assessment 2 version 3

Assessment 2 – Developing a problem solving plan

Jordan Bird

CS1320 – Problem Solving

November 24, 2014

Assessment 2 – Developing a problem solving plan | Jordan Bird

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Introduction and Method

Within this report I hope to produce a detailed description and in-depth explanation on my understanding of a problem solving process to understand and solve problems at hand, as well as show a ready skill in solving issues of this type of nature. The method I am going to follow according to the requirements of the task set1 to fill this criteria is as follows: 1. Firstly I will endeavor to understand the problems using Paul Vickers’ problem solving

process2 2. After I have developed a clear understanding of the problems, I am going to devise a plan in

solving these via Vickers’ method of planning, and then ultimately conclude with solutions to them.

3. While I do this, I will compare the two problems similarity and differences 4. - and also use this information to help with solving one another issue hand in hand. 5. Finally I will research throughout to strengthen my work via secondary research and

therefore reference research material via Vancouver Citations for readability, I feel the research needed would decrease readability with other referencing styles. The format I will be using is Harvard formatting.

The problems I endeavor to solve are: 1. The Man with the Fox, Chicken and Grain problem. A man must take all of the subjects

across a river, but there are constraints. The Fox and chicken cannot be alone as the chicken would be killed, the chicken and grain cannot be alone because the grain would be eaten. The problem is solved when they are all safely across the river. This is a very famous problem appearing in many ‘thinking activities’ such as Nintendo’s Professor Layton – a very successful best-selling game for the Nintendo DS.3 Which I believe goes to show the sheer popularity of this method of taking a mathematical problem and giving it a higher level of understanding.

2. 6 people – three missionaries and three cannibals are on one side of a river. A boat must take them all across the river but if the cannibals outnumber the missionaries, the missionaries will be killed and eaten. Again, the problem is solved with all subjects across the river.

1 Aston University, Birmingham. (2014). CS1320 - Problem Solving Assessment 2 – Developing a Problem Solving Plan. Available: https://vle.aston.ac.uk/bbcswebdav/pid-665885-dt-content-rid-2372172_1/courses/2014_CS1320/Assessments/CS1320CW02.pdf 2 Paul Vickers. (2008). Chapter 2. In: How to Think Like a Programmer, Problem Solving for the Bewildered. : Cengage Learning. ISBN-10: 1408065827, ch. 2 pp. 30-37 3 Level-5 Studios (2007/2008). Professor Layton (レイトン教授) and the Curious Village. Published Worldwide: Nintendo.


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Understanding the two problems

Before I can hope to solve the problems, I must understand them. According to the famous publication by Polya4, understanding the problem is a very neglected yet ironically important stage in the process – and can lead one to sure success or sure failure depending on whether it is completed or not. He considers a problem not fully solved if not fully understood. I will in light of Polya’s teachings being followed by academic teachers (Scholastic still to this day support these teachings after nearly 70 years of publication5) use his methods of solving the problem as well as Vickers, following both of their teachings to hopefully understand both problems.

(For this paper, the Chicken Fox and Grain problem will be dubbed ‘problem 1’ for readability, and the Missionary/Cannibal problem ‘problem 2’ for the same reason)

Please note: Due to my previous essay in the effectiveness of each of the problem solving

steps, I will be omitting some of the stages in Vickers’ original plan to form my own slightly different problem solving process which is far more suited to the problems at

hand. The reasons for this are detailed in the reviewing process of my previous report.

“Read it through several times” The first step taught by both Polya and Vickers was the re-reading of the problem to solidify the foundation of understanding. Reading through both of the problems gave me these conclusions – Problem 1 – First of all I saw this problem and simply thought I’d have to keep the opposites apart and therefore it would be a simple problem to solve without much input needed. I then on my second read through realized it would be foolish to consider a fox, chicken or a bag of grain to be able to control a boat. This reading led me to understand there was only one space on the boat, as the man must stay on to maneuver it each time. I noticed this from the phrase “Only one item can be taken in the boat at a time. So the man can take the fox or the chicken or the grain across the river in the boat but he cannot take any two together” 6 Problem 2 – The second problem requires simply the missionaries to always outnumber the cannibals. This seemed a simple method of solving at first until I realized that in fact, the exception that there can be +1 cannibals if the missionary count is zero on one side means that a solution may involve a larger number of cannibals on one side at a stage, but this would not affect the outcome as it wouldn’t be possible to kill and eat zero missionaries. On the other hand though, a lonely cannibal must bring the boat back, rendering the lonely cannibal a foolish first move that is unneeded and will not lead to the best solution. This I believe though has been

