Parabola - Issue 2
https://parabola.unsw.edu.au/2010-2019/volume-48-2012/issue-2
enVolume 48 Issue 2 Header
https://parabola.unsw.edu.au/content/volume-48-issue-2-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-48-2012/issue-2">Issue 2</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://parabola.unsw.edu.au/files/styles/volume_issue_header_image/public/promotional_images/Gateshead%20bridge-Vol48_1.jpg?itok=3Crrl_b_" width="640" height="250" alt="" /></figure></div></section><section class="field field-name-field-nav-pic-issue-number field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">Issue Number: </h2><ul class="field-items"><li class="field-item even"><a href="/issue/issue-2">Issue 2</a></li></ul></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-48">Volume 48</a></li></ul></section>Tue, 11 Feb 2014 04:38:49 +0000z9803847246 at https://parabola.unsw.edu.auSolutions to Problems 1381 - 1390
https://parabola.unsw.edu.au/2010-2019/volume-48-2012/issue-2/article/solutions-problems-1381-1390
<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>Q1381 </strong>It is commonly believed that the minute hand and the hour hand on a clock are in <em>exactly</em> symmetrical positions when the time is 10:08 and 42 seconds.</div><div> </div><div><span style="line-height: 1.5;">(a) Without detailed calculations, prove that this is wrong.</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://parabola.unsw.edu.au/files/articles/2010-2019/volume-48-2012/issue-2/vol48_no2_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%2Fparabola.unsw.edu.au%2Ffiles%2Farticles%2F2010-2019%2Fvolume-48-2012%2Fissue-2%2Fvol48_no2_s.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 08:05:47 +0000fcuadmin104 at https://parabola.unsw.edu.auProblems Section: Problems 1391 - 1400
https://parabola.unsw.edu.au/2010-2019/volume-48-2012/issue-2/article/problems-section-problems-1391-1400
<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>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? (<em>Hint</em>: see the solution of problem~1381 in this issue.)</div>
<div> </div>
<div><strong>Q1392 </strong>Find all real numbers $x$ which satisfy the equation</div>
<div>$$\lfloor x\rfloor-\{2x\}+\lceil3x\rceil=5\ .$$</div>
<div>As in problem 1384 we write $\lfloor x\rfloor$ for $x$ rounded to the integer below, and $\lceil x\rceil$ for $x$ rounded to the integer above; also, $\{x\}$ denotes rounding to the nearest integer, with halves rounding upwards. For example,</div>
<div>$$\{\pi\}=3\quad\hbox{and}\quad \{3{\textstyle\frac12}\}=4\ .$$</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://parabola.unsw.edu.au/files/articles/2010-2019/volume-48-2012/issue-2/vol48_no2_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%2Fparabola.unsw.edu.au%2Ffiles%2Farticles%2F2010-2019%2Fvolume-48-2012%2Fissue-2%2Fvol48_no2_p.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 08:03:01 +0000fcuadmin103 at https://parabola.unsw.edu.auHistory of Mathematics: The Vectors of Mind
https://parabola.unsw.edu.au/2010-2019/volume-48-2012/issue-2/article/history-mathematics-vectors-mind
<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"><p>My title for this column is that of an influential book, first published in 1935 by the American psychologist L. L. Thurstone. It is almost self-explanatory: although we are accustomed to see intelligence measured as a simple scalar, the IQ, most of us realize that this single number has to be a gross oversimplification of the underlying reality. Different people exhibit different mental capacities. Here is an obvious example: there are those of us who (like <em>Parabola</em>'s readers) are good at Mathematics, but there are also otherwise highly intelligent people who have no aptitude for it at all!</p></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://parabola.unsw.edu.au/files/articles/2010-2019/volume-48-2012/issue-2/vol48_no2_3.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%2Fparabola.unsw.edu.au%2Ffiles%2Farticles%2F2010-2019%2Fvolume-48-2012%2Fissue-2%2Fvol48_no2_3.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 08:01:05 +0000fcuadmin102 at https://parabola.unsw.edu.auNot Enough Monkeys
https://parabola.unsw.edu.au/2010-2019/volume-48-2012/issue-2/article/not-enough-monkeys
