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Andrius Kulikauskas
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Math Discovery Examples
Math Discovery Basic Three
Blank sheets* Independent trials. We may think of our mind as blank sheets, as many as we might need for our work. We shouldn't get stuck, but keep trying something new, if necessary, keep getting out a blank sheet. We can work separately on different parts of a problem. This relates also to independent events (in probability), independent runs (in automata theory) and independent dimensions (in vector spaces). If something works well, then we should try it out in a different domain. Sarunas Raudys notes that we must add a bit of noise so that we don't overlearn.9
 Avoid errorprone activity* Simplify .... We could multiply out all the terms, but it would take a long time, and we'd probably make a mistake. We need a strategy. pg.166 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2172
 Get your hands dirty* So we try another strategy, one of the best for beginning just about any problem: get your hands dirty. We try plugging in some numbers to experiment. If we are lucky, we may see a pattern. ... This is easy and fun to do. Stay loose and experiment. Plug in lots of numbers. Keep playing around until you see a pattern. Then play around some more, and try to figure out why the pattern you see is happening. It is a wellkept secret that much highlevel mathematical research is the result of lowtech "plug and chug" methods. The great Carl Gauss ... was a big fan of this method. In one investigation, he painstakingly computed the number of integer solutions to x**2+y**2<=90,000. ... Don't skimp on experimentation! Keep messing around until you think you understand what is going on. Then mess around some more. pg.7, 30, 36 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1412
 Get your hands dirty UCL problem solving technique 1 of 5
 Knowing when to give up* Sometimes you just cannot solve a problem. You will have to give up, at least temporarily. All good problem solvers will occasionally admit defeat. An important part of the problem solver's art is knowing when to give up. pg.16, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1414
 Mental toughness, confidence and concentration* But most beginners give up too soon, because they lack the mental toughness attributes of confidence and concentration. It is hard to work on a problem if you don't believe that you can solve it, and it is impossible to keep working past your "frustration threshold". ... You build upon your preexisting confidence by working at first on "easy" problems, where "easy" means that *you* can solve it after expending a modest effort. ... then work on harder and harder problems that continually challenge and stretch you to the limit ... Eventually, you will be able to work for hours singlemindedly on a problem, and keep other problems simmering on your mental backburner for days or weeks. ... developing mental toughness takes time, and maintaining it is a lifetime task. But what could be more fun than thinking about challenging problems as often as possible? pg.16, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1415
 Practice* Practice by working on lots and lots and lots of problems. Solving them is not as important. It is very healthy to have several unsolved problems banging around your conscious and unconscious mind. pg.25, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1423
 Toughen up* Toughen up by gradually increasing the amount and difficulty of your problem solving work. pg.24, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1424
 Vary the trials* The mouse in the trap ... threw himself violently against the bars, now on this side and then on the other, and in the last moment he succeeded in squeezing himself through ... We must try and try again until eventually we recognize the slight difference between the various openings on which everything depends. We must vary our trials so that we may explore all sides of the problem. Indeed, we cannot know in advance on which side is the only practicable opening where we can squeeze through. The fundamental method of mice and men is the same; to try, try again, and to vary the trials so that we do not miss the few favorable possibilities ... a man can vary his trials more and learn more from the failure of his trials than the mouse. pg.16, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc., quoting "Mice and Men" by George Polya, Mathematical Discovery, Volume II, 1965.1413
 Give up and try again differently. Dog with hula hoop in its teeth can't lift it over a step. So the dog lets go of it, picks it up in a different place, and tries again.
