Volume 29 MATHEMATICS COMPETITIONS

Mathematics Competitions Vol 29 No 1 20162A Typical Instruction: Dishing out a Collection ofFacts a la “Give a Man a Fish”Instruction is often reduced to memorization of a certain collection offacts: dates in history, theorems in mathematics, etc. While memorization and knowledge are of value, they seem to be overestimated ininstruction. I agree with the great Chinese Sage Lǎozı̌: giving a man afish will not solve man’s problem of survival.3Lǎozı̌ and a Skilled Approach to Life: “Teach aMan How to Fish”Lǎozı̌ proposes to teach a man fishing as a method of solving the problemof survival. This does go further than giving a man a fish. However, isit good enough in today’s world?4Beyond Lǎozı̌: Enable a Man to Learn How toSolve ProblemsNot every education is as good an investment as another. We ought togo beyond Lǎozı̌ and his universally celebrated lines. Is teaching skillsgood enough? Not quite, dear Sage, not in today’s rapidly changingworld. What if there is no more fish? What if the pond has dried outwhile your man has only one skill, fishing?A problem solver will not die if the fish disappears in a pond—he’lllearn to hunt, grow crops, solve whatever problems life puts in his way.And so, we will go a long way by putting emphasis not on training skillsbut on creating environment for developing problem solving abilities andattitudes. The proverb for today‘s world ought to be:Give a man skills, and you will feed him in the short run.Let a man learn solving problems, and you will feed him for alifetime.Mathematics & LifeEvery day we confront and solve a myriad of problems. Life is aboutsolving problems. And mistakes in solving life’s problems can be quite9

Mathematics Competitions Vol 29 No 1 2016costly. This is where mathematics comes in handy. Mathematics allowsus to learn how to think creatively, how to solve problems. And onceour student masters problem solving in mathematics, s(he) will be betterprepared to confront problems in any human endeavor.Are the Two Popular Approaches to Mathematics InstructionGood Enough?Today’s discussions of mathematical instruction seem to be reduced totwo competing approaches, “Embrace the Technology” vs. “Back to theBasics.” Should we not first ask, what are the goals of instruction?“Back to the Basics” is not the best solution, for it emphasizes mindnumbing drill, and treats students as robots, who need to be preprogramed with a set of skills. In the newer “Embrace the Technology” approach, I support taking a teacher off the lectern: one cannotteach mathematics, or anything for that matter. Students can learnmathematics only by doing it, with a gentle guidance of the teacher. Regretfully, this approach more often than not treats students like robots,and pre-programs them with skills of today. But technology nowadayschanges rapidly, as does the societal demands for particular skills.Providing public education is not only an ethical thing to do—it is aprofitable investment. Are there many jobs today for computer-illiteratepersons? And yet just one generation ago, computers were a monopolyof researchers, and one generation before that did not exist at all. Andso, we will go a long way by putting emphasis not on training skillsbut on creating atmosphere for developing problem solving abilities andattitudes.5The True Goal of Mathematics Instruction is toAllow Students to Discover What Mathematics Isand What Mathematicians DoStandardized three-letter tests, such as SAT, ACT, GRE, KGB, CIA(the latter two triples are from a different opera :-) can only inform ushow well a student does on these tests. Is this the goal of instruction?We ought to abandon standardized multiple choice testing of skills.10

Mathematics Competitions Vol 29 No 1 2016There are more important things to assess. Over the past 33 years, theColorado Mathematical Olympiad has been offering middle and highschool students 5 original problems and 4 hours to think, to invent,and to solve. We “test” predominantly not knowledge, not skills, butcreativity and originality of thought [7].Is the goal “teaching to the test,” as the past USA President GeorgeW. Bush believed? Not really. We all agree that problem solving isthe means of instruction. However, what is problem solving? A typicalsecondary school problem asks to find the hypotenuse of a right triangle,whose legs are 3 and 4, by using Pythagoras Theorem. No, not anymore, you would reply. Nowadays, at the Age of Technology, a typicalsecondary school problem asks to find the hypotenuse of a right triangle,whose legs are 3.1 and 4.2, by using Pythagoras Theorem and yoursmartphone.More generally, a secondary school problem has the structure A B,i.e., given A prove B by using theorem C. In real life, no one gives aresearch mathematician a B; it is discovered by intuition and is basedon experimentation. And of course, no one knows a C since nobodyknows what would work to solve the problem which is not yet solved: aresearch mathematician is a pioneer, moving along an untraveled path!And so, we ought to bring our secondary and college mathematics, whichoften looks so superficial, as close as possible to the ‘real’ mathematics.We ought to let our students experiment in our classroom-laboratory.We ought to let them develop intuition and use it to come up with aconjecture B. And we ought to let our students find those tools C thatdo the job of deduction proving the conjecture B. In my opinion, thetrue goal of mathematics instruction is to allow students discover in theclassroom what mathematics is, and what mathematicians do.6What Can Mathematical Olympiads Bringto Mathematics Instruction?In 1932, even the leader of mathematics David Hilbert observed thatfor most people mathematics is boring: “It is true, generally speaking,that mathematics is not a popular subject, even though its importance11