4 George Polya (1945). How To Solve It. Princeton University: Princeton University Press. ISBN 0-691-08097-6. 5 Scholastic Inc. (Date of publication unknown. Adapted from "Science World," November 5, 1993.). 4 Steps to Problem Solving. Available: http://teacher.scholastic.com/lessonrepro/lessonplans/steppro.htm. Last accessed 17 November 2014. 6 Aston University, Birmingham. (2014). CS1320 - Problem Solving Assessment 2 – Developing a Problem Solving Plan. Available: https://vle.aston.ac.uk/bbcswebdav/pid-665885-dt-content-rid-2372172_1/courses/2014_CS1320/Assessments/CS1320CW02.pdf


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an effective exercise as it has led me to reveal a wrong first move simply via reading through the problem’s brief. Similarities and differences – To differentiate between the two, the exceptions are slightly different. The read through result of problem 1 is somewhat of a limitation due to the man having to ferry people across. Whereas problem 2 encounters a positive exception as it means there could be a quicker way to solving the problem with the outnumbering in mind. To compare the two, they both have very similar issues yet one of them is hindering while the other one is helping. Can we use these similarities and differences to help us? – Once I have reached one conclusion, I will look at the difference made by the exception I noticed in the first problem, and then see if I can use it to my advantage in the other one. Although as stated the things I have noticed are slightly different, it will still most surely help me find the best solution with it in mind. The similarities have already helped me thus far, as noticing the first one led me to look into finding the second one, which I did. I believe due to this it has already helped me analyze the problem, and will go on to help me solve them in the best possible way. Evaluation – This point in the process was extremely useful to me. Without doing this planning step, I would have gone on with the misunderstanding of problem one in mind, and therefore would have failed to come up with a proper solution. For this reason I feel that step one is very vital when concerning our two problems. “Identify Principal Parts” and “Rephrase the problem” Melanie Pinola of LifeHacker explained to us that changing words around in a problem can give us a new perspective on it and therefore help everyone reach a better level of understanding of the puzzling situation at hand7. I believe that due to this research, as well as the fact that Grimm and Railsback of Princeton University said that the rephrasing of a problem “allows us to be explicit” negating any empty definition from one person to another8. In light of this research, I feel that the problems at hand are quite general and therefore must be addressed from a second perspective as to be made clearer – just because I feel I understand the problems to a large extent does not mean another could read my report and understand them just the same. - Problem 1 – Firstly I will take the problem and annotate it, making bold the principal parts of the problem. “A man is on the south bank of a river, with a fox, a chicken and a bag of grain, which he wishes to get to the north bank. There is a boat by the south bank. The constraints are - Only one item can be taken in the boat at a time. So the man can take the fox or the chicken or the grain across the river in the boat but he cannot take any two together (i.e. the fox and the chicken). If the fox is left alone with the chicken then the fox will eat the chicken. If the chicken is left alone with the grain then the chicken will eat the grain. The fox can be left alone with the grain since he is not interested in eating the grain.”9

7 Melanie Pinola. (2011). Redefine Problems By Changing the Words You Use to Describe Them. Available: http://lifehacker.com/5819153/redefine-problems-by-changing-the-words-you-use-to-describe-them. Last accessed 18 November 2014. 8 Volker Grimm, Steven F. Railsback (2013). Individual-based Modeling and Ecology. Princeton University: Princeton University Press. p25. 9 Aston University, Birmingham. (2014). CS1320 - Problem Solving Assessment 2 – Developing a Problem Solving Plan. Available: https://vle.aston.ac.uk/bbcswebdav/pid-665885-dt-content-rid-2372172_1/courses/2014_CS1320/Assessments/CS1320CW02.pdf


