Parabola - Issue 3
https://www.parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-3
en Volume 47 Issue 3 Header
https://www.parabola.unsw.edu.au/content/volume-47-issue-3-header
<section class="field field-name-field-nav-pic-volume-issue field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">Volume/Issue: </h2><ul class="field-items"><li class="field-item even"><a href="/2010-2019/volume-47-2011/issue-3">Issue 3</a></li></ul></section><section class="field field-name-field-nav-pic-image field-type-image field-label-above view-mode-rss"><h2 class="field-label">Image: </h2><div class="field-items"><figure class="clearfix field-item even"><img class="image-style-volume-issue-header-image" src="https://www.parabola.unsw.edu.au/files/styles/volume_issue_header_image/public/promotional_images/Hammock_2.jpg?itok=zjZ9sUMU" width="640" height="250" alt="" /></figure></div></section><section class="field field-name-field-nav-pic-volume-number field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">Volume Number: </h2><ul class="field-items"><li class="field-item even"><a href="/volume/volume-47">Volume 47</a></li></ul></section>Tue, 11 Feb 2014 04:47:36 +0000z9803847250 at https://www.parabola.unsw.edu.auSolutions to Problems 1361 - 1370
https://www.parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-3/article/solutions-problems-1361-1370
<div class="field field-name-field-article-author field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Various</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><div><strong>Q1361</strong> Find a six-digit number which can be split into three two-digit squares and also into two three-digit squares. (The first digit of a number cannot be zero.)</div><div><strong>SOLUTION</strong> <span style="line-height: 1.5;">Find a six-digit number which can be split into three two-digit squares and also into two three-digit squares. (The first digit of a number cannot be zero.)</span></div><div> </div><div>The number must begin with a three--digit square whose first two digits also form a square. So we seek a three--digit square of the form$$\def\\#1{{\rm#1}} 16\\X\quad\hbox{or}\quad25\\X\quad\hbox{or}\quad36\\X\quad\hbox{or}\quad49\\X\quad\hbox{or}\quad64\\X\quad\hbox{or}\quad81\\X\quad;$$ the possibilities are $169$, $256$ and $361$. The last of these digits must begin a two--digit square, which rules out $169$. The remaining options for our six--digit number are $$2564{\rm X}{\rm Y}\quad\hbox{and}\quad3616{\rm X}{\rm Y}\ .$$ Now $4{\rm X}{\rm Y}$ is a three--digit square beginning with $4$, and so we have ${\rm X}{\rm Y}=00,41$ or $84$; the first is ruled out by the conditions of the problem and the others are not squares. The only answer to the problem is $361625$.</div><div> </div><div><strong>Q1362</strong> Sandy leans a ladder against a wall in order to clean the gutter running a<span style="line-height: 1.5;">long the top of the wall. Sandy is worried that the foot of the ladder is going to slip away from the wall and therefore ties a tightly stretched string between the middle of the ladder and a nail which is located directly below the top of the ladder, at the point where the floor meets the wall. Assuming that the floor is perfectly horizontal and the wall is perfectly vertical, how much is this going to help?</span></div><div> </div><div><strong>SOLUTION</strong> If the foot of the ladder slips away from the wall then the middle of the ladder is always the same distance from the nail. (Why? Because the angle between the wall and the floor is a right angle, so the line from the nail to the middle is always a radius of the circle having the ladder as diameter.) So connecting these two points by a string is not going to help at all!!</div></div></div></div><div class="field field-name-field-article-article-pdf field-type-file field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><a href="https://www.parabola.unsw.edu.au/files/articles/2010-2019/volume-47-2011/issue-3/vol47_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%2F2010-2019%2Fvolume-47-2011%2Fissue-3%2Fvol47_no3_s.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 07:03:45 +0000fcuadmin92 at https://www.parabola.unsw.edu.auProblems Section: Problems 1371 - 1380
https://www.parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-3/article/problems-section-problems-1371-1380
<div class="field field-name-field-article-author field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Various</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><div><strong>Q1371 </strong>Consider shuffles of a standard $52$-card pack. (See the article in this issue for terminology and basic information.) Let $\mathop{rev}\nolimits$ be the shuffle which reverses the pack -- that is, the first card is swapped with the last, the second with the second last, and so on. As in the article, $\mathop{out}\nolimits$ denotes the outshuffle.</div>
<div> </div>
<div><span style="line-height: 1.5;">(a) Write a formula for $\mathop{rev}\nolimits(k)$ in terms of $k$; also, write $\mathop{rev}\nolimits$ as a product of cycles.</span></div>
<div>(b) Without any calculation, write down the cycle type of the composite shuffle $\mathop{rev}\circ\mathop{out}\circ\mathop{rev}$.</div>
<div>(c) Show that if we shuffle a pack of cards with both an outshuffle and a reverse shuffle, it makes no difference which one we do first.</div>
<div> </div>
<div><strong>Q1372 </strong>Show that a composition of $n-1$ cycles</div>
<div>$$(\,1\ 2\,)\circ(\,1\ 3\,)\circ(\,1\ 4\,)\circ\cdots\circ(\,1\ n\,)$$</div>
<div>can be written as a single cycle. Is</div>
<div>$$(\,1\ n\,)\circ\cdots\circ(\,1\ 4\,)\circ(\,1\ 3\,)\circ(\,1\ 2\,)$$</div>
<div>the same cycle? For any numbers $a_1,a_2,\ldots,a_m$, describe the shuffle</div>
