Number 79 Autumn 2022 Parallelograms in action – the Llanthony lift bridge at Gloucester dock This bridge was built in 1972 at a cost of £84,886.92. The first bridge at this site was a wooden swing bridge built to carry Llanthony Road over the new docks when they were constructed in 1794. This was replaced in the 1860s with an iron swing bridge. It is operated by a bridge keeper, who works between 9am and 6pm, thus restricting movement on the canal to those hours for any vessel that cannot pass under the brides when lowered. Think about the fixed points, parallelograms and levers when you examine these pictures! THE FIELDS MEDAL WINNERS It was recently announced that Maryna Viazovsa, James Maynard, June Huh and Hugo Duminil-Copin were awarded the Fields Medal, recognising outstanding mathematical achievement. The medals and cash prizes are awarded by a trust established by J C Fields at the University of Toronto. It was first awarded in 1936 and then every four years from 1950 onwards. Recipients must be under 40 at the start of the year and only two, three or four are awarded. Maryna is the first Ukrainian, and June the first person of Korean origin to receive this award. Maryna (Photo credit: Matteo Fieni) James (Photo credit: Ryan Cowan) June (Photo credit: Lance Murphy) Hugo (Photo credit: Matteo Fieni) Recent News
2 SYMmetryplus 79 Autumn 2022 EDITORIAL Welcome back to a new term! If you sat any examinations over the summer, or had any involvement with them, I hope they went well. My days of going into school to see how the students had done are now long gone – it will be my grandchildren who will now be subject to this outmoded method of assessment in a few years’ time. This is my final issue as editor, and Oli Saunders, who has worked with me on the past couple of issues is now ready to take on the editorship. I will pass on all the articles I have received in the past, but not yet published, to him for consideration in future issues. It has been a pleasure and great privilege to work on SYMmetryplus over the past decade and I would like to thank all authors who have submitted articles, Editorsin-Chief at the MA, HQ staff, the printers Steve and John at 4Word and anyone else who has contributed comments or helped in any way. Congratulations to the British International Mathematical Olympiad (IMO) team that participated in this year’s IMO in Norway. They did so well, with 6 medals including a Gold, and a ranking of 13th out of the 104 participating teams. The results are at https://www.imo-official.org/year_country_r.aspx?year=2022 . Mark King emailed to say there was mention on page 3 of the last edition of a 24-hour orbit of geostationary satellites. This over-simplification led to rather expensive mistakes by early engineers slavishly following the specification given, and there was a drift of about a degree a day until it was explained to the bosses that the earth goes around the sun and so the sidereal day of 23hrs 56 min 4.09 seconds is what was needed. Answers to questions from issue 61 onwards, errata, resource sheets and additional notes based on comments received from readers can be found at http://www.m-a.org.uk/symmetry-plus Please send contributions to symmetryplus@m-a.org.uk Peter Ransom A GEOMETRIC PROBLEM An equilateral triangle is formed by joining three points of a (regular) hexagonal grid as shown. Three congruent pentagons are shaded pink inside the equilateral triangle. Exactly what part of the equilateral triangle is shaded pink? Arsalan Wares A CRYPTARITHM Each letter stands for a different digit. S I X T H + F O R M M A T H S There is a hint at the bottom of page 13. I only own pairs of black and white socks. I have four more pairs of black socks than the number of pairs of white socks. If I draw two socks from my sock cupboard at random, the probability they match is ½. How many socks do I own? QUICKIE 42A
SYMmetryplus 79 Autumn 2022 3 THREE GAPS - 2 This article continues from part 1 in SYMmetryplus 78. Here is one case in nature where equal angles are stepped off. Many plants issue a single leaf bud, climb, issue a second leaf bud, climb, issue a third leaf bud, climb, and so on, and do so with constants shifts in angle. Since at any point in time the lower leaves will have grown more than the upper ones, seen in plan, the leaf tips follow a spiral. If the growth is linear, the spiral will be Archimedean: it goes out in direct proportion to how far it goes round. In the figure below I have given the plant leaves which are regular hexagons. It has just budded its 12th leaf. I have created a slider tool again to vary the angle between the leaves. ‘Hofmeister’s rule’ is the observation that new leaves appear in the biggest gaps. The plant’s strategy is clear: it needs to maximise the light flux on it and therefore the total leaf area exposed to the sun. Is Hofmeister’s rule consistent with budding at a constant angle? Yes, if the angle is right. The claim is that, in the ideal case, this angle would be the one which divides a whole angle in the golden ratio. This is (3 − √5) radians, about 137.5°. Fractions of 2 which are ratios of alternate numbers in the Fibonacci sequence: 1 2 ,1 3 , 2 5 , 3 8 , 5 13 , ... converge on this value, and examples abound of all those five particular ratios. Where does our hexagon plant fit in this scheme? We will take our slider over a range of values centred on the golden angle and measure the area in each case. We use the ‘points’ tool to mark vertices defining the region. We use the ‘polygon’ tool to span it. Finally, we use the ‘measure area’ tool.
