20102019/volume482012/issue3/vol48_no3_s.pdf">Click here to download the PDF file</a><iframe id="pdf_reader" src="https://docs.google.com/viewer?embedded=true&url=https%3A%2F%2Fwww.parabola.unsw.edu.au%2Ffiles%2Farticles%2F20102019%2Fvolume482012%2Fissue3%2Fvol48_no3_s.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>
Tue, 20 Aug 2013 08:24:31 +0000
fcuadmin
111 at https://www.parabola.unsw.edu.au

Problems Section: Problems 1401  1410
https://www.parabola.unsw.edu.au/20102019/volume482012/issue3/article/problemssectionproblems14011410
<div class="field fieldnamefieldarticleauthor fieldtypetext fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even">Various</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even"><div><strong>Q1401 </strong>Solve the recurrence relation</div>
<div>$$a(n)=6a(n1)9a(n2)\ ,\quad a(0)=2\ ,\quad a(1)=21\ .$$</div>
<div><strong>Comment</strong>. You can use the same method as in previous problems  see, for example, the solution to problem~1393  but at one point you will find that things are a little different.</div>
<div> </div>
<div><strong>Q1402 </strong>Suppose that the three lines</div>
<div>$$y=ax+b\quad\hbox{and}\quad y=cx+d\quad\hbox{and}\quad y=ex+f$$</div>
<div>all have different gradients. Find conditions on $a,b,c,d,e,f$ for the lines to intersect in a single point.</div>
<div> </div>
<div><strong>Q1403 </strong>Seven different real numbers are given. Prove that there are two of them, say $x$ and $y$, for which</div>
<div>$$\frac{1+xy}{xy}$$</div>
<div>is greater than $\sqrt3\,$.</div>
<div> </div>
<div><strong>Q1404 </strong>In the game of poker, a pack of cards (consisting of the usual $52$ cards) is shuffled and five cards are dealt to each player. A hand is referred to as ``four of a kind'' if it contains four cards of the same value and one other card. For example, $\spadesuit7,\,\heartsuit7,\,\diamondsuit7,\,\clubsuit7,\,\diamondsuit{\rm J}$ constitutes four of a kind. Suppose that I deal two fivecard hands from the same pack, one to myself and one to my opponent.</div></div></div></div><div class="field fieldnamefieldarticlearticlepdf fieldtypefile fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even"><a href="https://www.parabola.unsw.edu.au/files/articles/20102019/volume482012/issue3/vol48_no3_p.pdf">Click here to download the PDF file</a><iframe id="pdf_reader" src="https://docs.google.com/viewer?embedded=true&url=https%3A%2F%2Fwww.parabola.unsw.edu.au%2Ffiles%2Farticles%2F20102019%2Fvolume482012%2Fissue3%2Fvol48_no3_p.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>
Tue, 20 Aug 2013 08:21:18 +0000
fcuadmin
110 at https://www.parabola.unsw.edu.au

UNSW School Mathematics Competition Winners 2012
https://www.parabola.unsw.edu.au/20102019/volume482012/issue3/article/unswschoolmathematicscompetitionwinners2012
<div class="field fieldnamefieldarticleauthor fieldtypetext fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even">Editor</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even"><div style="textalign: center;"><strong>Competition Winners â€“ Senior Division</strong></div><div><strong>First Prize</strong></div><div><span style="lineheight: 1.5;">John Papantoniou Sydney Grammar School</span></div><div> </div><div><strong>Second Prize</strong></div><div><span style="lineheight: 1.5;">Jonathan Zheng </span>James Ruse Agricultural High School</div><div> </div><div><strong>Third Prize </strong></div><div>Nancy Fu <span style="lineheight: 1.5;">James Ruse Agricultural High School</span></div><div>Frank Fan Shore School</div></div></div></div><div class="field fieldnamefieldarticlearticlepdf fieldtypefile fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even"><a href="https://www.parabola.unsw.edu.au/files/articles/20102019/volume482012/issue3/vol48_no3_w.pdf">Click here to download the PDF file</a><iframe id="pdf_reader" src="https://docs.google.com/viewer?embedded=true&url=https%3A%2F%2Fwww.parabola.unsw.edu.au%2Ffiles%2Farticles%2F20102019%2Fvolume482012%2Fissue3%2Fvol48_no3_w.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>
Tue, 20 Aug 2013 08:19:46 +0000
fcuadmin
109 at https://www.parabola.unsw.edu.au

UNSW School Mathematics Competition Problems 2012
https://www.parabola.unsw.edu.au/20102019/volume482012/issue3/article/unswschoolmathematicscompetitionproblems2012
<div class="field fieldnamefieldarticleauthor fieldtypetext fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even">David Angell, Chris Angstmann, Peter Brown, Michael Cowling, David Crocker, Bruce Henry (Director), David Hunt, Tyrone Liang, Adrian Miranda</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even"><div><div><strong>Problem 1</strong></div><div>The infinite nested radical $$c=\sqrt{1+2\sqrt{1+2\sqrt{1+2\sqrt{1+2\sqrt{\ldots}}}}}$$ converges.</div><div> </div><div>Find $c$. </div><div> </div><div><strong>Solution 1</strong></div><div> </div><div>Note that if \( c=\sqrt{1+2\sqrt{1+2\sqrt{1+2\sqrt{\ldots}}}} \)</div><div> </div></div><p> </p></div></div></div><div class="field fieldnamefieldarticlearticlepdf fieldtypefile fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even"><a href="https://www.parabola.unsw.edu.au/files/articles/20102019/volume482012/issue3/vol48_no3_c.pdf">Click here to download the PDF file</a><iframe id="pdf_reader" src="https://docs.google.com/viewer?embedded=true&url=https%3A%2F%2Fwww.parabola.unsw.edu.au%2Ffiles%2Farticles%2F20102019%2Fvolume482012%2Fissue3%2Fvol48_no3_c.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>
Tue, 20 Aug 2013 08:17:32 +0000
fcuadmin
108 at https://www.parabola.unsw.edu.au

