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Breaking News the 2019 IMO Number 68 Spring 2019 La Force du Savoir (The Power of Knowledge) This 35 tonne granite sculpture was spotted at Sartilly, a French town near the Mont St Michel. It represents the range of books in a library and on the side shown you can see the Golden Ratio at the top. On the other side are seven words chosen by the children in the area: Liberte, Creation, Respect, Effort, Evolution, Imagination and Ecriture. Breaking News THE 2019 IMO This year the 60th International Mathematical Olympiad (IMO) will take place in Bath in July. The Opening Ceremony will take place on July 15 and the Closing Ceremony on July 21. The IMO is the World Championship Mathematics Competition for High School students and is held annually in a different country. The first IMO was held in Romania, with seven countries participating. It has gradually expanded to over 100 countries from five continents. The IMO Board ensures that the competition takes place each year and that each host country observes the regulations and traditions of the IMO. Last year the UK finished 12 out of 107 and Agnijo Banerjee was ranked 1st with a perfect score of six 7s. To find out more about the 2019 event, visit www.imo2019.uk/. Information about the IMO in general, with results over the years, visit www.imo-official.org/ EDITORIAL CROSSNUMBER Welcome to 2019. I hope this new year works out well 1 2 3 for you in whatever you do, wherever you are. On December 21st last year the largest known prime 4 number, 2182,589,933 − was discovered. It has 24,862,048 st digits and is the 51 Mersenne prime and was found by 5 6 Patrick Laroche on December 7. Read more about it at www.mersenne.org. 7 8 9 In this issue we start with another crossnumber from Paul Harris and then Paul Stephenson has been very 10 busy working with shapes, starting by dissecting even- sided polygons into rhombuses. Later on, he writes 11 12 about pentagon 15 that can be used to tile the plane and how it was discovered. It is interesting that he mentions glide reflections: at the start of this century these were part of the GCSE mathematics syllabus. Neil Across Walker sets us some interesting work on circle parts 1. A prime number times 49. and regions as well as providing us with two pages of 3. The largest two-digit prime number. short problems. I asked Jenny Ramsden to write about 4. The smallest number with four different prime Katherine Johnson as she (KJ) celebrated her century factors. last year and this leads on nicely to Graham Hoare’s 5. A prime number which equals n4 −3 for some value article on a rocket equation. That’s followed by a bit on of n . experimenting with rockets and the UK Youth Rocketry 7. This number equals the square of the sum of its Challenge. A big thank you to Matthew Mason who has digits. let us know how he solved the interesting squares 10. This number is a palindrome where the middle digit problem in the last edition of SYMmetryplus. is the sum of the other digits. The Mathematical Association publish a number of 11. Two dozen. journals, one of which, The Mathematical Gazette, 12. A power of 2. contains some student problems and I have included Down some here for those of you who like to be challenged. 1. The digits of this number form a geometric All the answers and an index can be found on the sequence. SYMmetryplus page on the MA website: 2. A perfect cube. http://www.m-a.org.uk/symmetry-plus. 3. One more than a perfect square. Articles and pictures on what inspire you are always 6. A power of 2 multiplied by 101. welcome! Please send them to me at 8. A Fibonacci number that is also a perfect square. [email protected]. 9. Three consecutive digits. Peter Ransom Paul Harris 2 SYMmetryplus 68 Spring 2019 RHOMBS IN POLYGONS As the title states, this piece is about polygons – in fact even-sided regular polygons – dissected into rhombuses. Everything of importance is contained in the following figures. Therefore, before I continue, study it carefully and see what ‘jumps out at you’. Fig. 2 The most important thing is the one you are least likely to spot. Start outside one of the pictures and follow a track across the rhombs, following this rule: Always exit one rhomb across a side parallel to the one across which you entered. You find you can cross the whole polygon and come out the other side. Even-sided regular polygons are examples of zonogons. A zone is a band