4 SYMmetryplus 79 Autumn 2022 Here is my plot, with much uncertainty about the size of the error in my area measurements. I have taken readings every 4° (the filled in circles) with a couple of extra ones around the golden angle (the open circles). For 12 leaves of our hexagon plant, we seem to have a peak at 142 ± 1°. As far as the arrangement of leaves (phyllotaxis) goes, this puts it in the same group as the poplar, which has a more or less circular leaf, and the pear, whose leaf is an ellipse with approximate aspect ratio 3 ∶ 2. Bearing in mind that the area of an ellipse is just , where a is half the long axis and b half the short one, confirm that the hexagon leaf area falls within 1% of the mean of the leaf areas of the poplar and the pear. Find a plant with a very different leaf shape and try to model it in the way described. (Though you can measure the areas of circles and ellipses with GeoGebra, to measure the area of overlapping shapes, you have to approximate them as polygons.) Paul Stephenson FACTOR PUZZLES Factor puzzles (for want of a better name) are a nice example of an inverse process being (much) harder than the direct process (Note 1). First, let us do the direct process: choose four positive integers and write them in some arrangement on adjacent sides of a 2 × 2 square: 3 8 4 6 Then fill in the four products in the four cells: 3 8 4 12 32 6 18 48 This is pretty straightforward multiplication tables practice. But suppose, instead, you had the inverse problem – you were just given the interior numbers? 12 32 18 48 Could you have worked out the four outside numbers that must have been used? This inverse process is much harder than the direct process. And can you be sure whether there might be more than one possibility? A systematic approach is to consider the common factors of each row and column, written below in italics: 1, 2, 3, 6 1, 2, 4, 8, 16 1, 2, 4 12 32 1, 2, 3, 6 18 48
SYMmetryplus 79 Autumn 2022 5 Then choose the row or column with the smallest number of common factors (i.e., row 1 above) and try each of these factors as an outside number (in bold below): 12 1 12 32 X 18 48 (i) 6 16 2 12 32 3 18 48 (ii) 3 8 4 12 32 6 18 48 (iii) In (i), trying the factor 1, we obtain 12 as an outside number, and then find that row 2 is impossible to complete, since 12 is not a factor of 18. However, we find that both 2 (ii) and 4 (iii) work as numbers at the top of the left-hand side, and so we obtain two solutions – the one we started with (iii) and another one (ii). Students may be surprised that there can be more than one solution. At this point there are many questions we might pose: 1. What other approaches are good for solving these? 2. When do we get more than one solution? 3. When do we get no solutions? These puzzles are easy to invent, but the challenge becomes not just finding ‘a’ solution but being sure how many solutions there are. Students might be challenged to invent puzzles with exactly two solutions or three solutions. What is the maximum number of solutions possible? It may not be immediately apparent, but these diagrams are the same in structure as what are sometimes called ‘ratio tables’. If we generalise the four outside numbers to p, q, r and s , then we have: p q r pr qr s ps qs Now we can see that the entries in the second column are times those in the first column, and the entries in the second row are times those in the first row, just as in any ratio table. We can also see that the ‘determinant’ must be zero, since the product of the entries on the main diagonal – × – has to be equal to the product of the entries on the other (anti-) diagonal – × . This means that there is definitely no solution to a factor puzzle if these two diagonal products are not equal, and it turns out that if they are equal then there is definitely at least one solution (Abusaris, & Alhami, 2022). There are lots to think about with factor puzzles, and it is possible to extend the idea to 2 × 3, 3 × 3 or other sizes of puzzle, as well as to include fractions, decimals and even algebraic expressions in the cells. For example, how would you go about solving this puzzle? 1 6 2 9 1 2 3 20 1 5 9 20 Note 1. These are related to ‘multiplication table puzzles’ like https://findthefactors.com/solve-find-thefactors/. Reference Abusaris, R., & Alhami, K. (2022). Factor puzzles from definition to applications [pre-print]. F1000Research 2022, 11, 727. https://doi.org/10.12688/f1000research.111241.1 Colin Foster My gross brother also only owns pairs of black and white socks. He has 33 pairs of black socks and some pairs of white socks. If he draws two socks from his sock cupboard at random, the probability they match is also ½. How many socks does he own? QUICKIE 42B