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I will now take this problem and using the key parts, negate jargon and create a rephrased problem. “A man must take a Chicken, Fox, and bag of Grain from one side of a river to another, he can only take one at a time in his boat. The Chicken and Grain cannot be left together, and the fox and chicken cannot be left together.” This is now far more manageable for not only others, but in actual fact for myself. I have realized that my understanding was not as advanced as I thought it was, as I now feel this issue is far easier to manage now it has been cut down. In light of this, I now feel that I should do this with problem 2. Problem 2 – Now I have realized how important this stage is with this kind of problem, I will now annotate and rephrase problem 2 accordingly. “There are three missionaries and three cannibals on the west bank of a river. They want to safely transport everyone to the other side of the river. There is a boat by the west bank that they can use. The constraints are. The boat can only carry two persons at a time. At any point in time, if the cannibals outnumber the missionaries, they will eat the missionaries (i.e. this means on either river bank and in the boat).”10 “Three missionaries and three cannibals need to be moved from one side of the river to another safely. Any of them can use the boat, which holds two people. If there are more cannibals than missionaries at any point, the missionaries will die.” I feel that with this short description in mind, the problem can be solved much more efficiently. I will be able to remember this shortened version and keep it in mind when I am thinking of the best solution to the problem. Similarities and Differences – I have slowly come to notice that the two issues are basically the exact same problem except for the constraints, which differ very slightly. The transport systems and the fact that some entities cannot stand with others in situations are identical, the failure criteria although does change between the two situations. Can we use these similarities and differences to help us? – After realizing the sheer similarity between the two problems, I feel that to come to a solution, I can create a system to solve it that can be applied to both issues and therefore solve them both. Inputs and outputs may be different, but I now feel that the system to create an output from this input can be the same and therefore solve these two problems, and therefore any problem of this kind of nature. It will also be a system that will point out unsolvable problems (for instance the infamous, “now you’ve solved the cannibal problem, try 4 cannibals and missionaries” – which is an impossible problem). Developing a system like this will increase efficiency, and if it turns out to solve all of these problems, will revolutionize my understanding of these problems as it means their similarities and differences can be used to solve one another. I will detail this system in my concluding sections.

10 Aston University, Birmingham. (2014). CS1320 - Problem Solving Assessment 2 – Developing a Problem Solving Plan. Available: https://vle.aston.ac.uk/bbcswebdav/pid-665885-dt-content-rid-2372172_1/courses/2014_CS1320/Assessments/CS1320CW02.pdf


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Evaluation – These questions were worded un-needingly complicated. The fact that my understanding is much clearer now after translating them to more simple terms goes to show that this point is very useful, although I would consider it second to that of the first point so far as I could still have gone on to complete the problems, but would have just taken slightly longer due to the complication in explanation. “Represent the problem” The practice of diagrammatic reasoning (an evolution of the simple diagram) dates back in academic just over three-hundred years ago. The work of Gottfried Leibniz (independent developer of calculus during the time of Newton’s work11) led him to describe his philosophy with diagrams to properly show work as well as the reasoning to back up points, not to mention allow a deeper understanding of the theory12. Due to this research, and the fact that both Vickers and Polya support the notion, I am going to use diagrammatic reasoning to give myself a deeper understanding of the two problems and the links between them just as Leibniz did and many others who followed him. Creating diagrams will help me personally to visualize the problem and come to a much better understanding, as I will be able to solve it visually (the method I am most comfortable with). To create the diagram, I will use Microsoft Paint13. - Problem 1 – As seen below, the diagram I have created for the first problem shows the Fox, Chicken and Grain (F, C, G respectively) on one side of the river. It shows the man (M) on the boat, and the usable single space as represented by the red rectangle.