Symmetry. Symmetry group We unify internal and external points of view, link time and space, by considering a group of actions in time acting on space. Some aspects of the space are invariant, some aspects change. Actions can make the space more or less convoluted. At this point, we have arrived at a selfstanding system, one that can be defined as if it was independent of our mental processes. Our problem has become "a math problem". Analogously, in real life, after projecting more and more what we mean in general by people, including ourselves and others, we finally take us for granted as entirely one and the same and instead make presumptions towards a universal language by which we might agree absolutely.13
 Axiom schema of specification* Wikipedia: If z is a set, and P is any property which may characterize the elements x of z, then there is a subset y of z containing those x in z which satisfy the property. The "restriction" to z is necessary to avoid Russell's paradox and its variants. I think this relates to the idea that we can focus on the relevant symmetry and the relation between the locations affected or not by the symmetry group and the actions of that group.1170
 A bank of useful derivatives of "functions of a function"* We conclude our discussion of differentiation with two examples that illustrate a useful idea inspired by logarithmic differentiation. ... Logarithmic differentiation is not just a tool for computing derivatives. It is part of a larger idea: developing a bank of useful derivatives of "functions of a function" that you can recognize to analyze the original function. If a problem contains or can be made to contain the quantity f'(x)/f(x), then antidifferentiation will yield the logarithm of f(x), which in turn sheds light on f(x). pg.300 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2244
 Algebraic symmetry* Sequences can have symmetry, like this row of Pascal's Triangle: 1, 6, 15, 20, 20, 15, 6, 1 ... In just about any situation where you can imagine "pairing" things up, you can think about symmetry. pg.74, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1627
 Combination of techniques* We shall end the chapter with an exploration of the diophantine equation x**2 + y**2 = n ... where n is a prime p. Our exploration will use several old strategic and tactical ideas, including the pigeonhole principle, Gaussian pairing, and drawing pictures. The narrative will meander a bit, but please read it slowly and carefully, because it is a model of how many different problemsolving techniques come together in the solution of a hard problem. pg.274 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2229
 Complex Numbers* Complex numbers are the crossover artist's dream: like light, which exists simultaneously as wave and particle, complex numbers are both algebraic and geometric. You will not realize their full power until you become comfortable with their geometric, physical nature. This in turn will help you to become fluent at translating between the algebraic and the geometric in a wide variety of problems. ... We strongly urge you to read at least the first few chapters of our chief inspiration for this section, Tristan Needham's Visual Complex Analysis. This trailblazing book is fun to read, beautifully illustrated, and contains dozens of geometric insights that you will find nowhere else. ... If z=a+bi, we define the conjugate of z to be zbar = abi. Geometrically, zbar is just the reflection of z about the real axis. Complex numbers add "componentwise" ... Geometrically, complex number addition obeys the "parallelogram rule" of vector addition ... Multiplication by the complex number rCisTheta is a counterclockwise rotation by Theta followed by stretching by the factor r. So we have a third way to think about complex numbers. Every complex number is simultaneously a point, a vector, and a geometric transformation, namely the rotation and stretching above! pg.131134, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2162
 Exploit underlying symmetry in polynomials* Algebra problems with many variables or of high degree are often intractable unless there is some underlying symmetry to exploit. ... Solve x**4 + x**3 + x**2 + x**1 + 1 = 0 ... we will use the symmetry of the coefficients as a starting point to impose yet more symmetry, on the degrees of the terms. Simply divide by x**2 yielding x**2 + x + 1 + 1/x + 1/x**2 then make the substitution u := x + 1/x. pg. 75, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1629
 Fixed objects* When pondering a symmetrical situation, you should always focus briefly on the "fixed" objects which are unchanged by the symmetries. For example, if something is symmetric with respect to reflection about an axis, that axis is fixed and worthy of study (the stream in the previous problem played that role). pg. 72 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1585
 Four bugs chasing each other* a classic problem which exploits rotational symmetry along with a crucial fixed point ... Four bugs are situated at each vertex of a unit square. Suddenly, each bug begins to chase its counterclockwise neighbor. If the bugs travel at 1 unit per minute, how long will it take for the four bugs to crash into one another? pg.71 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1586
 Geometric symmetry* The simplest geometric symmetries are rotational and reflectional. pg. 71 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1582
 Fetching water for Grandma* Your cabin is 2 miles due north of a stream which runs eastwest. Your grandmother's cabin is located 12 miles west and 1 mile north of your cabin. Every day, you go from your cabin to Grandma's, but first visit the stream (to get fresh water for Grandma). What is the length of the route with minimum distance? pg.71 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1584
 Square inscribed in circle inscribed in square* A square is inscribed in a circle which is inscribed in a square. Find the ratio of the areas of the two squares. pg.70 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1583