6 SYMmetryplus 79 Autumn 2022 153: MORE THAN JUST A NUMBER There are many numbers of interest. This article concerns itself with the mathematical properties of 153. Perhaps this will provide the interest to explore a number of your own choice – there are plenty to choose from … The first mention of 153 can actually be found in the New Testament. This is the story where the meaning of 153 is referred to as the miraculous draught of large fish caught on the shore of the Sea of Tiberius. The precision of the number of fish in this narrative has long been considered peculiar, and many scholars have argued that 153 has some deeper significance. Figurate numbers 153 is the sum of the first 17 integers and is therefore triangular. 17(17+1) 2 = 153. The reverse of 153 is 351, which is the 26th triangular number. 26(26+1) 2 = 351. Since it is triangular, it is also a hexagonal number, because it is of the form 2 (2 −1) 2 , where = 9. 2×9(2×9−1) 2 = 153. For any triangular number ( ), × 1225 + 153 = 35 +17. 1 × 1225 + 153 = 1378 = 52, 3 × 1225 + 153 = 3828 = 87, 6 × 1225 + 153 = 7503 = 122, 10 × 1225 + 153 = 12403 = 157. Note that 1225 is a triangular square number. The sum of the digits of 153 is a perfect square: 1 + 5 + 3 = 9 = 32. Ruth-Aaron pairs District prime factors are the prime factors that are different from each other so although 153 = 32 × 17, the distinct prime factors of 153 are 3 and 17, and these add up to 20. Interestingly, there are three distinct prime factors of 154 that also add to 20. That is 2, 7 and 11. The fact that 153 and 154 form such a pair, means they are a Ruth-Aaron pair. In mathematics a Ruth-Aaron pair consists of two consecutive integers for which the sums of the prime factors of each integer are equal. Such pairs are named after the baseball players, Babe Ruth and Hank Aaron, whose home run records were 714 and 715 respectively. Niven numbers A number is said to be a Niven, or Harshad number, if the number is divisible by the sum of the digits of the number. 153 is a case in point. 1 + 5 + 3 = 9, 153 ÷ 9 = 17. Niven numbers are named after Ivan M. Niven (1915 – 1999). He was a Canadian-American mathematician who specialised in number theory. He is remembered for his work on Waring’s problem and proving to be an irrational number (1947). For many years he was a professor at the University of Oregon.
SYMmetryplus 79 Autumn 2022 7 Narcissistic numbers 153 is also the smallest 3-digit narcissistic number. In number theory, a narcissistic number is one that is the sum of its own digits each raised to the power of the number of digits. In this case: 13 + 53 + 33 = 1 = 125 + 27. Other than the trivial 1-digit numbers, 153 is the smallest example. A number chain An interesting feature of the number 153 is that it is the limit of the following algorithm. Take a random positive integer that is divisible by 3: * split that number into its base 10 digits; take the sum of their cubes; repeat from *. For example, starting with the number 87. 83 + 73 = 512 + 343 = 855 83 + 53 + 53 = 512 + 125 + 125 = 762 73 + 63 + 23 = 343 + 216 + 8 = 567 53 + 63 + 73 = 125 + 216 + 343 = 684 63 + 83 + 43 = 216 + 512 + 64 = 792 73 + 93 + 23 = 343 + 729 + 8 = 1080 13 + 03 + 83 + 03 = 1 + 512 = 513 53 + 13 + 33 = 125 + 1 + 27 = 153. Vesica Piscis The number 153 is associated with the geometric shape known as the Vesica Piscis. The vesica piscis is a type of lens in shape, a mathematical shape formed by the intersection of two disks that have the same radius, intersecting in such a way that the centre of each lens lies on the perimeter of the other. In Latin, “vesica piscis” literally means “bladder of a fish”, reflecting the shape’s resemblance to the conjoined dual air bladders (“swim bladder”) found in most fish. Archimedes, in his Measurement of a Circle, referred to this ratio (265/153), as constituting the "measure of the fish", this ratio being an imperfect representation of √3. Today this shape is more commonly seen in Venn diagrams, works of art and jewelry. Activities based on 153 1. What is the result of the sum of the aliquot divisors of 153? (Aliquot divisors of a number are all the divisors of that number excluding the number itself but including 1). 2. Find the next three 3-digit narcissistic numbers that are less than 1000. 3. Find the first pair of Ruth-Aaron numbers. (This can be calculated easily on paper or in your head). 4. Apply the algorithm given in A number chain to other numbers that are divisible by 3. Observe anything of interest. 5. Find an image of the Coat of Arms of Guam. Describe the shape. Neil Walker 265 153 My daughter also only owns pairs of black and white socks. She has a total of 400 socks, of which more are black than white. If she draws two socks from her sock room at random, the probability they match is also ½. How many black socks does she own? QUICKIE 42C
8 SYMmetryplus 79 Autumn 2022 NEUSIS - PART 4 In SYMmetryplus 76, I wrote that when it came to geometric constructions, the Ancient Greeks had an order of preference for the means they used: 1. Straight edge and compass, 2. Conic sections (e.g., parabolas), and, an absolute last resort, 3. Neusis (noy-sis). By this term they meant the process of converging on a value by adjusting the position of a construction aid to coincide with set marks. So ‘adjustment’ is a fair translation. They handed down to us four famous geometrical problems (FGP) which they failed to solve by straight edge and compass. For good reason. Two millennia on these were proved insoluble by this means. (David S. Richeson’s book Tales of Impossibility, published in 2020 and available for Kindle, explains why. Only the last two chapters use mathematics beyond A level.) Now I will l look at the fourth and final FGP. FGP 4 Constructing a square with the same area as a given circle I start with a physical experiment. I take a circle (disk) of unit radius, whose area is therefore × 12 = square units. I roll it along a line (track) until it has executed a quarter-turn, thus unrolling 1 4 of its circumference, 2 . This gives me the blue rectangle, whose area is 2 × 2 = square units also. What I need is therefore a square with