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Mathematics Competitions Vol 29 No 1 2016may be generally conceded.” This is where Olympiads come to therescue. Olympiads allow us to introduce secondary students to topics,ideas, and methods of ‘real’ mathematics in the context and terminology of secondary mathematics, in the form that is digestible by them.Problems of Mathematical Olympiads—as not much else—demonstratebeauty and elegance of mathematics. At the age of 14, I switched fromwriting and performing piano music to mathematics due exclusively tothe Moscow Mathematical Olympiads. In March 1989, Paul Erdős toldme that “the Olympiads create a new enthusiasm toward mathematics,and in this sense are very valuable.”For the 33 years of Colorado Mathematical Olympiad existence, we havebeen often asked a natural question: how does one create a Mathematical Olympiad? This and other related questions are clarified by theUniversity of Colorado, which produced the film “Thirtieth ColoradoMathematical Olympiad—30 Years of Excellence,” which can be foundon the Olympiad’s homepage http://olympiad.uccs.edu/.7The Moral Foundation Is CriticalThere is an opinion shared by many of my colleagues that all thatmatters is mathematics, Mathematik über Alles, if you will, above allmoral concerns. In my opinion, there is no good science or good artunless it is built on the foundation of high ethical principles. LuitzenEgbertus Jan Brouwer, a great Dutch mathematician and philosopherwrote in his 1929 letter: “It is my opinion that the tiniest moral matteris more important than all of science, and that one can only maintainthe moral quality of the world by standing up to any immoral project.”We have seen in history time and again how evil the usage of sciencecould be if it is not built on high moral foundation. Atrocities of NaziGermany alone provide countless examples of how science, technologyand even art can be used for ill deeds. My new book [9] is dedicatedto moral dilemmas of a scholar in the Third Reich and in the world oftoday. Lessons of history ought to enter our classrooms and guide ourstudents today. I value education, however, I must admit thatFine education does not guarantee high culture,And high culture does not guarantee humanity.13

Mathematics Competitions Vol 29 No 1 2016In order for creative work to be good, it must also serve the good.It ought to be humane. It has to be grounded in morality, empathy,compassion, and kindness. The Great Russian poet Alexander Pushkin(1799–1837) beautifully wrote about it. Let me translate his lines foryou:And people will be pleased with me for years to come,For I awakened kindness with my lyre,For in my cruel age I Freedom praised and sangAnd urged I mercy for the fallen people.And so we ought to pass to our students the baton of mercy and humanity, so that our students by their creative work contribute to the highculture of our small endangered planet.Part IIExamples & Aspects OfImplementaionAlright, but what kind of problems should we offer our students? Whatapproaches should we present in our classrooms? Permit me, to illustrateseven essential components of the state-of-the-art classroom.8Experiment in MathematicsFirst of all, we ought to set up a mathematical laboratory, where studentsconduct mathematical experiments, develop inductive reasoning and aninsight needed to create conjectures. Some examples can be found from[5].For example, a short experiment allows us to conjecture a formula for14