<div>$$(\,a_1\ a_2\ \cdots\ a_m\,)\circ(a_m\ \cdots\ a_2\ a_1\,)\ .$$</div></div></div></div><div class="field field-name-field-article-article-pdf field-type-file field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><a href="https://www.parabola.unsw.edu.au/files/articles/2010-2019/volume-47-2011/issue-3/vol47_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%2F2010-2019%2Fvolume-47-2011%2Fissue-3%2Fvol47_no3_p.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 07:01:55 +0000fcuadmin91 at https://www.parabola.unsw.edu.auUNSW School Mathematics Competition Winners 2011
https://www.parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-3/article/unsw-school-mathematics-competition-winners-2011
<div class="field field-name-field-article-author field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Editor</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><div style="text-align: center;"><strong>Competition Winners – Senior Division</strong></div><div><strong>First Prize</strong></div><div>Edmond Cheng Newington College</div><div>Declan Gorey Sydney Boys High School</div><div><strong>Second Prize </strong></div><div>Timothy Large <span style="line-height: 1.5;">Sydney Grammar School</span></div><div>Jinghang Luo <span style="line-height: 1.5;">James Ruse Agricultural High School</span></div><div><strong>Third Prize </strong></div><div>Nancy Fu James Ruse Agricultural High School</div><div>Allan Zhang <span style="line-height: 1.5;">James Ruse Agricultural High School</span></div></div></div></div><div class="field field-name-field-article-article-pdf field-type-file field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><a href="https://www.parabola.unsw.edu.au/files/articles/2010-2019/volume-47-2011/issue-3/vol47_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%2F2010-2019%2Fvolume-47-2011%2Fissue-3%2Fvol47_no3_w.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:57:29 +0000fcuadmin90 at https://www.parabola.unsw.edu.auUNSW School Mathematics Competition Problems 2011
https://www.parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-3/article/unsw-school-mathematics-competition-problems-2011
<div class="field field-name-field-article-author field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">David Angell, Chris Angstmann, Peter Brown, David Crocker, Bruce Henry (Director), David Hunt and Dmitriy Zanin</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><div style="text-align: center;"><strong>Junior Division - Problems and Solutions</strong></div><div> </div><div> </div><div><strong>Problem 1</strong></div><div><span style="line-height: 1.5;">A second-cousin prime $n$-tuple is defined as a set of $n$ prime numbers $\{p, p+6, \ldots p+6(n-1)\}$ with common difference six. Each number in the set is a prime and consecutive members of the set differ by six.</span></div><div>For example 2011 is a member of a second-cousin prime 2-tuple.</div><div> </div><div>Show that there is one and only one second-cousin prime 5-tuple and there are no second-cousin prime 6-tuples.</div><div> </div><div><span style="line-height: 1.5;"><strong>Solution 1</strong></span></div><div><span style="line-height: 1.5;">Clearly if $p$ is not equal to five and is a member of a second-cousin prime $n$-tuple then the last digit of $p$ must be one of one, three, seven or nine. Suppose it ends</span></div><div>in one, then the next member of the second-cousin prime $n$-tuple ends with a seven, the next member a three, the next member a nine and then the next number that differs by six ends in a five and</div><div>is therefore non-prime. Thus there are no second-cousin prime $n$-tuples with $n>4$ if the first prime in the set is not equal to five. It remains to consider a second-cousin prime $n$-tuple starting with $p=5$.</div><div>By construction the largest second-cousin prime $n$-tuple is the second-cousin prime 5-tuple $(5,11,17,23,29)$ and there are no second-cousin prime 6-tuples.</div></div></div></div><div class="field field-name-field-article-article-pdf field-type-file field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><a href="https://www.parabola.unsw.edu.au/files/articles/2010-2019/volume-47-2011/issue-3/vol47_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%2F2010-2019%2Fvolume-47-2011%2Fissue-3%2Fvol47_no3_c.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:54:37 +0000fcuadmin89 at https://www.parabola.unsw.edu.auShuffling Along and Cycling Around
https://www.parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-3/article/shuffling-along-and-cycling-around
<div class="field field-name-field-article-author field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">David Angell</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><div>Many readers will at some time have played games with a pack of cards. In most games one begins by shuffling the cards so as to randomise their order. There are various different ways of shuffling, one of the most popular being the <em>riffle shuffle</em>. In this shuffle, the pack is split into two roughly equal parts, one is held in each hand, with the edges adjacent <span style="line-height: 1.5;">and the thumbs are used to flip through the two parts so that they are merged together again, cards falling more or less from each hand alternately. If you haven't seen a riffle shuffle before, this description probably didn't help much!</span></div></div></div></div><div class="field field-name-field-article-article-pdf field-type-file field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><a href="https://www.parabola.unsw.edu.au/files/articles/2010-2019/volume-47-2011/issue-3/vol47_no3_3_0.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%2F2010-2019%2Fvolume-47-2011%2Fissue-3%2Fvol47_no3_3_0.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:50:28 +0000fcuadmin88 at https://www.parabola.unsw.edu.auSubliminal Deduction?