this area. The side of the square must be the geometric mean of the rectangle’s side lengths, 2 and 2: √2 × 2 = √ . This I can construct, using the circle property 2 = . The red square therefore has the same area as the original white circle, . There is a big difference between this solution and solutions for the previous three FGPs. In the latter cases the degree of accuracy is visible. We can perform the operations as accurately as we could a straightedge-and-compass construction. In this case we have no way to check the disk has maintained point-forpoint contact with the track: it might have slipped. In fact, I do not think the Ancient Greeks would have accepted my attempt as an example of neusis at all. Unsurprisingly, Archimedes made a better fist of it. The spiral which bears his name grows outwards ( ) from the pole of a polar grid in direct proportion to how far it goes round ( ). In polar coordinates therefore it has an equation of the form = . So, the constant is the rate of change of with . We shall make = 1 2 so that, after one complete circuit, = 1 2 × 2 = 1. We draw a circle centred on the pole with this radius.
SYMmetryplus 79 Autumn 2022 9 k is also the slope of the tangent to the curve, its gradient. After one circuit Archimedes could therefore draw a tangent which defined the two similar right triangles shown right. As you see, the length of the middle side of the larger triangle is the circumference of the circle. From there Archimedes could proceed by ruler and compass, using the circle property we applied above, to ‘square the circle’. You meet two kinds of calculus in the sixth form: differential and integral. Archimedes is celebrated for anticipating the latter. But, by drawing the tangent to his spiral, he can also be said to have anticipated the former. Mathematicians have occasionally amused themselves by taking a close rational approximation to and seeking a straight-edge-and-compass solution. (This is always possible in such cases.) Write out the first three odd numbers twice: 1 1 3 3 5 5. Split the set in half and put the second half over the first: 355 113 . Take the earth’s circumference at the equator as 40,000 km. Calculate the earth’s diameter: first by dividing by your calculator’s value, (in my case this is accurate to 9 significant figures: 3.14159265); second by using 355 113 . Roughly how far out does that leave you? Go to the Wikipedia page ‘Crockett Johnson’. Read the entry but make sure you go down to ‘References’ and click on no. 1: ‘National Museum of American History page on Crockett Johnson with images of paintings’. Paul Stephenson ANIMAL PARABOLA? Watching young children play and organise their toys is fascinating. I noticed this after our 2-yearold grandson had been playing with his animals.
10 SYMmetryplus 79 Autumn 2022 LOUIS POINSOT When Gustave Eiffel designed his tower for the 1889 Exposition Universelle in Paris, which was to celebrate the 100th year anniversary of the French Revolution, he paid tribute to scientists of the 18th and 19th centuries by inscribing the names of 72 of them on the facades of the Tower, in the cantilever located above the beam used to seal the large arches. On the side of the tower that faces South-East (the Military Academy side), you will find the name of Poinsot, who was a mathematician and engineer. Poinsot was the inventor of geometrical mechanics; he investigated how to resolve a system of forces acting on a rigid body into a single force and a couple. Louis Poinsot was born on January 3rd, 1777, in Paris. He attended Louis-le-Grand school in the city and, aged 17, he sat the entrance examination for the École Polytechnique. Despite performing poorly in the algebra assessment, he was admitted and studied there for three years, after which he moved to the École des Ponts et Chaussée, in order to become an engineer. However, while undertaking the course for engineering, he became interested in abstract mathematics and neglected his more practical studies. Poinsot decided to change his choice of career and instead became a mathematics teacher, with his first teaching position being at the Lycée Bonaparte in Paris, where he taught from 1804 to 1809. He then moved into higher education, being appointed to the position of Inspector General of the Imperial University of France in 1809 by Napoleon I. The Imperial University had been established in France in 1806 by Napoleon I and opened in 1808. It was one of the main elements of Napoleon I’s institutional reorganisation, and focused on the training of skills that the country needed, particularly in law and medicine, but also in other sciences. The Imperial University had its main buildings on the Champs-de-Mars and the Île des Cygnes (not to be confused with the Île aux Cygnes) in Paris. It controlled education in all the universities in France and also had control over primary and secondary education. The scope of the university’s work meant that Poinsot travelled extensively across France in his new role. Poinsot’s main areas of mathematical interest were geometry, mechanics and statics. He published his first work in 1803 while he was still studying to be an engineer, entitled Eléments de statique. In this book, he approached the subject of mechanics without any explicit reference to dynamics. The book also contained for the first time the idea of “torque”, or a couple, which he later applied in his 1834 publication Théorie nouvelle de la rotation des corps, where he considered the nature of rotational motion by means of an ellipsoid rolling on a fixed plane. Previous work on rotational motion had taken an analytical approach which had not allowed students to visualise the motion, whereas Poinsot’s geometrical approach meant motion of a rigid body could be considered as if it were a single point moving through space. He continued his mathematical research while he was teaching at the Lycée, publishing two memoirs in 1806, on the composition of moments and the composition of
