Now you've found your solution (or not), let's have a look at the solutions I received and see what they tell us about the problem solving process. In particular, I want to think about something Paul Zeitz calls the crux move, the key move that helps us solve a problem. Where does the crux move come from? As mathematicians, how can we improve our chances of finding the crux move? As teachers, how can we help others increase their chances of finding it?
The first solution was from cavmaths (@srcav), who talks about his solution in his blog here. The crux move in his solution is here:
The second solution is from @bodilUK:
I received the third solution in instalments from Martin Noon (@letsgetmathing). The twitter conversation went like this:
Making sense of these solutions
In a previous blog I talked about the idea that problem solving can be split into tactics and strategies. We need to learn both in order to increase our chances of success - and then try lots of problems with which to practice using them. Through repeatedly using various tactics and strategies, we can increase our chances of solving a given problem, which gives us confidence to try different approaches the next time we encounter a problem, and the perseverance to keep going when nothing seems to work.
We can see this split between tactics and strategies in the above solutions to this problem. Each person used tried and tested mathematical techniques (tactics) such as similar triangles, trigonometry or Pythagoras to arrive at the final answer.
But to me the more interesting part of this problem is the crux move that unlocked the problem, and the strategies that people used to find it. It would appear that each person who solved the problem drew a diagonal in the titled rectangle, and drew a perpendicular from H. Where did the ideas to draw these diagonals or perpendiculars come from? Did it come from divine intervention? Did it come from just randomly trying things out? Did each of these mathematicians stumble across the crux move, or did they have a strategy for finding it?
Mathematician George Polya tells this story about a mouse:
The landlady hurried into the backyard, put the mousetrap on the ground (it was an old-fashioned trap, a cage with a trapdoor) and called to her daughter to fetch the cat. The mouse in the trap seemed to understand the gist of these proceedings; he raced frantically in his cage, threw himself violently against the bars, now on this side and then on the other, and in the last moment he succeeded in squeezing himself through and disappeared in the neighbour's field. There must have been on that side one slightly wider opening between the bars of the mousetrap ... I silently congratulated the mouse. He solved a great
problem, and gave a great example.
That is the way to solve problems. We must try and try again until eventually we recognize the slight difference between the various openings on which everything depends. We must vary our trials so that we may explore all sides of the problem. Indeed, we cannot know in advance on which side is the only practicable opening where we can squeeze through.
The fundamental method of mice and men is the same: to try, try again, and to vary the trials so that we do not miss the few favorable possibilities. It is true that men are usually better in solving problems than mice. A man need not throw himself bodily against the obstacle, he can do so mentally; a man can vary his trials more and learn more from the failure of his trials than a mouse.
Is Polya suggesting that problem solving is just a case of trying things until they work? I don't think so, I think he is just saying that we have to fail and fail again until we succeed, which is true for any problem that is worth solving - if a problem is worth solving, the crux move will be hidden from us for some time and we have to search until we find it.
So how do we go about searching for the crux move efficiently? It is the development of strategies, as well as tactics, that will help us become better problem solvers. In this website I will continue to present problems, analyse solutions, and identify strategies in order to try to get to the essence of problem solving.
And for the record, here is my solution to the problem: