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With or Without May 21, 2004 Klára Pintér– János Karsai With or Without With and Without Computing & Problem Solving

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With or Without May 21, 2004 Motto Why should we solve every problem immediately? Cannot we enjoy the problem itself? Miért kell minden problémát azonnal megoldani? Nem élvezetnénk magát a problémát?

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With or Without May 21, 2004 An introductory problem Problem 1 Uncle Joe is walking home at 0.7 m/s speed. His dog is running to him at 2 m/s speed. When they meet, they are at 10 m from the house. Then the dog runs back to the house, and then again to his owner, and so on. How long distance did the dog while his owner got home? Solution 1 Both are moving under the same time. Their velocities are known… The properties are investigated without knowing the motion itself.

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With or Without May 21, 2004 An introductory problem Problem 1 Uncle Joe is walking home at 0.7 m/s speed. His dog is running to him at 2 m/s speed. When they meet, they are at 10 m from the house. Then the dog runs back to the house, and then again to his owner, and so on. How long distance did the dog while his owner got home? Solution 1 Solution 2 Both are moving under the same time. Their velocities are known… The properties are investigated without knowing the motion itself. Describe the motion of the dog and sum the length of the pieces. Analogous to the billiard (here the wall is moving). Theory: Impulsive systems Experiments

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With or Without May 21, 2004 Problem 2 Another problem Uncle Joe was walking to his hous along a straight road at the speed 1 m/s. The neighbor’s dog watching at 20 m distance from the road and him observed and tried to catch him such that the dog was running at 1.4 km/h speed to the moving uncle Joe. How long distance did the dog take and how much time elapsed while the dog could catch uncle Joe? Solution: Desribe the motion of the dog. Find the path and calculate the length. Known problem: the equation of the motion can be easily given. Question: Can the eqn. be solved formally? Experiments Animation

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With or Without May 21, 2004 Another problem Generalization: the path of the missile (Robinson and the cannibal) What is the orbit of the missile if it flies to the target? What happens if the target is controlled and its orbit is general? What happens with a slow missile?. Is there an optimal orbit? Are there catching or excaping strategies? What cases can be handled formally? Experiments Animation

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With or Without May 21, 2004 What is a problem? Pólya : Finding route Jackson: problem = target + difficulty Problem Task Problem

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With or Without May 21, 2004 The flow of the problem-solving (after Pólya) The flow Phenomenon Solving process Problems Solutions Summary, discussion

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With or Without May 21, 2004 The control of problem-solving (after The control of problem-solving (after Neumann) Control The problem The Mind (Controller) Library Solutions Knowledge bases............ Languages Computerized In mind

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With or Without May 21, 2004 Example 1. I can see apples on the tree. I do not pick apples from and do not leave apples on the tree. How many apples were on the tree? The language The importance of the languages

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With or Without May 21, 2004 The language The importance of the languages Example 2. Uncle Joe has 8 horses: 4 brown, 3 gray and 1 black. What is the probability of that any randomly chosen horse can say about itself that uncle Joe has another horse of the same color?

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With or Without May 21, 2004 The language The importance of the languages Example 3. (x-a)(x-b)(x-c)…(x-z)=0

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With or Without May 21, 2004 The language The importance of the languages Example 4. Take an arbitrary point on each of two adjacent sides of a square. Connect them with the opposite vertices. The green or red region is bigger?

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With or Without May 21, 2004 The language The importance of the languages Example 4. Take an arbitrary point on each of two adjacent sides of a square. Join them with the opposite vertices. The green or red region is bigger? Hint:

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With or Without May 21, 2004 The language Real manipulation Mathematical formulation Visualization Some more examples! Some languages

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With or Without May 21, 2004 Applicability Algorithmic problemAlgorithmic problem „A New Kind of Science (Wolfram)”„A New Kind of Science (Wolfram)” Visualization, explorationVisualization, exploration What is missed Heuristic methodsHeuristic methods IntuitionIntuition Theoretical basisTheoretical basis Singular casesSingular cases The computerized knowledge bases (Programming) language + knowledge formulated in the language Computerized knowledge bases Examples Features Typical formulations Can be given → Give itCan be given → Give it Exists → Construct itExists → Construct it For all … → ???For all … → ??? VisualizeVisualize Main features Symbolic, numeric operationsSymbolic, numeric operations Data handling, structure operationsData handling, structure operations VisualizationVisualization Well defined languageWell defined language

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With or Without May 21, 2004 Simple mathematical constructions Example 1 Give a function f(x) for which f’(0)=0, but zero is neither extremal nor inflection point. Construction Computer applications of basic level Experiments and hand in the manual work Result: deeper understand, illustrations, new problems

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With or Without May 21, 2004 Simple mathematical constructions Example 2 Take the powers 2 n. What is the probability of that the first digit of 2 n in the decimal system is 1,2,3,…,9. Experiment Computer applications of basic level Experiments and hand in the manual work Result: deeper understand, illustrations, new problems

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With or Without May 21, 2004 Problem: Consider the differential equation x’’+a(t)x’+x=0. If, then the equation has a solution that tends to zero as t →∞. The problem is still open for the equation x’’+a(t)x’+x n =0 (n 1). The method of phase-mapping General problem: Consider the family of functions x(t,x 0 ) (x(0,x 0 )=x 0 ) such that x 0 H 0. The question is: How do the properties of the phase maps H t ={x(t,x 0 ), x 0 H 0 } depend on the time? Linear system Animation Nonlinear system

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With or Without May 21, 2004 New Kind of Science Computer applications of sophisticated level Take use of that Computing is a science. Deep mutual influence Result: „A New Kind of Science” (interdisciplinarity)