SYMmetryplus 79 Autumn 2022 11 areas, and on the general theory of equilibrium and of movement in systems. In 1809 Poinsot published a memoir on polygons and polyhedra, following his discovery of four new regular polyhedra, namely the small stellated dodecahedron, the great stellated dodecahedron, the great icosahedron and the great dodecahedron. The first two of these had previously been described in Kepler’s Harmonice Mundi in 1619, but Poinsot was unaware of this and had discovered them independently. Two-dimensional representations of these polyhedra are shown below: All of these memoirs were well-received by mathematicians and the wider scientific community. Poinsot gained himself a strong reputation for publishing fully-developed results that were presented in a clear and elegant manner. The quality of his research led to him being appointed as Assistant Professor of Analysis and Mechanics at the École Polytechnique, which he undertook alongside his Inspector General role. Poinsot retained his position at the École Polytechnique until the school was reorganised in September 1816, but he only taught there for three years, arranging in 1812 for Reynaud and subsequently Cauchy to take over his lecturing duties, so that he had more time for his other interests and commitments. In 1813 he was elected to the mathematics section of the Académie des Sciences. After the reorganisation of the École Polytechnique, Poinsot became its admissions examiner for 10 years, and on several occasions after 1830 he also worked with its Conseil de Perfectionnement, which ensured coherence of the various programmes of study. Poinsot retained his role as Inspector General until 1824. He was appointed to the Bureau des Longitudes from 1839, eventually becoming its President, which he held until his death. He was nominated in 1840 to the Conseil Royal de l’Instruction Publique, which dealt with higher education issues. This was a moderately political position, which resulted in him being nominated in 1846 to the Chambre de Paris, the same year in which he was made an Officer of the Légion d'honneur by King Louis Phillipe I. When the Senate was formed in 1852, Napoleon III made him a member. Poinsot continued his mathematical research throughout his life, investigating topics in number theory as well as geometry and mechanics. His investigations in number theory including looking at primitive roots, certain Diophantine equations, and how to express an integer as the difference of two squares. The quality of his research and its findings led to him being elected as a Fellow of the Royal Society in 1858, the citation for which noted that he was “distinguished for his important original researches in Mathematics”. Poinsot died in Paris on December 5th, 1839. He was buried in the Père Lachaise cemetery in the eastern part of the city, and his gravestone gives details of his various achievements in mathematics and the honours that were bestowed on him. The inscription notes that he discovered new principles of one of the theories known in ancient times, invented by Archimedes and perfected by Galileo Galilei. Honours bestowed on Poinsot posthumously include Rue Poinsot in the Montparnasse district of Paris, in the 14th Arondissement (1864), and Crater Poinsot, on the northern part of the far side of the Moon (1970). Jenny Ramsden
12 SYMmetryplus 79 Autumn 2022 DIFFERENCE OF TWO SQUARES Many mathematicians, including Louis Poinsot, have considered the properties of squares at some point during their careers. If we look at the difference of two squares, some of which are set out in the table below, we can begin to see some patterns: n2 12 22 32 42 52 62 72 82 92 102 12 0 3 8 15 24 35 48 63 80 99 22 3 0 5 12 21 32 45 60 77 96 32 8 5 0 7 16 27 40 55 72 91 42 15 12 7 0 9 20 33 48 65 84 52 24 21 16 9 0 11 24 39 56 75 62 35 32 27 20 11 0 13 28 45 64 72 48 45 40 33 24 13 0 15 32 51 82 63 60 55 48 39 28 15 0 17 36 92 80 77 72 65 56 45 32 17 0 19 102 99 96 91 84 75 64 51 36 19 100 The table has been completed for the first ten squares, looking at the positive difference each time (the smaller subtracted from the larger). From the results, it can be conjectured that every odd number greater than 1 can be expressed as the difference of two consecutive squares (for example, 3 = 22 − 12, 5 = 32 − 22, 17 = 92 − 82, and so on, as shown on the table in the yellow cells). We can test this algebraically: Let represent a positive integer, so that ≥ 1. We can express consecutive squares as being ( + 1)2 and 2. Their difference is ( + 1)2 − 2. If we expand the brackets and simplify: ( + 1)2 − 2 = 2 + 2 + 1 − 2 = 2 + 1. With ≥ 1, the values that the expression 2 + 1 generates are 3, 5, 7, 9, 11 and so on, which are the odd positive integers greater than 1. Our conjecture was accurate; every odd number greater than 1 can be expressed as the difference of two squares. But what about the