​Idea Seeds #03 - Problem solving, the Questions

Joni Mitchell - Both Sides, Now [Original Studio Version, 1969]

Learning to Learn

The words from Joni Mitchell’s songs:

“I've looked at life from both sides now;

From up and down and still somehow;

It's life's illusions I recall;

I really don't know life at all.”

seem an appropriate starting point for this, the third article, written in the hope that some of the ‘idea-seeds’, the ones written in italics, find a place to germinate and help to further the debate round ‘learning’ and ‘learning to learn’. The focus thus far has been to convince and encourage you to start developing your own ‘problem solving rituals and checklist’ if you haven’t already done so.

The words ‘both sides’ in the song implies only two sides. In the real world there are, more often than not, many more sides to look at. If my father could rewrite the song, he would change the word ‘both’ to ‘all’ and ‘up and down’ to ‘every angle’. The song would still rhyme and better match the real world and what he is hoping to get you to think about.

But first, in keeping with good practice, a quick summary of where we have come from and where we have got to in explaining the development of the ‘problem solving checklist’ that he uses. When presented with a new problem he asks the following questions:

  • “Do I really understand what the problem is about?
  • Have I identified everything that has been given?
  • Am I clear about what it is that I have been asked to do?
  • Have I identified all the ‘limits’ and ‘constraints’ that apply,

or put another way,

  • have I identified the boundaries within which I have to find a solution?”.

He has asked you to solve a variety of puzzles that demonstrate the benefits the answers to these questions can bring in helping to get to a solution more effectively.

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Looking from another angle

The next puzzle demonstrates the need for yet another question you should be thinking about adding to your checklist.

Many big tennis tournaments are played on a knock-out basis to find the champion. Names are drawn randomly to decide on who plays against whom in the first round. The winners from the first round matches then go forward into the second round. This process is repeated until there is a champion. This is the puzzle: If thirty one players enter a knock-out tennis tournament, how many matches must be played before there is a champion?

Please try and work this out now by yourself. Once again, remember to note the ‘processes’ you use to get to your solution, before reading any further.

Because of the odd number of players, one player cannot take part in the first round which leaves 15 pairs to play 15 matches in the first round. There are now 16 players in the second round, 15 winners from the first round and the player who was given a bye in the first round; 8 pairs needing 8 more matches to decide the 8 winners to go forward into the third round. 4 more matches in the third round, 2 more matches in the fourth, and 1 more match in the fifth. The total number of matches then is the sum of 15+8+4+2+1=30. So if you start a tournament with 31 players it requires 30 matches to be played to get to a champion. I hope you got this answer!

If you haven’t already done so, look at the same puzzle but from another angle. We know the end result is one winner so there must be thirty losers. You can only become a loser if you have played a match so thirty matches have to be played to get thirty losers and a champion.

Did you get the easy answer by looking at the problem from another angle?

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Look at the Problem from Every Angle”

This is the sort of thinking that you should be cultivating to improve your problem solving skills. You should be adding the following questions to your ‘problem solving checklist’:

  • “Are there any other starting points I can start from?”,

and,

  • “Have I really looked at the problem from ‘every angle’?”

Somewhere on Planet Earth

The following puzzle is another example of where these questions help and are therefore important. “Imagine you are positioned somewhere on Planet Earth. From that position you walk 10 kilometres due south. You turn and then walk 10 kilometers due east. You turn again and walk 10 kilometres due north and find, when you get there, that it is the very same place you started from. Is this a unique place on Planet Earth or are there other places on it where this happens?” Think about the puzzle for a few minutes before reading further.

Be assured that, as they say on the quiz show ‘Who wants to be a millionaire’: “There are no-trick-questions”, I too do not use trick questions or puzzles with trick answers. The puzzles have been chosen to illustrate ‘processes’, not tricks. So if you think there is only one answer to the above puzzle; ‘the North Pole’, you need to think again but please try and do so holistically by inspecting all the ‘limits’.

Imagine yourself as the walker and then ’visualize’ what you would actually be doing as you carry out the instructions. If necessary find a ball or an orange and a felt-tip pen and make a model to help you ‘visualize’ what it is you are doing. Choose a point to start from that is not near any limits. If you had gone through your ‘problem solving checklist’ you would have already identified the equator and the North and South Poles as limits. Start, let’s say, on a line of latitude 60 degrees north of the equator, then when you walk south you will be doing so on a line of longitude. When you turn and walk east you will be walking on a line of latitude. When you turn and walk north you will again be walking on a line of longitude. If the distance you walk south is the same when you walk north then you will get back to the same line of latitude that you started from but to the east of your starting point. Importantly, you would be walking south from a line of latitude with a smaller circumference to one with a larger circumference.

If you now start moving your starting point nearer and nearer to the North Pole, one of the ‘limits’, the circumference of the line of latitude you start from decreases and the distance along the line of latitude between start and end points gets closer and closer. In the ‘limit’ this distance becomes zero. A line of latitude with a circumference of zero must be a point and that point, in this case, is the North Pole − one solution to the puzzle.

When you start moving the starting point south from the line of latitude at 60 degrees north, as you approach the equator, another ‘limit’, the start and end points get further and further apart but always less than 10 km. If you start from a line of latitude 10 km north of the equator, when you walk east it will be along the equator. There is nothing noteworthy about this limit in terms of a solution.

The next, not so obvious limit, you get to is when you start from a line 5km north of the equator and walk to the latitude 5 km south of the equator, east along it and then back north to a point exactly 10 km to the East of your starting point. Any starting points further south from this line of latitude results in distance between the start and end points moving farther and farther apart, the distance always being greater than 10 km. Noteworthy is that you will now be going from a latitude with a larger circumference to one with a smaller circumference.

Infinite solutions

At the equator the circumference is approximately 40 000 km (24 900 miles), at the poles it is zero. So somewhere between these two there will be a line of latitude with a circumference of exactly 10 km. Start from a point 10 km north of the line of latitude with the circumference of 10 km, walk 10 km south down the line of longitude to this latitude, then 10 km east and you will be back at the line of longitude you walked down on, walk 10 km up it and you will be back where you started from.

There are therefore an infinite number of starting points on the line of latitude 10 km north of the line of latitude with a circumference of exactly 10 km. Here again one of the students came after class to tell my father that there were even more solutions. If the circumference of the line of latitude is 5 km then you could walk east from the line of longitude you walked down on, twice round the circumference always going east and get back to the line of longitude. Working the ‘limits’ as he did you could walk east 10 times round a line of latitude with a 1 km circumference, 20 times with one half a km and so on to infinitum. A bright fellow you must agree! Thus we can see the link between ‘Geometry’ and ‘Visualization’; they go hand in hand and are both on my father's list of critically important ‘idea-seeds’ but that’s enough to think about for now!

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