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Idea Seeds #02 - Problem Solving, the Ritual
Know your limits
Limits in problem solving
Belt and Braces
In Article One my father, the retired professor of mechanical engineering, outlined the first step in his ‘problem solving ritual’ − he reflects back to himself, or to whoever has posed the problem, what he understands has been given and what it is he is expected to do.
After reflecting back to an actual person, and that person agrees with what he has said, he then knows that he is starting with the right information and by doing so will avoid all those:
“Oh I thought that is what you meant”, or “I didn’t see it that way”, situations that can cause so many hassles when things go awry in particular when real money and reputations are involved.
The five-square match puzzle outlined in that article (see link below: Seed ideas#1 -Problem Solving 101) demonstrates very clearly the benefits of developing a ‘problem solving ritual’.
After going through the questions: “What has been given?” and “What has to be done?” and finding that sixteen matches have been given to make four squares with one-match-per-side it becomes obvious the pattern sought must have four stand-alone squares. To find the two matches that have to be moved from where they are situated in the five-square pattern is then a simple task.
Seed Ideas # -.Problem Solving 101
- Problem solving in Life
I have a wonderful father, a university professor who has dedicated his life to education and thinking about thinking. I have been encouraged to work things out from basic principles.
The Nine Dot puzzle
Two More Puzzles
He uses the following two puzzles to demonstrate the next step in his ‘problem solving ritual’. The first puzzle is old and well known and has been used by psychologists to give them insight into how their patients think. He draws nine dots on the black-board, as shown here, and asks the students to copy them onto a sheet of paper so they can try and solve the problem by themselves.
The task, he tells them, is to draw the least number of straight lines through the nine dots without lifting their pencils. he demonstrates the latter by drawing a five line solution through the nine dots without lifting the chalk as shown here. Again, he reminds them to make a note of the ‘processes’ they used to get to the solution.
They should have reflected back to themselves: “I have been given nine dots arranged in a pattern of three rows and three columns. My task is to find the least number of straight lines that can be drawn through the nine dots without lifting the pencil and while doing so, to make a note of the ‘processes’ I have used to get to the solution.”
Please try and solve the puzzle now, before reading further.
The EquilateralTriangle Puzzle
The second puzzle is another puzzle involving matches. He sets up six matches on the overhead as shown below and tells students the task is to make four equilateral triangles with one-match-per-side using only six matches.
You should have reflected back to yourself: “I have been given six matches. My task is to use them to make four equilateral triangles with one-match-per-side and while doing so to make a note of the ‘processes’ I use to get to the solution.”
Please try and solve the puzzle now, before reading further.
Thinking Out of the Box
These two puzzles highlight the need for the third question you should ask and add to your own ‘problem solving ritual checklist’. “Are there any ‘limits’ or ‘constraints’ that must be taken into account?” In the nine-dot puzzle most people wrongly make the assumption that they have to stay within the four boundaries of the outer eight dots even though no such constraint has been specified. The four-line solution, shown here, requires you to think outside the proverbial box and the reason, I suspect, why it is of interest to psychologists!
Limits or Not?
Solution To the Six Match Puzzle
In the six-match puzzle, if you arrange the matches in the shape of a three-dimensional triangular ‘pyramid’, also known as a ‘tetrahedron’, the requirements for a solution are met. The solution is shown here. To make four ‘stand alone’ equilateral triangles would require twelve matches. Given only six matches, each match must act as a side in two adjacent triangles. Most people make the assumption that the problem is two-dimensional, 2D, and fail to find a solution. Had they asked the right questions and defined the ‘limits’ or ‘constraints’ properly they would have known to look for three-dimensional solutions, 3D, as well.
So when considering what ‘limits’ or ‘constraints’ apply when ‘analyzing’ real world problems, issues such as those that follow need to be decided on before any thought is given to finding solutions: 2D or 3D solution space?; time?; money?; accessibility to resources?; infrastructure?; legal implications?; are but a few from many.
The nine Dot Puzzle Solution
As a short in between, my father tells the story of when he had given a new intake of students the nine-dot puzzle. At the end of the lecture a student came up to him and showed him a sheet of paper on which he had drawn the nine dots and said: “If you are happy with these nine dots then I can do the problem using only three lines”. He proceeded to draw the solution shown here.
Very excited about this bit of lateral thinking and, after exploring the limits question again − with thoughts of infinity and of all those parallel lines meeting there churning in his head, he showed this solution to John Web, the person responsible for starting the Maths Olympiad and a Professor in the Department of Mathematics at UCT. He took one look at the problem and the student’s solution and said: “Then it can be done with one line. Roll the paper into a cylinder and draw a helical line three times round the cylinder and on each turn, draw the line through a row of three dots.” I wonder what the psychologists would have said if one of their patient’s had come up with this solution? I may yet ask one day!
The helical solution unrolled would look like that shown here. I suggest you cut it out, roll it into a cylinder so you can see what the helix looks like for yourself. (Hint: When you make the join, make sure the left hand end of the top line joins to the right hand end of the middle line and the left hand end of the middle line to the right hand end of the bottom line.)