even numbers? Looking again at the table, we have a set of multiples of 4 being generated, as indicated in the blue cells, this time from squares that are alternate to each other (for example, 12 = 42 − 22, 16 = 52 − 32, 28 = 82 − 6, and so on). We can investigate this algebraically: Let represent a positive integer, so that ≥ 1. Alternate squares can be represented as ( + 1)2 and 2. Looking at their difference, expanding the brackets and simplifying: ( + 1)2 − ( − 1)2 = 2 + 2 + 1 − ( 2 − 2 + 1) = 4 . With ≥ 1, the values of 4 are 4, 8, 12, 16, 20, and so on, which are the positive multiples of 4. However, looking back at the table, we see that we are not able to generate 4 itself, as the first even number that occurs in the table is 8. This is because, if we go back to our algebra, a value of = 1 in the original formula would give the value of ( − 1)2 as zero, and we have only been considering squares from 12 onwards. Therefore, we can generate every positive multiple of 4 that is strictly greater than 4 from the difference of two squares. Considering differences of two squares generally gives a multitude of further outcomes. The reader might want to explore why some numbers can be expressed as a difference of two squares in more than one way (for example, 56 can be expressed as 92 − 5, and also as 152 − 132). Other considerations might include what pairs of squares will yield a difference that uses exactly one occurrence of each digit. Some examples of these are given below: 123456789 = 111152 − 2942 987654321 = 11111911612 − 291885602 152976384 = 123722 − 3002. Warning – there are 272158 numbers involving all nine digits used once only that can be expressed as the difference of two squares, so it could take quite a long time if anyone wants to find them all. Jenny Ramsden
SYMmetryplus 79 Autumn 2022 13 CROSSNUMBER 5 NUMBERS ACROSS (UPPER CASE) A Factor of g B Arrangement of M C i / D D n – C E F – 2n F Prime factor of G G Palindromic arrangement of i > i H Factor of e I Prime number > 90 J Arrangement of i K Palindromic cubic number M Arrangement of B N Palindromic arrangement of n Q Arrangement of r R Multiple of H T Factor of j numbers down (lower case) a Factor of g b 10h c Arrangement of i d Palindromic number e Palindromic arrangement of n f 7a g aA h Square number i 7 × 113 × 13 j Factor of e k Multiple of t m n inverted n 2 × 3 × 11 × 17 q Arrangement of r, or R r Palindromic square number t Factor of M Mike Rose A a b B c Cd e D f E g F h G i Hj Ik J m n Kq M N r Q R t T This page can be found as a separate resource sheet at Mathematical Association - SYMmetryplus (m-a.org.uk)
14 SYMmetryplus 79 Autumn 2022 MARGARET BRYAN – A GEORGIAN POLYMATH Margaret Bryan: frontispiece to her Lectures on Natural Philosophy, 1806 Margaret Bryan (née Haverkam) was born in 1759 and became a prominent teacher and writer by the end of the 18th century. In her preface to A Compendious System of Astronomy (1797) she published the letter she received from Charles Hutton (see Jenny Ramsden’s article in SYMmetryplus 54 Summer 2014), with whom she corresponded. The 3rd 1805 edition of this book included an advertisement for her school at Blackheath. This edition was dedicated to her pupils who are said to have asked for the frontispiece portrait of Margaret with her two daughters, shown below. It is thought the ages of her daughters might be around 13 to 15 years old. Note the display of mathematical instruments, an important sign of a mathematical practitioner. These are an armillary sphere, dividers, globe, quadrant and her telescope, likely to be the 4inch reflector with a 2-foot brass tube on an equatorial mount that she purchased from W. & J. Jones of Holborn. Both her books on astronomy and natural philosophy contained subscription lists (names of people who had pre-paid for the book, often at a lower than published price). The numbers of women subscribers were 125 (33%) and 155 (42%) respectively, which may seem surprisingly high, but this was in fact quite usual for women authors. Knowledge of women’s interest in mathematics cannot be judged solely by reference to book subscription lists: this sample is biased in many ways, but could be an area worthy of research. Her writings suggest that she completed 8 years of study before having 7 years of practical experience and
SYMmetryplus 79 Autumn 2022 15 the publication of her first book. By 1790 she had started teaching at Northumberland House, Northdown Road, Margate. She continued her astronomical interests and wrote to William Herschel (1738-1822) on October 3 1811, meeting the Herschel family in Slough at some point. It seems highly likely that she would have met Caroline Herschel at that point. As well as the two books mentioned, Margaret also published a board game: Science in Sport, or, the Pleasures of Astronomy in 1804, shown here. Image Courtesy YCBA Her last book was an Astronomical and Geographical Class Book for Schools, a thin octavo published in 1815. The date of her demise is unknown. Peter Ransom PULL THE PULLEYS Remember those pulley problems in mechanics? Did you ever experience them in a practical? If you visit the National Waterways Museum in Gloucester docks, you can! So, assuming the mass of each sack to be M kg, the pulleys to be frictionless, and the rope to be fixed at an angle to the vertical, calculate the force necessary in each of the three cases to raise the bags. Peter Ransom Hint: The MA is perfect.