Mathematics Competitions Vol 29 No 1 2016the sum of cubes of consecutive integers:13 1213 23 3213 23 33 6213 23 33 43 102We observe that the sums of consecutive cubes are perfect squares. Butthe square of what number? If you are not able to develop a conjectureyet, continue to experiment: 13 23 33 43 53 152 . You will soonnotice that 13 23 33 43 53 (1 2 3 4 5)2 . This kind ofequality hold for all the values in our experiment, and the conjecture isready13 23 · · · n3 (1 2 · · · n)2 .We can now prove, for example, by mathematical induction, that boththe left side and the right side of the conjectured equality is equal to212 n(n 1) .9Construction of Examples in MathematicsConstruction of counterexamples is almost non-existent in secondaryeducation and even university, whereas counterexamples play a majorrole in mathematics, amounting to circa 50 % of its results. In fact, theGreat Russian mathematician Israel M. Gelfand once said, “Theoriescome and go; examples live forever.”You would agree that practically the entire school mathematics consistsof analytical proofs. In order to bring instruction closer to the ‘real’mathematics we ought to include in education construction of examplesand counterexamples. Let me share one example, where a constructionsolves the problem [7].Positive2 (18th Colorado Mathematical Olympiad, A. Soifer, 2001).Is there a way to fill a 2001 2001 square table T with pluses andminuses, one sign per cell of T , such that no series of interchanging all15

Mathematics Competitions Vol 29 No 1 2016signs in any 1000 1000 or 1001 1001 square of the table can fill Twith all pluses?Solution. Having created this problem and its solution for the 2001 Colorado Mathematical Olympiad, I felt that another solution was possibleusing an invariant, but failed to find it. Two days after the Olympiad, onApril 22, 2001, the past double-winner of the Olympiad Matthew Kahle,now a Professor at Ohio State University, found the solution that eludedme. It is concise and beautiful.Figure 1Define (see Fig. 1) Φ {the set of all cells of T , except those in themiddle row}. Observe that no matter where a 1000 1000 square S isplaced in the table T , it intersects Φ in an even number of cells, becausethere are 1000 equal columns in S. Observe also that no matter wherea 1001 1001 square S is placed in T , it also intersects Φ in an evennumber of unit squares, because there are 1000 equal rows in S (onerow is always missing, since the middle row is omitted in S.)Now we can easily create the required assignment of signs in T thatcannot be converted into all pluses. Let Φ have any assignment with anodd number of signs, and the missing in Φ middle row be assignedsigns in any way. No series of operations can change the parity of thenumber of pluses in Φ, and thus no series of allowed operations can createall pluses in Φ.16

Mathematics Competitions Vol 29 No 1 201610Utilizing AnalogyA sense of analogy could be a powerful tool. Here is one example.Problem 2 Prove that a map formed in the plane by finitely manycircles can be 2-colored (Fig. 2).Figure 2Proof. We partition regions of the map into two classes (Fig. 3): thosecontained in an even number of circles (color them gray), and thosecontained in an odd number of circles (leave them white). Clearly,neighboring regions got different colors because when we travel acrosstheir boundary line, the parity changes.Figure 3I am sure you realize that the shape of a circle is of no consequence. Wecan replace circles in problem 2 by simple closed curves. However, canwe replace simple closed curves by straight lines?17

Mathematics Competitions Vol 29 No 1 2016Problem 3 Prove that a map formed in the plane by finitely manystraight lines is 2-colorable (Figure 4).Figure 4An inductive proof is well known, but, as is usually the case with proofsby mathematical induction, it does not provide an insight. Decades ago Ifound a ‘one-line’ proof that takes advantage of similarity between simpleclosed curves and straight lines.Proof. Attach to each line a vector perpendicular to it (Fig. 5). Call thehalf-plane inside if contains the vector, and outside otherwise. Repeatthe proof of problem 2 word-by-word to complete the proof (Fig. 6).Figure 5Figure 618

Mathematics Competitions Vol 29 No 1 201611Method and Anti-MethodTiling with Dominoes (Method) Can a chessboard with two diagonally opposite squares missing, be tiled by dominoes (Figure 7)?Figure 7Solution. Color the board in a chessboard fashion (Figure 8). No matterwhere a domino is placed on the board, vertically or horizontally, it wouldcover one black and one white square. Thus, it is necessary for tileabilityto have equal numbers of black and white squares in the board—but theyare not equal in our truncated board. Therefore, the required tiling doesnot exist.Figure 8It is impressive and unforgettable for a student to see for the first timehow coloring can solve a mathematical problem. However, I noticed19