of rhombs like the one you have just traversed. If our polygons have 2n sides, then n in Fig. 1 is 4 – it is Fig. 1 an octagon; in Fig. 2, 5 – it is a decagon. In the second case we can arrange the rhombs in a ‘lotus flower’ SYMmetryplus 68 Spring 2019 3 pattern (top) and rearrange them in a ‘star’ pattern These angles decrease by an odd number of units each (bottom). In Fig. 1, we cannot do this unless we shrink time. If we are to reach the centre, the gap between them first! Because we have to scale them by ½, there adjacent rhombs must close up, i.e. the angle must are four times as many tiles on the second figure as on become exactly 0. This will only happen if n is itself an the top one. odd number. So it works for the decagon but not the octagon. So, the question is: Why is this necessary in the first case but not the second? In the ‘lotus flower’ arrangement in a dodecagon below we have marked supplementary angles through a zone Experiment on this hypothetical polygon. Start at the to show how the same rhombs get repeated but in perimeter and work your way inwards, adding rhombs different orientations. Here n = 6, an even number, so as you go and working out angles. These are the facts the angles go from 1 to 5, an odd number. This means you need: that we will always have one rhomb with equal 1. The interior angle. Using Seymour Papert’s TTTTT supplementary angles in the middle - a square! (The Total Turtle Trip Theorem), this exterior angle is Notice also that, counting the rhombs from the top, we 2ππ = . 2nn have 1 + 2 + 3 + 4 + 5, the fifth triangular number, which 5+ (5 1) is =15 of them. Write the formula in terms of n. 2 It is not possible to dissect an even-sided regular polygon into fewer rhombs, though we shall not ππ … so this, the interior, angle is π− = (n − 1) . nn attempt to prove this. π It will be handy to make your unit angle. n 2. Opposite angles of a rhombus are equal. 3. Adjacent angles of a rhombus are supplementary – so, in our case, the straight angle is n, so we might have n – t and t as our two supplementary angles. 4. A whole angle is equal to 2n. Confirm the sizes of the angles I have marked as I have followed a zone: Go to the Wikipedia entry ‘Regular polygon’ and down to section 2.5, ‘Dissections’. Paul Stephenson Editor’s note: the plural of rhombus (or rhomb) can be rhombuses, rhombi (or rhombs). 4 SYMmetryplus 68 Spring 2019 KNIGHTS’ TOURS only once. A possible knight’s tour for a 3 by 4 board is given below, the number 1 showing where the knight starts, 2 showing its second position, and so on. 3 6 9 12 8 11 2 5 1 4 7 10 Clearly in boards with only 1 or 2 rows, the knight does not have enough space to turn around adequately so a In chess a knight moves in an L shape such that it knight’s tour is not possible (apart from trivially in a 1 moves two squares along a row or column followed by by 1 board where the tour has already been one square in a direction perpendicular to this. completed!) Normally a chess board is 8 squares by 8 squares but What about boards with 3 rows? for our purposes in this article the board could be We already know that the 3 by 1 and 3 by 2 are not rectangular of any number of rows and columns. going to work, and a knight sitting in the middle of a 3 Below are shown some knights, shown by the letter N by 3 board will not have any squares to go to! (often used in chess to distinguish it from the King, However, the 3 by 4 board gives us some hope that which uses K), with some asterisks marking the squares future boards may be possible. to which each knight could move. However, this does not occur immediately. Why not? N Let us examine a 3 by 5 board. * * * * * * x N * * * * It will be noticed that in travelling along a tour, the colour of the square that the knight occupies alternates * between what are traditionally referred to as black and * * white. In particular, on the 3 by 5 board shown this N * means that any knight’s tour must start and finish on a * N black square, given the way I have shaded the board, as there are 8 black squares and 7 white squares. * * Given that the tour goes through every square once and only once, it must go through the square on the far A knight’s tour is a route which in the knight, in making right of the middle row in which I have placed an x for legitimate knight moves, visits every square once and the xth position occupied by the knight.
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