16 SYMmetryplus 79 Autumn 2022 WHEN IS A NUMBER NOT A NUMBER? I expect you know what numbers are, and how they work. We use them to count and to measure, and we can add, subtract, multiply and divide them. Well, sometimes. Think about telephone numbers. We call them numbers, and they certainly look like numbers, being composed of digits, but are they really numbers? They don’t count or measure anything, and a big hint that they are not really numbers is that a leading 0 is significant. 52 and 052 mean the same as numbers, albeit that 052 is a funny way to write it, but try missing the initial 0 off a telephone number and see where it gets you. That is why if you enter a telephone number into a spreadsheet or database, you have to use a text field not a numeric one, or you will lose any initial zeroes. So, Excel does not think a telephone number is a number. Furthermore, you cannot add or subtract two telephone numbers and get anything meaningful: you certainly cannot expect the answer to be another telephone number. That is even more true of multiplication and division. Not even the size is significant: as a telephone number 352074 is no bigger, heavier or better than 120356; it is just different. So, I put it to you that telephone numbers are not really numbers at all, they are just identifiers, and so are credit card numbers, bank account numbers and lots if other things we call numbers that are composed of digits, but that serve only to provide a unique name for something. It would not make any difference to their function if they had letters in them, and in fact some of the things we call numbers do have letters in them, such as National Insurance numbers, and driving licence numbers. As an extreme example, vehicle registration numbers, as written on what we call number plates, are mostly letters. These “numbers” with letters in them are obviously not really numbers, but my contention is that a telephone number is really no more a number than a National Insurance number is. I do not think English really has a good word for these things, or if it does, we do not use it. I am going to call them identifiers for the rest of this article, numeric identifiers if they consist entirely of digits. A Vehicle Registration “Number” is mostly letters What got me interested in numeric identifiers is how we say them. If you say the telephone number 352074 out loud, you do not say “three hundred and fifty-two thousand and seventy-four”, which is how you would say it as a number. You treat is like the meaningless string of symbols it is, just reading the symbols out one by one: “three five two oh seven four”. That “oh” is a bit of a giveaway that we do not really think of these things as numbers. Perhaps it should be “zero”, but it does not really matter. As we have seen it would not make any difference to the usefulness of telephone numbers if they contained letters, so there is really no harm in calling that round symbol “oh”. Actually, the rules for pronouncing numeric identifiers are a bit more complicated than that, particularly for short ones. Consider road numbers. Here in Gloucestershire, we pronounce the A4019 “ay four oh one nine” just as the rule above suggests, although my car’s satnav fails the Turing test here by pronouncing it “ay four thousand and nineteen”. However even we humans pronounce the A38 “ay thirty-eight”, and in fact I think two-digit numeric identifiers are always spoken as numbers. Three-digit numeric identifiers ending with that round symbol are another exception, the A430 being “ay four thirty”, so neither “ay four hundred and thirty”, nor “ay four three oh”. I do not think we do that when there are more than three digits. There are longer exceptions too, for example the A1000 is “ay one thousand”. Then there are the doubles – the telephone number 516344 might be pronounced “five one six three double four”.