I will use this diagram to solve the problem, as I personally am most comfortable with looking at the visual side of things. Although this may be personal to me, Fleming wrote that 75% of people were visual learners as oppose to auditory and kinesthetic14, which places this method not only as positively enforced to myself, but also the vast majority of my audience. It is also thankfully

11 Smith, David Eugene (1929). A Source Book in Mathematics. New York: McGraw-Hill Book Company, inc. 12 Leibniz, G (Original Paper:1704 Open Court Book:1949). New Essays Concerning Human Understanding. LaSalle: Open Court Publishing. 13 Microsoft. (1985 - Present). Microsoft Paint. Available: http://windows.microsoft.com/en-gb/windows/using-paint#1TC=windows-7. Last accessed 18 November 2014 14 K.G, Fleming (Publication:1960, Online Publishing:2006). Journal of Philosophy of Education: Criteria of Learning and Teaching. : Academic Paper. p39 onwards.


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the easiest reported method of solution. This goes to show the sheer aid that visualizing a problem can give, and therefore this again reinforces my reasoning of why I used this method. Problem 2 – Again, as we can see below, I have represented the problem in diagram form. The cannibals are represented as red squares, and the missionaries as blue. This time though, there are two spaces on the boat instead of just one. I will again use the diagram to aid in solving the problem as it is the visual method I am most comfortable with.

Similarities and differences – It is obviously no secret that to complete this stage of the process, I more or less used the exact same diagram, using the base I created as seen here below. I created this base for the first problem, and noticed quite quickly it would be the exact same case for the second problem. With this in mind, we further see links develop between the two, the fact that they are in effect the same problem.

- The blank base used for both problems which I created.

Can we use these similarities and differences to help us? – I believe that in this stage, I have shown how this can help us specifically. The fact that less work needed to go into the second problem on this stage, due to half of it already being done for the first one, just goes to show improvement of the problem solving process. My process is more efficient, as I do not need to repeat my work between problems. This will lower the time it takes to come up with the correct solution, and also solidify the fact that it is the correct solution (one diagram used twice without issue shows a positive side to said diagram), which for me is most definitely helping in the problem solving process and the creation of a final solution.


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Evaluation – This step for me is extremely effective. As mentioned, I personally have great difficulty with solving a problem without visualizing it first. Therefore I feel this step has been very effective in helping my understanding the problem as it allows for me to actually see it. On the other hand though, it may be less effective for other people who prefer to think in other ways than visualization. “What are the knowns and unknowns of the problems?” (Possibilities in this case) It was Savransky who spoke about the simplicity of problems being based on the variables involved. He speculated that the more known variables in the problem, the easier it is to solve.15 Therefore I feel in effect to this research, I feel that if I bring into account all of the facts I know about the problem in an easy to read form, then the process of creating the system to solve them will be far easier. Problem 1 – Below I have created a list of the knowns and their results: - Fox, Chicken, Grain = start - Fox, Chicken = Chicken dies, fail if not accompanied - Fox, Grain = Neither fail nor complete - Chicken, Grain = Grain eaten, fail if not accompanied - Fox, Chicken, Grain = finish I can use this list of knowns to aid me in the process. It shows the first move to the problem must be the removal of the Chicken, as it is the only way to begin without failing on move one. Problem 2 – where Cannibal = C and Missionary = M, and River = -

- CCCMMM = Start - CCCMM - M = Failure - CCCM - MM = Failure - CCC - MMM = No failure – point of success but not finish, all missionaries have made it over. - CCMMM - C = No failure - CMMM – CC = No failure - CM - CCMM = No failure - CCMM - CM = No failure - CCCMMM = Finish

Again, we are now given combinations that will not result in failure, which we can use later on to help us develop our system. Similarities and Differences – As we can see, both of the problems have three separate outcomes. No failure, failure and finish (start too, but start is equal to the finish combination). We must avoid the failure at all costs, and follow a path of combinations that are equal to ‘no failure’ to reach the finish. Can we use these similarities and differences to help us? – Yes, we can. As we can see, the second problem is far more complicated than the first one, it will obviously take more moves to cross the stepping stones of ‘no failure’ if you will. Therefore to help us with this issue, we will pay attention to primarily solving the first problem, and then use our developed method to solve the second more complicated one almost automatically. We are helped via an interchangeable system based around a simple problem, applied to a more complicated one.