https://www.parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-3/article/subliminal-deduction
<div class="field field-name-field-article-author field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Michael A. B. Deakin</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><div>Let me begin by recounting a story I first heard almost fifty years ago. This is, of course, a long time and over the intervening years I have lost contact with the people involved. So it may well be that I have misremembered parts of it and the actual event may well have been somewhat different in its details. However, I will tell it as I recall it, and its general gist is enough to introduce my topic for this column.</div><div> </div><div>Two graduate students, contemporaries of mine, married, saw the birth of a son and went together to study abroad. On one occasion, the wife went to the dentist, leaving the youngster in the care of his father, a somewhat absent-minded man. Sitting in the dentist's chair, the wife suddenly had an overpowering feeling that something awful had happened to the boy. So strong was this feeling that she got up at once and rushed home, to find that he had indeed fallen down a flight of stairs and suffered concussion, without the father's being aware of the event.</div></div></div></div><div class="field field-name-field-article-article-pdf field-type-file field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><a href="https://www.parabola.unsw.edu.au/files/articles/2010-2019/volume-47-2011/issue-3/vol47_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%2F2010-2019%2Fvolume-47-2011%2Fissue-3%2Fvol47_no3_2.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:47:58 +0000fcuadmin87 at https://www.parabola.unsw.edu.auLarge Splines
https://www.parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-3/article/large-splines
<div class="field field-name-field-article-author field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Bill McKee</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><div>This article is fundamentally about the calculations behind the ways in which computers draw graphs. In the era before computers (unknown to most of you, but very familiar to me!) graphs were drawn on paper. Typically, the data points were plotted. Next, pins were placed in the paper at these data points and a thin flexible piece of wood was threaded around these points to produce a nice smooth shape which was then traced by hand. This piece of wood was known as a <em>spline</em>. The word itself apparently derives from a dialect word from the East Anglia region of England for a strip of wood and is related to the word <em>splinter</em>. This article will present a computational method which approximates the behaviour of a spline. This method, and generalisations thereof, underlie much of computer graphics.</div><div> </div><div>An earlier article in <em>Parabola Incorporating Function</em> (Volume 44, Number 3) showed how to construct a polynomial which passed exactly through some data points. Splines provide a different way of constructing a smooth curve which also passes exactly through the given data points and avoids some of the pitfalls which can be sometimes associated with the earlier procedure.</div></div></div></div><div class="field field-name-field-article-article-pdf field-type-file field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><a href="https://www.parabola.unsw.edu.au/files/articles/2010-2019/volume-47-2011/issue-3/vol47_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%2F2010-2019%2Fvolume-47-2011%2Fissue-3%2Fvol47_no3_1.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:43:04 +0000fcuadmin86 at https://www.parabola.unsw.edu.auEditorial
https://www.parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-3/article/editorial
<div class="field field-name-field-article-author field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">B. I. Henry</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><div><span style="line-height: 1.5;">Welcome to a packed issue to close the year in 2011.</span></div><div> </div><div>Congratulations to all of the students, their parents and teachers who had success in the 50th Annual UNSW School Mathematics Competition. The competition problems, solutions and names of prizewinners are included in this issue.</div><div> </div><div>Mathematical problems might seem far removed from everyday life but really this is not the case. In fact, as Mrs Fibonacci says in the wonderful picture book by Jon Scieszka and Lane Smith, <em>Maths Curse</em>: ``YOU KNOW, you can think of almost everything as a maths problem''. </div><div> </div></div></div></div><div class="field field-name-field-article-article-pdf field-type-file field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><a href="https://www.parabola.unsw.edu.au/files/articles/2010-2019/volume-47-2011/issue-3/vol47_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%2F2010-2019%2Fvolume-47-2011%2Fissue-3%2Fvol47_no3_e.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:39:04 +0000fcuadmin85 at https://www.parabola.unsw.edu.au