Wilkinson Polynomials
https://www.parabola.unsw.edu.au/20102019/volume482012/issue3/article/wilkinsonpolynomials
<div class="field fieldnamefieldarticleauthor fieldtypetext fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even">Bill McKee</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even"><div><span style="lineheight: 1.5;">This article is about a family of polynomials introduced by James H. Wilkinson some five decades ago, which have the peculiar property that some of their zeros are extremely sensitive to small changes in the values of one or more of the coefficients.</span></div><div> </div><div><span style="lineheight: 1.5;">If $n$ is a nonnegative integer, we define the Wilkinson polynomials $W(x,n)$ by</span></div><div> </div><div>$$W(x,0) = 1$$</div><div>$$(x,1) = x1$$</div><div>$$W(x,2) = (x2)(x1)$$</div><div>$$(x,3) = (x3)(x2)(x1)$$</div><div> </div><div>and so on to give $$W(x,n) = (xn) W(x,n1) \quad \mbox{for} \quad n>0.$$</div><div> </div></div></div></div><div class="field fieldnamefieldarticlearticlepdf fieldtypefile fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even"><a href="https://www.parabola.unsw.edu.au/files/articles/20102019/volume482012/issue3/vol48_no3_2.pdf">Click here to download the PDF file</a><iframe id="pdf_reader" src="https://docs.google.com/viewer?embedded=true&url=https%3A%2F%2Fwww.parabola.unsw.edu.au%2Ffiles%2Farticles%2F20102019%2Fvolume482012%2Fissue3%2Fvol48_no3_2.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>
Tue, 20 Aug 2013 08:14:56 +0000
fcuadmin
107 at https://www.parabola.unsw.edu.au

History of Mathematics: Winning strategies
https://www.parabola.unsw.edu.au/20102019/volume482012/issue3/article/historymathematicswinningstrategies
<div class="field fieldnamefieldarticleauthor fieldtypetext fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even">Michael A. B. Deakin</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even"><div>This column's topic will be a theorem in the mathematical theory of games. It concerns what could be thought of the simplest possible type of game we can imagine.</div><div> </div><div>It is played by 2 players who move alternately.</div><div> </div><div>It is a game of perfect information; there are no hidden data such as would occur with a card or dice game; both players are fully aware at all times of the state of the game.</div><div> </div><div>The game ends within a finite number of moves.</div><div> </div><div>It is impossible to have a drawn game; one or the other of the players must win.</div></div></div></div><div class="field fieldnamefieldarticlearticlepdf fieldtypefile fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even"><a href="https://www.parabola.unsw.edu.au/files/articles/20102019/volume482012/issue3/vol48_no3_1.pdf">Click here to download the PDF file</a><iframe id="pdf_reader" src="https://docs.google.com/viewer?embedded=true&url=https%3A%2F%2Fwww.parabola.unsw.edu.au%2Ffiles%2Farticles%2F20102019%2Fvolume482012%2Fissue3%2Fvol48_no3_1.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>
Tue, 20 Aug 2013 08:12:07 +0000
fcuadmin
106 at https://www.parabola.unsw.edu.au

Editorial
https://www.parabola.unsw.edu.au/20102019/volume482012/issue3/article/editorial
<div class="field fieldnamefieldarticleauthor fieldtypetext fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even">Bruce Henry</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even"><div><span style="lineheight: 1.5;">As we go to press there has been a lot of excitement in mathematics about the proof of the ABC conjecture. The conjecture, which was posed by two prominent number theorists Joseph OesterlĂ© and David Masser in 1985, is considered to be one of the most important conjectures in number theory and the proof of the conjecture is a notoriously hard problem. It may be that the proof will be revealed to be invalid but the mathematican who came up with the proof, Sinichi Mochizuki, is very highly regarded so it may well be valid. Understandably it will take experts quite some time to wade through the 500 or so pages of proof.</span></div><div> </div><div>So, what is the ABC conjecture? There are many different ways of stating the conjecture, some easier to understand than others. To begin, you might try the following exercise: take any two positive integers $a$ and $b$ having no common factor and add them to obtain a larger integer $c$; write down all the prime factors of $a,b$ and $c$, ignoring any repetitions; multiply these primes to obtain a product $P$. The aim is to choose $a$ and $b$ in such a way that $c$ is larger than $P$: you will usually find that this is not the case. Here are a few examples.</div></div></div></div><div class="field fieldnamefieldarticlearticlepdf fieldtypefile fieldlabelhidden viewmoderss"><div class="fielditems"><div class="fielditem even"><a href="https://www.parabola.unsw.edu.au/files/articles/20102019/volume482012/issue3/vol48_no3_e.pdf">Click here to download the PDF file</a><iframe id="pdf_reader" src="https://docs.google.com/viewer?embedded=true&url=https%3A%2F%2Fwww.parabola.unsw.edu.au%2Ffiles%2Farticles%2F20102019%2Fvolume482012%2Fissue3%2Fvol48_no3_e.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>
Tue, 20 Aug 2013 08:09:04 +0000
fcuadmin
105 at https://www.parabola.unsw.edu.au