15 Savransky, Semyon D. (29 Aug 2000). Engineering of Creativity: Introduction to TRIZ Methodology of Inventive Problem Solving. CRC Press.


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Evaluation – This step was quite useful as it showed the different known outcomes of the problem. Therefore they can be applied later on and used as stepping stones to reach the solution.

Devising a plan

Following Vickers’ teachings in accordance with my own ideas that I have developed throughout the report, I must devise a plan. Vickers lays out a few steps that must be followed to sufficiently devise a proper plan to solve the problem: Identify and solve all sub-problems – The sub problem of the main problems, is getting across in the least moves. To solve this sub problem, I would have to create a plan that will solve all of the possible outcomes, then pick out the shortest one from it. The task is to solve the two problems and use their similarities to aid each other in solving – but I believe we can take this a step further and not only help one another problem, but develop a system to solve every problem of this nature. Simplify the problems – The problems seem quite simple in themselves, but I believe we can pick out a sub-problem to solve help us along the way. I believe the ‘stepping stones’ we discovered earlier will aid us on our way. To show this, I have taken all of the possible combinations for both problems that will not result in a fail.

- Fox, Chicken, Grain = start - Fox, Chicken = Chicken dies, fail (no failure if man is accompanying) - Fox, Grain = Neither fail nor complete - Chicken, Grain = Grain eaten, fail (no failure if man is accompanying) - Fox, Chicken, Grain = finish

- CCCMMM = Start - CCC - MMM = No failure – point of success but not finish, all missionaries have made it over. - CCMMM - C = No failure – ILLOGICAL FIRST MOVE - CMMM – CC = No failure - CM - CCMM = No failure - CCMM - CM = No failure - CCCMMM = Finish

We can take this idea of ‘stepping stones’ discovered in problem 2 and applied to problem 1, to solve the sub-problems of where to start and which course to take. From the first one the sub-problem is solved by showing us that the first move must be the Chicken being transported as it will be the only move that doesn’t result in a failure. As for the second one, we see the combinations that are possible, and that we must use them to reach “CCC-MMM” then “CC-CMMM” in which case victory is assured by the Cannibal being used to single-handedly ferry across the other two from one side to another. This simplification of the problems will give us a starting point for our system to base itself on.


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Make a drawing or sketch – For this step, I will use my earlier sketch of the playing boards to help myself visualize the problem once again before solving it. I will do this to solidify the thinking process required.

I will take these two sketches and refer to them when I am solving the problem, as it will allow me to imagine the situation, and not a combination of letters. To reiterate, this system of visualization is something I find very useful. Piggott and Woodham16 wrote that visualizing a problem in this way is an important factor throughout the whole problem solving process, most notably deriving a solution – which is the stage we are currently heading through and goal we are about to reach with our plan, and therefore is a logical piece of research to follow. I am now going to sketch out my actual plan of action. My plan is to design a visual system, due to the reasons explained previously, that will go through possibilities in the ‘stepping stone’ fashion I described. The plan will go through in a branch-type fashion throughout possibilities of each state of play. The possibilities that do not end in fail will continue to branch off, whereas those that fail will go red and stop branching.

16 Jennifer Piggott and Liz Woodham. (First published 1996). Thinking Through, and By, Visualising. Available: http://nrich.maths.org/6447. Last accessed 20th October 2014.


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As we can see here, the start and solution bubbles represent the beginning and end of the problem solving process. The red bubbles result in a fail and therefore they are not followed by possibilities. The green bubbles do not result in fail and therefore they are followed by another branch of possibility. This method will reveal every single solution, and the shortest one will quite literally be the shortest one in size on the image.

Carrying out the plan

To carry out the plan, I will develop a system specifically for problem 1 – but one that is unaffected by the differences in the problems, put it into action for both of the problems in an effort to use their similarities and differences to solve one another, output solutions, and conclude by taking a look back at my original criteria of success.

