Parabola - Issue 2
https://parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-2
enVolume 47 Issue 2 Header
https://parabola.unsw.edu.au/content/volume-47-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-47-2011/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/Hammock_1.jpg?itok=dCJDOt-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-47">Volume 47</a></li></ul></section>Tue, 11 Feb 2014 04:46:57 +0000z9803847249 at https://parabola.unsw.edu.auSolutions to Problems 1351 - 1360
https://parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-2/article/solutions-problems-1351-1360
<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>Q1351 </strong>A city consists of a rectangular grid of roads, with $m$ roads running east--west and $n$ running north--south. Every east--west road intersects every north--south road. A construction vehicle travels around the city, visiting each intersection once (and only once) and finally returning to its starting point. As it travels it builds a fence down the middle of each road it uses: thus it constructs, in effect, a single long fence which eventually loops back on itself. How many city blocks are now inside the fence?</div><div> </div><div><strong>SOLUTION</strong> If we assume that each block is a square with side length $1$ unit, then the number of blocks inside the fence is equal to the area enclosed by the path. Since the path is a polygon with lattice points for its vertices we can use Pick's Theorem:</div><div>$$A=I+{\textstyle\frac{1}{2}}P-1\ ,$$</div><div>where $A$ is the area enclosed, $I$ is the number of points inside the path and $P$ is the number of points on the path. In this case there are $mn$ points altogether, and they are all on the path (because we are told that every intersection was visited). Therefore $P=mn$ and $I=0$, and the number of blocks inside the path is $\frac{1}{2}mn-1$.</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://parabola.unsw.edu.au/files/articles/2010-2019/volume-47-2011/issue-2/vol47_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-47-2011%2Fissue-2%2Fvol47_no2_s.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:36:00 +0000fcuadmin84 at https://parabola.unsw.edu.auProblems Section: Problems 1361 - 1370
https://parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-2/article/problems-section-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> </div>
<div><span style="line-height: 1.5;"><strong>Q1362 </strong>Sandy leans a ladder against a wall in order to clean the gutter running along 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><span style="line-height: 1.5;"><strong>Q1363 </strong>Find the smallest possible value of $x^2+y^2$, if $x$ and $y$ are real numbers for which $y\ge2+3x$ and $y\le7\sqrt x\,$.</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-47-2011/issue-2/vol47_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-47-2011%2Fissue-2%2Fvol47_no2_p.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:33:34 +0000fcuadmin83 at https://parabola.unsw.edu.auThe Body at the Bottom of the Cliff
https://parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-2/article/body-bottom-cliff
<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>In the latter part of year 2009, I attended a scientific talk at Sydney University about the path of a body exiting a cliff. The position at which the body lands from the base of the cliff depends on exit velocity. We show here that for small exit velocities and small cliff heights the landing position is not impacted significantly by the angle of inclination and air resistance.</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-47-2011/issue-2/vol47_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-47-2011%2Fissue-2%2Fvol47_no2_3.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:28:43 +0000fcuadmin82 at https://parabola.unsw.edu.auThe First Atomic Test
https://parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-2/article/first-atomic-test
<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>Here I revisit a topic I have written on several times before, and which furthermore continues the theme of my last column. In the third issue of <em>Parabola Incorporating Function</em> in 2007, I discussed a technique called <em>dimensional analysis</em> and briefly touched on its application by G I Taylor to discover the energy released in the very first atomic blast (the so-called <em>Trinity Test</em>) in 1945: this happened at a time when that figure was still classified as top secret by the US military. I had earlier given a fuller account of the calculation involved in <em>Function</em> (February 1986, with a follow-up in February 1995). I also used this material in a set of course notes at Monash in 1996.</div><div> </div><div>In all these, I was more concerned with the actual Mathematics rather than with the historical detail. Here by contrast, I want to examine that detail more carefully. The story is an intricate one and, as I unravel it, it will emerge that what I wrote earlier, and what is still popular belief, is not entirely accurate. Most of the material that is widely available gives a greatly simplified account of the full situation.</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-47-2011/issue-2/vol47_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-47-2011%2Fissue-2%2Fvol47_no2_2.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:24:46 +0000fcuadmin81 at https://parabola.unsw.edu.auSudoku, Logic and Proof
https://parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-2/article/sudoku-logic-and-proof
<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">Catherine Greenhill</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>As you probably know, a Sudoku puzzle is a $9\times 9$ grid divided into nine $3\times 3$ subgrids. Some of the cells in the grid contain a symbol: usually the symbols are the numbers $1,2,\ldots, 9$. The idea is to put a symbol into each empty cell so that every symbol appears exactly once in each row, exactly once in each column and exactly once in each of the nine $3\times 3$ subgrids. An example of a Sudoku puzzle is given below: it was the daily puzzle on <a href="http://sudoku.com.au">http://sudoku.com.au</a> [3] on 9 June 2011, when I wrote this article.</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-47-2011/issue-2/vol47_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-47-2011%2Fissue-2%2Fvol47_no2_1.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:21:16 +0000fcuadmin80 at https://parabola.unsw.edu.auEditorial
https://parabola.unsw.edu.au/2010-2019/volume-47-2011/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">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;">Since the last issue of </span><em style="line-height: 1.5;">Parabola Incorporating Function</em><span style="line-height: 1.5;"> the mathematics world has been surprised by the appearance of a preprint, dated May 2011, by the German mathematician Gerhard Opfer who claimed to have proved the Collatz Conjecture. And then, some weeks later, other mathematicians reported that the proof was flawed and Opfer updated his preprint on 17 June 2011 with ``the statement `that the Collatz conjecture is true' has to be withdrawn, at least temporarily". The Collatz Conjecture is one of those remarkable mathematical problems that is easy to state and easy to check but difficult to prove. Starting with any natural number $n_0>1$, multiply the number by three and then add one if it is odd, or divide the number by two if it is even to construct another natural number $n_1$. Then apply the same rule to $n_1$ to construct another natural number $n_2$ and so on, but stop if you ever reach the number 1. The Collatz Conjecture says that the sequence $n_0, n_1, n_2, \ldots$ will always reach the number 1. Try it out, suppose we start with seven, then we obtain the sequence:</span></div><div>$$7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1$$</div><div>The Wikipedia page on the Collatz Conjecture is very informative.</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-47-2011/issue-2/vol47_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-47-2011%2Fissue-2%2Fvol47_no2_e.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:18:30 +0000fcuadmin79 at https://parabola.unsw.edu.au