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With or Without May 21, 2004 Examples Substitution, pattern recognition a_ → f(a) List rotations: {a,b,c} → {c,a,b} {a,b,c} → {b,c,a} (Substitute anything by anything) Improve and extend the symbolism of structure operations {a,b,c} {x,y,z} ={a x,b y,c z } Two ideas and a construction New Kind of Science

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With or Without May 21, 2004 Examples: Strange behavior of a system Take the list of n real numbers: X={x 1,x 2,…,x n } Define the mapping T: R n →R n T(X)={x 1 - x n, x 2 -x 1,…, x n -x n-1 }=|X-Shift(X)| Iterate T! (a discrete dinamical system) ExperimentsStatements: For odd n’s, the iteration can give periodic sequence For even n’s, almost every sequence becomes zero after finite steps, but…

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With or Without May 21, 2004 We have a 10x10 meter size square guarden. How many of unit square turves are needed to grass the garden, if an empty square will be grassed if it has at least two grassed neighbours. Statement: 10 squares are sufficient. Examples: Grassing

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With or Without May 21, 2004 Statement: 9 squares are not sufficient. A smart method: During the grassing the circumference cannot increase. Invariance principle !!! (see energy conservation, perpetum mobile, Ljapunov method, etc.) Examples: Grassing

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With or Without May 21, 2004 Statement: 9 squares are not sufficient. A smart method: During the grassing the circumference cannot increase. Algorithmic method: See all the possible cases Understand the mechanism of the process. Grass can fill out only the covering square. Invariance principle !!! (see energy conservation, Ljapunov method, etc.) Examples: Grassing Simulations

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With or Without May 21, 2004 Simple Generalizations: Simulations NxN sized squareNxN sized square How the grass diffuse if 1,2,3,4 grassed neighbours are needed to occupy an empty square.How the grass diffuse if 1,2,3,4 grassed neighbours are needed to occupy an empty square. What is the role of the initial shape?What is the role of the initial shape? How many grassed turvesHow many grassed turves can garantee full grassing for any initial shape? Examples: Grassing

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With or Without May 21, 2004 Further generalizations Theory: life games, cellular automata, dinamical systems That is why S. Wolfram created Mathematica What is the case of the torus and the sphere?What is the case of the torus and the sphere? The neighborhood is N, N-E, E, S-E, S, …The neighborhood is N, N-E, E, S-E, S, … Examples: Grassing Simulations

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With or Without May 21, 2004 Even more generalizations : stochastic diffusion Even more generalizations : stochastic diffusion Theory: stochastic cellular automata, theoretical ecology Solution: Stochastic nonlinear models only with experimental results. How the grass is spreading if the probability of grassing is P(i), where i is the number of grassed neighbors?How the grass is spreading if the probability of grassing is P(i), where i is the number of grassed neighbors? How can we handle the extinction?How can we handle the extinction? What about weeds among the valuable grass?What about weeds among the valuable grass? Examples: Grassing

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With or Without May 21, 2004 Aunty bought a hen. It layed two eggs and then aunty cooked it. Either cocks or hens can hatch out of the eggs. The cocks are eaten, each hen lays two eggs and eaten after. Once, the barn-yard empties. If we know that 2004 ccoks are eaten, can we say the number of hens? Aunty bought a hen. It layed two eggs and then aunty cooked it. Either cocks or hens can hatch out of the eggs. The cocks are eaten, each hen lays two eggs and eaten after. Once, the barn-yard empties. If we know that 2004 ccoks are eaten, can we say the number of hens? Examples: hens and cocks

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With or Without May 21, 2004 Examples: hens and cocks Smart proof: Chickens=eggs +1 H+C=2H+1 Aunty bought a hen. It layed two eggs and then aunty cooked it. Either cocks or hens can hatch out of the eggs. The cocks are eaten, each hen lays two eggs and eaten after. Once, the barn-yard empties. If we know that 2004 ccoks are eaten, can we say the number of hens? Aunty bought a hen. It layed two eggs and then aunty cooked it. Either cocks or hens can hatch out of the eggs. The cocks are eaten, each hen lays two eggs and eaten after. Once, the barn-yard empties. If we know that 2004 ccoks are eaten, can we say the number of hens?

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With or Without May 21, 2004 Examples: hens and cocks Algorithmic study Give an algorithm for the egg- laying process! Construction Smart proof: Chickens=eggs +1 H+C=2H+1 Aunty bought a hen. It layed two eggs and then aunty cooked it. Either cocks or hens can hatch out of the eggs. The cocks are eaten, each hen lays two eggs and eaten after. Once, the barn-yard empties. If we know that 2004 ccoks are eaten, can we say the number of hens? Aunty bought a hen. It layed two eggs and then aunty cooked it. Either cocks or hens can hatch out of the eggs. The cocks are eaten, each hen lays two eggs and eaten after. Once, the barn-yard empties. If we know that 2004 ccoks are eaten, can we say the number of hens?

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With or Without May 21, 2004 Further questions: What is the probability of stopping after n steps?What is the probability of stopping after n steps? What is the expected value of the time of stopping?What is the expected value of the time of stopping? What happens if there are mutant hens lying more or less eggs?What happens if there are mutant hens lying more or less eggs? What are the methods of the construction and investigation of trees?What are the methods of the construction and investigation of trees? Construction of a tree Examples: hens and cocks

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With or Without May 21, 2004 What is this? Examples: Only a question

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With or Without May 21, 2004 The Internet in 1998. Examples: Only a question See: http://research.lumeta.com/ches/map/

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