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The argument is generalized to n by n fields with n-1 weed parcels, derives a necessary condition for full growth, and applies recursion/backtracking to find all arrangements leading to full 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is the weed-growth rule used in the problem?","Question",{"text":63,"@type":64},"A parcel becomes overgrown only if it is surrounded on its sides by at least two adjacent weed parcels after some time.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What conclusion does the essay reach for the 10m x 10m field with 9 initial weed parcels?",{"text":68,"@type":64},"With the given conditions, the essay concludes that the whole field cannot grow into weed.",{"name":70,"@type":61,"acceptedAnswer":71},"How is the problem generalized beyond the original 10m x 10m case?",{"text":72,"@type":64},"The essay generalizes the result to fields of dimensions n x n with n-1 weed parcels, finds a necessary condition for full overgrowth, and then extends the approach to determine all winning initial 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Šljivančanin  \nMay 2007  \nMathematics  \nAbstract  \nIn my extended essay I analysed,solved and extended a problem,which appeared in aMathematical Olympiad.The text of the problem is as follows:  \nA piece of land of a square shape with dimensions 10mx 10m is divided into 100square parcels with dimensions 1mx 1m.Initially,9 of the parcels are overgrown byweed.If a parcel is surrounded by at least 2 parcels with weed from its sides aftersome time that parcel will be overgrown by weed.Can the whole piece of land growinto weed after some time?  \nIn order to answer the question,I firstly analyzed some equivalent,simpler versionsof the same problem and tried to solve them by system of exhaustion.This allowedme to construct my hypothesis:that with the given conditions the whole piece of landcannot grow into weed.The hypothesis was then proved by the system of following ofchange in a certain variable.  \nAfter the hypothesis was proved,the problem itself was generalised and it was shownthat the hypothesis holds for even more complex,equivalent cases,when having afield with dimensions nx n and n-1 parcels with weed.Furthermore,a generalcondition necessary for the whole piece of land to grow into weed was found.  \nIn the second part of the essay,this necessary condition was used to extend the initialproblem.The extension was about finding all the possible arrangements of 10 parcelswith weed which would result in the whole piece of land growing into weed.  \nRather than drawing all the solutions,the system for finding all of the arrangementswas developed by principle of recursion and backtracking.The system was thengeneralized for the piece of land with dimensions nxn and n weed parcels.  \n*1  \nContents  \nIntroduction……………………………………………………………………………………………………………………p1  \nApproaching the problem…………………………………………………………………………………p2  \nField of dimensions 3mx 3m…………………………………………………………………………………………p3  \nField of dimensions 4mx 4m………………………………………………………………………………………p5  \nHypothesis…………………………………………………………………………………………………………p8  \nProving the hypothesis………………………………………………………………………………………………p9  \nConclusion 1…………………………………………………………………………………………………………………p11  \nField with dimensions nxn and(n-1)weed parcels………………………………………p12  \nExtending the initial problem…………………………………………………………………………p12  \nConclusion 2…………………………………………………………………………………………………………………………………p21  \nReferences………………………………………………………………………………………………………………………p22  \n# Introduction\n\nThe problem I am solving originally appeared in one of the Mathematical Olympiads inthe1990s.The problem was found on the website:  \nhttp://www.matf.bg.ac.yw~matic/competitions/dodatne/ZKOMB_mll.pdf  \nThe text of the problem is:  \nA piece ofland of a square shape with dimensions 10mx 10m is divided into 100 squareparcels with dimensions 1mx 1m.Initially,9 of the parcels are overgrown by weed.If aparcel is surrounded by at least 2 parcels with weed from its sides after some time thatparcel will be overgrown by weed.Can the whole piece of land grow into weed after sometime?  \n1  \n   \n# Approaching the problem\n\n   \nIn order to solve this problem,I will make use of the indirect approach to it.Namely,sincethe number of all the possibilities to arrange 9 parcels with weed in the 10x10 field is thenumber of ways of choosing 9 parcels out of 100 parcels:  \nit is obvious that an attempt to draw all the possible arrangements does not lead to thesolution(unless one lives 3619162.497 years and draws one arrangement each minute,without having any breaks).Therefore,the way of solving this problem will involve findinga pattern that will enable us to arrange the parcels with the weed in a certain way,whichwilleventually lead to the whole piece of land growing into weed after some time.Failingto find such arrangement would mean that under the given conditions the whole piece ofland can never grow into weed.In the further parts of the solution the parcels which areovergrown by weed will b","cbCaikNIN2SUnobU","https://ap.wps.com/l/cbCaikNIN2SUnobU","pdf",8715635,25,"English","# Introduction\n## Approaching the problem\n## Field of dimensions 3m x 3m\n## Field of dimensions 4m x 4m\n## Hypothesis\n## Proving the hypothesis\n## Conclusion 1\n## Field with dimensions n x n and (n-1) weed parcels\n## Extending the initial problem\n## Conclusion 2\n## References","[{\"question\":\"What is the weed-growth rule used in the problem?\",\"answer\":\"A parcel becomes overgrown only if it is surrounded on its sides by at least two adjacent weed parcels after some time.\"},{\"question\":\"What conclusion does the essay reach for the 10m x 10m field with 9 initial weed parcels?\",\"answer\":\"With the given conditions, the essay concludes that the whole field cannot grow into weed.\"},{\"question\":\"How is the problem generalized beyond the original 10m x 10m case?\",\"answer\":\"The essay generalizes the result to fields of dimensions n x n with n-1 weed parcels, finds a necessary condition for full overgrowth, and then extends the approach to determine all winning initial arrangements.\"}]","Mathematics - Extended Essay - May 2007 - Jovana Šljivančanin | PDF",1790054063,63]