The developed system, and solving the problems

In accordance to the understanding of the problem above, I am now going to use the similarities of the two problems to create a system that will generate a solution to them both. This is for two main reasons 1. Firstly, the second problem is much more complicated. The three cannibals and three

missionaries as oppose to the simple three subjects in the first problem cause it to be a


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longer process. Therefore, if I create a universal system based on the first, simpler problem, then I will automatically create a system for the second one.

2. Secondly, it is no small fact that this problem comes in many shapes and sizes, forms and descriptions. Ivars Peterson of Science News discussed in detail about the different forms that this problem comes in, speculating they are all in fact the same and follow a very specific pattern of solution17. With this academic paper in mind, I further feel that the best way to not only solve these two problems, but all problems of their nature, I will develop my own system to solve them all due to their sheer similarities.

System for solving both problems (in accordance to their similarities and differences) – I have noticed throughout that both problems and their likenesses in other problems are all exactly the same. I have taken all of the similarities and differences of the two problems (and therefore all of the other types of this problem) and created a system to solve all of them. It is a system which again, takes into account the links that join all of these problems, and also a system that is not affected specifically by their differences. I call this the Branch Solving Method. I will first show the two problems being solved actively with the branch method, and then conclude them afterwards. This is my primary method for solving all of the problems, solving the simplest one – problem 1. It has been placed on a new page due to size.

17 Peterson, Ivars. (2003). Tricky Crossings. Available: https://www.sciencenews.org/article/tricky-crossings. Last accessed 18 November 2014.


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C, F, GStart

C, F - GFox kills Chicken

C, G - FChicken eats Grain

F, G - CFox does not eat grain

C, F, Gback to start

AloneMan returns alone

G - F, CMan takes fox back

C, G - FMan takes

chicken back

C - F, GMan takes grain

AloneOnly logical move to avoid loop

C, G, FFinish

AloneFox kills Chicken

C,G,FPointless loop

F - G, CWILL COMPLETE IN THE SAME

AMOUNT OF MOVES AS SOLUTION


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From the above diagram, we can see the branching method of possibility, although due to software constraints, alignment is negatively affected on a visual level. The method goes through each possibility, and the hierarchy ends at each red point as it results in failure. We can see here that there are two solutions, and if we take away the looping moves, the shortest possible way to solve this problem is with 7 moves. This branching problem works with all possibilities of a given problem, therefore it will negate the differences between the two problems and provide us with a method to solve any of them, making use of the similarities between them that I have noticed. I am now going to take this method for solving a simple problem, and complete the ultimate step in helping with a solution due to the links between the problems by applying it to the next problem straight away.

NOTE: Incorrect moves have been omitted from the diagram for readability, but other moves would surely lead to missionaries

being killed and red outcomes, or would lead to an illogical loop. This is due to the ‘knowns’ section explaining all other situations

would lead to failure.


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MMMCCC

MMMC -CC

MMMCC -C

MMM -CCC

MMMC -CC

MC -CCMM

CCMM -CM

CC -MMMC

CCC -MMM

C -MMMCC

CC -MMMC

MMMCCC


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As we can now see, my system is not negatively affected by the differing constraints in the two problems. Due to their sheer similarities I have picked up in them throughout, the branching method developed for problem one can be adapted to solve the second problem – not only have I come up with the solutions to the two problems, but I have in fact used their similarities to develop a single system which will solve all of the “tricky crossing” puzzles. My original task was to solve the two with help from their similarities, which has most definitely been done here, I believe that the system of deriving a solution goes above and beyond this criteria as it can be applied to many other problems of the same nature too.

Derived solutions (< and > annotate boat direction) 1. > The man takes the chicken across the river

< Goes back alone > The man takes the fox across the river < The man takes the chicken back with him > The man takes the corn across, leaving the chicken alone < Goes back alone > Takes the chicken back across 7 moves

2. > MMMC – CC < MMMCC – C > MMM – CCC < MMMC – CC > MC – CCMM < MMCC – CM > CC – MMCM < CCC – MMM > C – MMMCC < CC – MMMC > MMMCCC 11 moves

Final conclusion To conclude whether I have completed my goals, here are my criteria from the introduction and their reasoning –

Firstly I will endeavor to understand the problems using Paul Vickers’ problem solving process - I used Vickers’ and Polya’s teachings throughout as referenced in my report, and I believe that with each step of the process, I grew more of an understanding of the problem to the point that the solution seemed clearly obvious by the time I came to work it out. I developed a thorough understanding throughout by firstly ironing out any misunderstanding, and then solidifying knowledge by making it easier to process and understand.