<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">Farid Haggar</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>A popular classical problem can be stated as follows:</div><div> </div><div>There are five monks, one monkey and pile of coconuts on a desert island. One monk goes to the pile of coconuts, gives one to the monkey, removes a fifth of the remaining coconuts, buries them and goes to sleep.</div><div> </div><div>The second monk then wakes up, goes to the pile of coconuts, gives one to the monkey, buries a fifth of what remains and goes to sleep. The other monks do likewise. In due course all five monks wake up and go to the pile of coconuts which they then succeed in sharing equally among them.</div><div> </div><div>What is the smallest possible number of coconuts that the pile originally contained?</div><div> </div><div>If after the final division there is still one coconut left for the monkey, what is the smallest possible number in the original pile?</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://parabola.unsw.edu.au/files/articles/2010-2019/volume-48-2012/issue-2/vol48_no2_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%2Fparabola.unsw.edu.au%2Ffiles%2Farticles%2F2010-2019%2Fvolume-48-2012%2Fissue-2%2Fvol48_no2_2.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 07:57:57 +0000fcuadmin101 at https://parabola.unsw.edu.auSolving Second Order Recurrences
https://parabola.unsw.edu.au/2010-2019/volume-48-2012/issue-2/article/solving-second-order-recurrences
<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>First consider a <strong>sequence</strong>, by which we mean a list of numbers which goes on forever. An example is the well-known <em>Fibonacci sequence</em></div><div>$$0, 1, 1, 2, 3, 5, 8, 13, 21, 34,...$$</div><div>in which every number (except for the first two) is the sum of the previous two. If we refer to these numbers, using function notation, as $a(0),a(1),a(2),a(3)$ and so on, the sequence is specified by the <strong>recurrence relation</strong></div><div>$$a(n)=a(n-1)+a(n-2)\quad\hbox{for}\quad n\ge2 (1)$$ <span style="line-height: 1.5;">together </span><span style="line-height: 1.5;">with the </span><strong style="line-height: 1.5;">initial conditions</strong></div><div>$$a(0)\quad\hbox{and}\quad a(1)=1\ .$$</div><div>Equation $(1)$ is referred to as a <strong>second order</strong> recurrence because each number is determined by the previous two numbers. If we wish, we can easily make up a third or higher order recurrence: for example, we might decide to write down a list in which each number is equal to the previous number, plus $14$ times the one before that, minus $24$ times the previous one again. That is,</div><div>$$a(n)=a(n-1)+14a(n-2)-24a(n-3)\quad\hbox{for}\quad n\ge3\ .$$</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://parabola.unsw.edu.au/files/articles/2010-2019/volume-48-2012/issue-2/vol48_no2_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%2Fparabola.unsw.edu.au%2Ffiles%2Farticles%2F2010-2019%2Fvolume-48-2012%2Fissue-2%2Fvol48_no2_1.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 07:54:21 +0000fcuadmin100 at https://parabola.unsw.edu.auEditorial
https://parabola.unsw.edu.au/2010-2019/volume-48-2012/issue-2/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">Bruce 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 this issue of </span><em style="line-height: 1.5;">Parabola incorporating Function</em><span style="line-height: 1.5;">. The issue starts with David Angell's article on solving a class of second order recurrences. Recurrence relations can look disarmingly simple because it is often a simple exercise to substitute in successive values to reveal the solution, a sequence comprised of these values. In solving recurrence relations you are seeking an algebraic expression that can be used to evaluate terms in the sequence directly, rather than recursively. In many cases it is not possible to find such an algebraic expression. Here is a rather famous example:</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://parabola.unsw.edu.au/files/articles/2010-2019/volume-48-2012/issue-2/vol48_no2_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%2Fparabola.unsw.edu.au%2Ffiles%2Farticles%2F2010-2019%2Fvolume-48-2012%2Fissue-2%2Fvol48_no2_e.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 07:51:19 +0000fcuadmin99 at https://parabola.unsw.edu.au