Klára Pintér– János Karsai With or Without With and Without Computing & Problem Solving

1. Klára Pintér– János Karsai With or Without With and Without Computing & Problem Solving

2. 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?

3. 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? An introductory problem Solution 1 Both are moving under the same time. Their velocities are known… The properties are investigated without knowing the motion itself.

4. 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? An introductory problem Solution 1 Solution 2 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 Both are moving under the same time. Their velocities are known… The properties are investigated without knowing the motion itself. Experiments

5. 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? Another problem Problem 2 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

6. 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

7. Pólya : Finding route Jackson: problem = target + difficulty Problem  Task Problem What is a problem?

8. The flow The flow of the problem-solving (after Pólya) Phenomenon Problems Solving process Solutions Summary, discussion

9. Control The control of problem-solving (after Neumann) The problem Knowledge bases The Mind (Controller) In mind . . . Library . . . Solutions Computerized Languages

10. The language The importance of the languages 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?

11. 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?

12. The language The importance of the languages Example 3. (x-a)(x-b)(x-c)…(x-z)=0

13. 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?

14. 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:

15. The language Some languages Mathematical formulation Real manipulation Visualization Some more examples!

16. (Programming) language + knowledge formulated in the language Computerized knowledge bases Examples The computerized knowledge bases Typical formulations • Can be given → Give it • Exists → Construct it • For all … → ??? • Visualize Main features • Symbolic, numeric operations • Data handling, structure operations • Visualization • Well defined language What is missed • Heuristic methods • Intuition • Theoretical basis • Singular cases Applicability • Algorithmic problem • „A New Kind of Science (Wolfram)” • Visualization, exploration Features

17. Simple mathematical constructions Computer applications of basic level Experiments and hand in the manual work Result: deeper understand, illustrations, new problems Example 1Give a function f(x) for which f’(0)=0, but zero is neither extremal nor inflection point. Construction

18. Simple mathematical constructions Computer applications of basic level Experiments and hand in the manual work Result: deeper understand, illustrations, new problems Example 2Take the powers 2n. What is the probability of that the first digit of 2n in the decimal system is 1,2,3,…,9. Experiment

19. The method of phase-mapping 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’+xn=0 (n 1). General problem: Consider the family of functions x(t,x0) (x(0,x0)=x0) such that x0 H0 . The question is: How do the properties of the phase maps Ht={x(t,x0), x0 H0} depend on the time? Animation Nonlinear system Linear system

20. 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)

21. New Kind of Science Two ideas and a construction Substitution, pattern recognition a_ → f(a) (Substitute anything by anything) Improve and extend the symbolism of structure operations {a,b,c} {x,y,z} ={ax,by,cz} List rotations: {a,b,c} → {c,a,b} {a,b,c} → {b,c,a} Examples

22. Examples: Strange behavior of a system Take the list of n real numbers: X={x1,x2,…,xn}Define the mapping T: Rn→RnT(X)={x1- xn, x2-x1,…, xn-xn-1}=|X-Shift(X)|Iterate T! (a discrete dinamical system) Statements: For odd n’s, the iteration can give periodic sequence For even n’s, almost every sequence becomes zero after finite steps, but… Experiments

23. Statement: 10 squares are sufficient. Examples: Grassing 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.

24. Examples: Grassing 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.)

25. Examples: Grassing Statement: 9 squares are not sufficient. Algorithmic method: See all the possible cases A smart method: During the grassing the circumference cannot increase. Understand the mechanism of the process. Grass can fill out only the covering square. Invariance principle !!! (see energy conservation, Ljapunov method, etc.) Simulations

26. Examples: Grassing NxN sized 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? How many grassed turves can garantee full grassing for any initial shape? Simple Generalizations: Simulations

27. Examples: Grassing What is the case of the torus and the sphere? Further generalizations The neighborhood is N, N-E, E, S-E, S, … Theory: life games, cellular automata, dinamical systems That is why S. Wolfram created Mathematica Simulations

28. Examples: Grassing 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? What about weeds among the valuable grass? Even more generalizations : stochastic diffusion Solution: Stochastic nonlinear models only with experimental results. Theory: stochastic cellular automata, theoretical ecology

29. Examples: hens and cocks 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?

30. Examples: hens and cocks 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? Smart proof: Chickens=eggs +1 H+C=2H+1

31. Examples: hens and cocks 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? Algorithmic study Give an algorithm for the egg-laying process! Smart proof: Chickens=eggs +1 H+C=2H+1 Construction

32. Examples: hens and cocks Further questions: • What is the probability of stopping after n steps? • What is the expected value of the time of stopping? • What happens if there are mutant hens lying more or less eggs? • What are the methods of the construction and investigation of trees? Construction of a tree

33. Examples: Only a question What is this?

34. Examples: Only a question The Internet in 1998. See: http://research.lumeta.com/ches/map/