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After I have developed a clear understanding of the problems, I am going to devise a plan in solving these via Vickers’ method of planning, and then ultimately conclude with solutions to them. – I developed my plan throughout. The first steps were mental clarity to start the planning process, which I then used effectively in developing my branch solving process. While I do this, I will compare the two problems similarity – At each step, I compared the similarities and differences of the two problems, comparing the information that the step in question had made clear to me. - and also use this information to help with solving one another issue hand in hand. – I took into account the similarities and differences to develop a system that was simple to create, but could solve these problems of any difficulty. The effort put into making a system to solve the first problem was very little and considerably easy to do. But this system could then be applied to the second problem with ease even though it was a more complicated problem. I feel I have effectively brought together the similarities and differences to aid with solving problems hand in hand, as the system could be applied to any number of thousands of Cannibals and Missionaries. A whole brood of chickens, a troop of foxes and tonnes of grain could all be managed in the branching system with very little effort from the user. I feel that with all of these points in mind, this report has been successful and has fulfilled my criteria that was set out accordingly, if not led to an above and beyond situation of problem solving due to the possibility of the problem solution giving coverage to not just two, but all river crossing problems.


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References

Aston University, Birmingham. (2014). CS1320 - Problem Solving Assessment 2 – Developing a Problem Solving Plan. Available: https://vle.aston.ac.uk/bbcswebdav/pid-665885-dt-content-rid-2372172_1/courses/2014_CS1320/Assessments/CS1320CW02.pdf

K.G, Fleming (Publication:1960, Online Publishing:2006). Journal of Philosophy of Education: Criteria of Learning and Teaching. : Academic Paper. p39 onwards. Leibniz, G (Original Paper:1704 Open Court Book:1949). New Essays Concerning Human Understanding. LaSalle: Open Court Publishing.

Level-5 Studios (2007/2008). Professor Layton (レイトン教授) and the Curious Village. Published Worldwide: Nintendo.

Microsoft. (1985 - Present). Microsoft Paint. Available: http://windows.microsoft.com/en-gb/windows/using-paint#1TC=windows-7. Last accessed 18 November 2014

Peterson, Ivars. (2003). Tricky Crossings. Available: https://www.sciencenews.org/article/tricky-crossings. Last accessed 18 November 2014. Piggott, Jennifer. Woodham, Elizabeth. (First published 1996). Thinking Through, and By, Visualising. Available: http://nrich.maths.org/6447. Last accessed 20th October 2014. Pinola, Melanie. (2011). Redefine Problems By Changing the Words You Use to Describe Them. Available: http://lifehacker.com/5819153/redefine-problems-by-changing-the-words-you-use-to-describe-them. Last accessed 18 November 2014. Polya, G. (1945). How to Solve It. Princeton University: Princeton University Press. ISBN 0-691-08097-6.

Savransky, Semyon D. (29 Aug 2000). Engineering of Creativity: Introduction to TRIZ Methodology of Inventive Problem Solving. CRC Press. Scholastic Inc. (Date of publication unknown. Adapted from "Science World," November 5, 1993.). 4 Steps to Problem Solving. Available: http://teacher.scholastic.com/lessonrepro/lessonplans/steppro.htm. Last accessed 17 November 2014. Smith, David Eugene (1929). A Source Book in Mathematics. New York: McGraw-Hill Book Company, Inc. Vickers, Paul. (2008). Chapter 2. In: How to Think Like a Programmer, Problem Solving for the Bewildered. : Cengage Learning. ISBN-10: 1408065827, ch. 2 pp. 30-37

Volker Grimm, Steven F. Railsback (2013). Individual-based Modeling and Ecology. Princeton University: Princeton University Press. p25.

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