Mathematics Competitions Vol 29 No 1 2016that once a student learns a coloring idea, s(he) always resorts to itwhen a chessboard and dominoes are present in the problem. This iswhy I created the following ‘Anti-Method’ Problem and used it in theColorado Mathematical Olympiad [7].The Tiling Game (Anti-Method) (6th Colorado MathematicalOlympiad, A. Soifer, 1989). Mark and Julia are playing the followingtiling game on a 1988 1989 chessboard. They in turn are putting 1 1square tiles on the board. After each of them made exactly 100 moves(and thus they covered 200 squares of the board) a winner is determinedas follows: Julia wins if the tiling of the board can be completed withdominoes. Otherwise Mark wins. (Dominoes are 1 2 rectangles, whichcover exactly two squares of the board.) Can you find a strategy for oneof the players allowing him to win regardless of what the moves of theother player may be? You cannot? Let me help you: Mark goes first!Solution. Julia (i.e., the second player) has a strategy that allows herto win regardless of what Mark’s moves may be. All she needs is abit of home preparation: Julia prepares a tiling template showing oneparticular way, call it T , of tiling the whole 1988 1989 chessboard. Fig.9 shows one such tiling template T for an 8 13 chessboard.Figure 920

Mathematics Competitions Vol 29 No 1 2016The strategy for Julia is now clear. As soon as Mark puts a 1 1 tileM on the board, Julia puts her template T on the board to determinewhich domino of the template T contains Mark’s tile M . Then she putsher 1 1 tile J to cover the second square of the same domino (Fig.10).Figure 1012Synthesis and Combinatorial GeometrySecondary school mathematics consists predominantly of problems withsingle-idea solutions, found by analysis. We ought to introduce a sense ofmathematical reality in the classroom by presenting synthesis, by offeringproblems that require for their solution ideas from a number of mathematical disciplines: geometry, algebra, number theory, trigonometry,linear algebra, etc.And here comes Combinatorial Geometry for the rescue. It offers anabundance of problems that sound like a ‘regular’ secondary school geometry, but require for their solutions synthesis of ideas from geometry,algebra, number theory, trigonometry, ideas of analysis, etc. See forexample [4]; [6]; and [2]. Moreover, combinatorial geometry offers usopen-ended problems. And it offers problems that any geometry student can understand, and yet no one has yet solved! Let us stop thisdiscrimination of our students based on their young age, and allow themto touch and smell, and work on real mathematics and its unsolved problems. They may find a partial advance into solutions; they may settle21

Mathematics Competitions Vol 29 No 1 2016some open problems completely. And they will then know the answerto what ought to become the fundamental questions of mathematicaleducation: What is Mathematics? What do mathematicians do?In fact, I would opine that every discipline is about problem solving.And so the main goal of every discipline ought to be to enable studentsto learn how to think within the discipline, how to solve problems ofthe discipline, and finally what that discipline is about, and what theprofessionals within the discipline do. And mathematics to sciences doeswhat gymnastics does to sports: Mathematics is gymnastics of the mind.Doing mathematics develops a universal approach to problem solvingand intuition that go a long way in preparing our students for solvingproblems they will face in their lives.13Open Ended and Open ProblemsAs a junior at the university, I approached my supervisor ProfessorLeonid Yakovlevich Kulikov with an open problem I liked—he was mysupervisor ever since my freshman year. He replied, “Learn first, thetime will come later to enter research.” He meant well, but politicallyspeaking, this was a discrimination based on my young age. Seeing mydisappointment, Kulikov continued, “It does not look like I can stop youfrom doing research. Alright, whatever results you obtain on this openproblem, I will count as your course work.” Soon I received my firstresearch results, and my life in mathematics began.We ought to allow our students to learn what mathematicians do by offering them not just unrelated to each other exercises but rather series ofproblems leading to a deeper and deeper understanding. And we ought tolet students ‘touch’ unsolved problems of mathematics, give them a tasteof the unknown, a taste of adventure and discovery. Combinatorial geometry serves these goals well by providing us with easy-to-

Mathematical Olympiad and Further Explorations: From the Mountains of Colorado to the Peaks of Mathematics. He has founded and for 32 years ran the Colorado Mathematical Olympiad. Soifer has also served on the Soviet Union Math Olympiad (1970–