 Harmony* An informal alternate definition of symmetry is "harmony". ... If you can do something that makes things more harmonious or more beautiful, even if you have no idea how to define these two terms, then you are often on the right track. pg. 70 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1580
 Invariant with respect to transformations* this topic [symmetry] is logically contained within the concept of invariants. If a particular object (geometrical or otherwise) contains symmetry, that is just another way of saying that the object itself is an invariant with respect to some transformation or set of transformations. For example, a square is invariant with respect to rotations about its center of 0, 90, 180 and 270 degrees. pg. 103, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1646
 Not quite symmetrical* The strategic principles of peripheral vision and rulebreaking tell us to look for symmetry in unlikely places, and not to worry if something is almost, but not quite symmetrical. In these cases, it is wise to proceed as if symmetry is present, since we will probably learn something useful. pg. 70 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1581
 Completing the square by trying to symmetrize* x**2 + a*x = x*(x + a) = (x + a/2 a/2)*(x + a/2 + a/2) = (x + a/2)**2  (a/2)**2 Above is a way to discover the completingthesquare formula by trying to symmetrize the terms, then adding zero creatively. pg. 163, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc. 1381
 Roots of Unity* The zeros of the equation x**n = 1 are the nth roots of unity. These numbers have many beautiful properties that interconnect algebra, geometry and number theory. One reason for the ubiquity of roots of unity in mathematics is symmetry: roots of unity, in some sense, epitomize symmetry... pg.131134, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2163
 Search for order* Many fundamental problemsolving tactics involve the search for order. Often problems are hard because they seem "chaotic" or disorderly; they appear to be missing parts (facts, variables, patterns) or the parts do not seem connected. ... we will begin by studying problemsolving tactics that help us find or impose order where there seemingly is none. pg. 69 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1524
 Symmetrize the coefficients* Solve the system of equations ... The standard procedure for solving systems of equations by hand is to substitute for and/or eliminate variables in a systematic (and tedious) way. But notice that each equation is almost symmetric, and that the system is symmetric as a whole. Just add together all five equations; this will serve to symmetrize all the coefficients ... Now we can subtract this quantity from each of the original equations to immediately get ... pg.166167 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2175
 Symmetry* Symmetry involves finding or imposing order in a concrete way, for example, by reflection. ... We call an object symmetric if there are one or more nontrivial "actions" which leave the object unchanged. We call the actions that do this the symmetries of the object (Footnote: We are deliberately avoiding the language of transformations and automorphisms that would be demanded by a mathematically precise definition.) ... Why is symmetry important? Because it gives you "free" information. If you know that something is, say, symmetric with respect to 90degree rotation about some point, then you only need to look at onequarter of the object. And you also know that the center of rotation is a "special" point, worthy of close investigation. pg. 6970 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1579
 The Gaussian pairing tool* Gauss, as a child, added up the sum 1 + 2 + 3 + ... + 100, presumably by pairing up the number 1 and 100, 2 and 99, 3 and 98, ... 50 and 51, yielding 50 pairs of 101 for a total of 5,050. Paul Zeitz notes this as an example of symmetry and calls it the Gaussian pairing tool. pg. 75, The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.1628
 Tilt the picture* we will present, with a "handwaving" proof, one important theoretical tool which will allow you to begin to think more rigorously about many problems involving differentiable functions. We begin with Rolle's theorem, which certainly falls into the "intuitively obvious" category. If f(x) is continuous on [a,b] and differentiable on (a,b), and f(a) = f(b), then there is a point u in (a,b) at which f'(u) = 0. The "proof" is a matter of drawing a picture. There will be a local minimum or maximum between a and b, at which the derivative will equal zero. Rolle's theorem has an important generalization, the mean value theorem. If f(x) is continuous on [a,b] and differentiable on (a,b), then there is a point u in (a,b) at which f'(u) = (f(b)  f(a))/(ba). ... the proof is just one sentence: Tilt the picture for Rolle's theorem! pg.297298 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc2242
 Transformation* The pattern of superposition points out a path from a leading special case (or from a few such cases) to the general case. There is a very different connecting path between the same endpoints with which the ambitious problemsolver should be equally acquainted: it is often possible to reduce the general case to a leading special case by an appropriate transformation. ... For a suggestive discussion of this topic see J. Hadamard, Lecons de geometrie elementaire. Geometrie plane, 1898; Methodes de transformation, pp. 272278. "Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving" by George Polya, 1962, John Wiley & Sons.2253
 Kindiak Math. Simple Geometry Stumps Almost Everyone. Morphism. Apply a morphism to lift the addition problem to a different domain. The addition can be understood to take place in the geometry itself. It can be understood to be meaningful in trigonometry (if we think of the addition formula). And it can be understood to be meaningful in complex numbers (especially if we know that angles can be defined as the arguments of a complex number.)