SYMmetryplus 79 Autumn 2022 17 Altogether it is much more difficult to codify the way numeric identifiers are spoken than it is for numbers. On the other hand, listening to someone say a string of digits is the best way to determine whether they are thinking of it as a number or a numeric identifier. There is, by the way, a fascinating feature of the way we say numbers, as opposed to numeric identifiers, that you may not have noticed. We never have to say “zero” or “nought” except when saying the number 0. The zeroes we write in non-zero numbers are place markers, indicating a power of ten no multiple of which is being used in this number. For example, we need a 0 in 207 to distinguish it from 27, and we need two of them in 2007. But when we pronounce numbers, we do not need those place markers, because all the powers of ten are pronounced differently: “twentyseven”, “two hundred and seven” and “two thousand and seven”. We have no more need for the placemarker zero when speaking numbers than the ancient Romans did in their written numbers: XXVII, CCVII and MMVII are quite different. Turning back to numeric identifiers, we write them differently from the way we write numbers too. For numbers it is fairly standard, with a comma separator every three digits, or sometime a thin space, for example 76,543 or 76 543. Numeric identifiers on the other hand are written in all manner of ways. For example, local telephone numbers are written without a separator however long they are, but if the area code is included, a space separates it. Bank card numbers are written in groups of four digits separated by spaces, and so on. Online it is a bit of a muddle: some web sites insist you put the separators in, others insist you do not. One last thought. Bank card numbers are an interesting halfway house between numbers and numeric identifiers, because the last digit is a checksum calculated from the first fifteen (look up the Luhn algorithm if you would like to know how that works). So, although the first fifteen digits form an identifier, it is important that the symbols are digits not letters, in order to be able to calculate the checksum. It should perhaps be thought of as 16 1-digit numbers rather than one 16-digit number. Similarly, an ISBN (International Standard Book Number) is printed on the back of title page of every book, and it too has a checksum, which Wikipedia article on ISBN explains. There are two sorts of ISBN, and for the 10-digit version, the last symbol is usually a digit, but is sometimes X. In a sense that is a Roman numeral X, because the checksum is calculated using mod 11 arithmetic, so sometimes comes out as 10, which is written X. How weird is that? Also see how strangely it is written, with hyphen separators. Jim Simons MARY SOMERVILLE Mary Somerville was a Scottish 19th century polymath, described at the time as the Queen of Science. Like Margaret Bryan, she was a mathematician and astronomer. November 29 this year marks the 150th anniversary of her demise. For more details, see Jenny Ramsden’s article on Mary in SYMmetryplus 43.
18 SYMmetryplus 79 Autumn 2022 SIMPLY SOLVE - 12 Here are some familiar and not so well-known problems. How many can you solve? How many cubes? (The cubes are not glued in any way.) 1 Two angles are supplementary. One angle is 39° less than twice the other angle. Find both angles. 2 Two numbers and 16 have a least common multiple of 48 and a highest common factor of 8. Find . 3 What number comes next? 2, 2, 4, 12, 48, __? 4 Two dice are rolled, and the numbers added. What is the probability that the total score is a prime number? 5 The Bob’s age is five times that of his son James. The sum of their ages is 72 years. What is the age difference? 6 It takes 414 digits to number the pages of a book with integers. If the page numbering starts with 1 and continues sequentially, what is the final page number? 7 8 In your mind, visualise a cube. Now imagine that a knife slices away each vertex with a straight plane cut. How many edges are there now on the cube? 9
SYMmetryplus 79 Autumn 2022 19 Neil Walker Create an equation that uses once only the digits 2, 3, 4 and 5 and the symbols + and =. 10 An apple farmer goes to the nursery to buy 10 apple trees. The owner says that if the farmer can plant the 10 trees in 5 rows of 4, he can have the trees for free. How does the farmer do it? 11 Olivia and Emma took turns multiplying numbers. Olivia picked the number four. Emma multiplied that by 4 to get 16. Olivia multiplied that by four to get 64. Emma multiplied that by 4 to get 256. This continued until one of them got 1,048,576. Who came up with this number? 12 Jack, Emily, and Jade can finish painting a fence in 2 hours. If Jack does the job alone, he can complete it in 5 hours. Emily, working alone can finish it in 6 hours. How long will it have taken Jade to complete the job on her own? 13 Ricky drove for 3 hours at a speed of 50 miles per hour and for 2 hours at 60 miles per hour. Find his average speed for the whole journey. 14 How many ways can 20 be made by adding just three of the numbers below? Each number can be used more than once. 2 4 6 8 10 12 14 15 A set of football matches is to be organised in a “round-robin” fashion, meaning that each team participating plays a match against every other team just once. If 91 matches in total are played, how many teams participated? 16 Two numbers are in the ratio 3 ∶ 2. If 2 is added to the first number and 6 is added to the second number, the ratio becomes 4 ∶ 5. Find the two numbers. 17
20 SYMmetryplus 79 Autumn 2022 THE BLETCHLEY EQUATORIAL SUNDIAL If you ever visit Bletchley Park, make sure you see the magnificent sundial at the Brunel Roundabout. Made by Wendy Taylor in 1982, the gnomon points true North and is inclined at 52°, the latitude of the sundial. ©2022 The Mathematical Association All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the copyright owner. Requests for permission to copy should be sent to the Editor-in-Chief at the MA office above. The views expressed in this journal are not necessarily those of The Mathematical Association.
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