Context* O Context If you read the problem carefully, if you understand and follow the rules, then you can also relax them, bend them. You can thus realize which rules you imposed without cause. You can also change or reinterpret the context. These are the holes in the cloth that the needle makes. I often ask my new students, what is 10+4? When they say it is 14, then I tell them it is 2. I ask them why is it 2? and then I explain that it's because I'm talking about a 12hour clock. This example shows the power of context so that we probably can't write down all of the context even if we were to know it all. We can just hope and presume that others are like us and can figure it out just as we do. Analogously, in real life, it's vital to obey God, or rather, to make ourselves obedient to God. (Or if not God, then our parents, those who love us more than we love ourselves, who want us to be alive, sensitive, responsive more than we ourselves do.) If we are able to obey, then we are able to imagine God's point of view and even make sense of it.16
 10 + 4 = 2* I ask my students, What is 10+4? and they answer 14, and then I say, No, it is 2! Do you know why? Because I'm thinking about a clock. 10 o'clock plus 4 o'clock is 2 o'clock on a clock. What the example shows is that meaning ultimately depends on the context which we interpret. Any explanations that we write may also be misinterpreted. Thus there is no way to explicitly assure that somebody means what we mean. However, the context may indeed coincide in all that is relevant to us, either explicitly or implicitly. That's why existentialism is important, because it's important for us that our words and concepts be grounded in the questions relevant to our existence. Gospel Math.1840
 Axiom schema of replacement* Wikipedia: Less formally, this axiom states that if the domain of a definable function f is a set, and f(x) is a set for any x in that domain, then the range of f is a subclass of a set, subject to a restriction needed to avoid paradoxes. I think this relates to the idea that context allows us to "substitute variables" in different ways and perhaps with different results, different meanings, thus yielding flexibility of interpretation.1169
 Matchsticks. Given an equilateral triangle made of match sticks. How to move one stick and make it a square? You change the context: You make the square number 4 by moving the rightmost matchstick.
 Kindiak Math. Simple Geometry Stumps Almost Everyone. We are given a row of three squares and three angles {$\alpha = \textrm{arctan}(1), \beta = \textrm{arctan}(\frac{1}{2}), \gamma = \textrm{arctan}(\frac{1}{3})$} and we are asked for {$\alpha + \beta + \gamma$}. I got stumped because I supposed that there was no fraction expressing the last term. Indeed, I confused myself by thinking {$\beta$} was 30 degrees. What I failed to do was to calculate the sum, however approximately. Then I would have realized that it was {$\frac{\pi}{2}=90^{\circ}$}. At that point, I would have understood that there must be a geometric way to think about the sum. Rethinking the real number as a rational number would have driven me to persist.
 Given equation {$1^x=2$} go beyond the real numbers, think in terms of complex numbers.
 Grothendieck's method. Crack a nut by immersing it in water. nLab: The Rising Sea. Rethink the context for all of geometry or all of math, as relevant.
 Change the context  change the rules. Art changes the rules.
 A page of a book can mean two different things. "A child was playing with a book and tore out the pages 7, 8, 100, 101, 222 and 223. How many pages did the child tear out?" Mind Your Decision
