Posts from the ‘logic’ Category

This is a delightful way to revise Venn Diagrams with older students: A Venn Diagram of mythical creatures.

I was recently invited to do an IQ test. One of the questions was as follows: “You walk five miles north, five miles west, then five miles south. How far are you from where you started?” The answer that they were looking for was 5 miles.

Perhaps our use of maps convince us of this logic; a logic based on Cartesian 2-dimensional geometry. Unfortunately, we do not live on a Cartesian plane!

How should we understand this IQ test question when we correctly consider that we live on a sphere? A good way to consider this is to think about what happens at the two poles!

The South Pole

Start at the south pole. Travel five miles north. Travelling west is to travel parallel to the equator, so then when travelling five miles west, you get no further from the pole. Then travel five miles south. You arrive where you started, with no distance between where you started and where you are now.

The North Pole

The north pole example is more difficult to imagine, and some may think there’s a trick here. If you start five miles and a bit south of the north pole, move five miles north then five miles west in many tight circles around the pole ending up exactly opposite to where you were when you originally arrived near the pole. Then move five miles south. More or less, you are now ten miles away from where you started.

The trick here is the ‘bit’ which ensures two things: firstly that you can in fact travel west; secondly that having travelled five miles west you finish up exactly opposite where you were when you started moving west. It might be argued that if you ended up exactly at the pole then you would be unable to move west at all. The ‘bit’ ensures that there is a trivially small circle around the pole that you can travel west around. It is also necessary to end up exactly opposite where you started to maximise the resultant difference between the two starting positions. If you imagine the bit as a radius of a circle around the pole, then it can be calculated as any r such that 5 miles = (2n+1)*pi*r where n is a positive integer.

Summary

Thinking about a problem often involves thinking around its extremes or limits. When thinking about compass bearings on a sphere, the poles offer places where their odd relationship to each other are most apparent. The IQ problem assumes that we live somewhere where the relationship between the compass bearings closely resembles the relationship between Cartesian axes. At the poles this similarity breaks down most markedly. By thinking about moving to and and from the poles, it transpires that if you move five miles north, five miles west, and five miles south, you may end up a distance of x miles from where you started, where x is such that: 0 miles <= x < 10 miles.

Now, if the IQ test was testing for this as an answer, I would have been suitably impressed!

Philosophical Footnote

The earliest known argument against the earth being flat comes from Aristotle, who argued that the shadow that the Earth casts onto the moon during lunar eclipses is always circular. The only object which casts a circular shadow irrespective of its orientation is a sphere, and since night and day convince us that the earth does not have a constant orientation with respect to the moon and the sun, the Earth must be spherical. (Aristotle De Caelo, 297b31-298a10)

Our natural instincts about the world are perhaps that it exists on a plane that is looped; the outside of a cylinder with the poles at the top and bottom of the cylinder. This is a practical simplified model of the earth because until you get into the arctic and antarctic more or less, two people moving north are moving more or less parallel to each other, and the consequences of longitude and latitude working on quite separate principles need not be considered.

We could of course change the way that north and south work, and make them akin to east and west. Perhaps the great circle through Grenwich could be the East-West equator, as in some senses it is, and we could therefore define an east and west pole, somewhere on the equator! It is an interesting thought experiment. Would our concepts be more easily understandable if we did this? Why did North and South become defined as it is?

Scientific-Computing.Com’s recent blog articles "Beyond the Prisoner’s Dilemma" and "Global warming and the Prisoner’s Dilemma" are interesting examples of using the logical structures of game theory as starting points for logical thought at younger levels.

The blog conversation starts with the big environmental issue of global warming and this friendly video where the protagonist explains the application of minimax to the global warming debate. He argues, in a nutshell, that given uncertain future consequences of humanity’s impact on the planet, and given also a choice of decisions about how to act against those potential consequences, it makes the most logical sense to exclude the catastrophic choice of doing nothing to prevent global warming by acting as if global warming were a certainty.

I like how the minimax principle here facilitates students’ understanding by offering a powerful structure for considering different possibilities. Because minimax is so clearly structured and relatively easy to grasp, it is the sort of idea, like the Venn Diagram, which ought to be a constantly recurring feature of students’ education. In Mathematics, it bears close resemblance to the Carroll diagram, which is a similarly undervalued structure for understanding issues.

Bloxorz is another interesting puzzle-based game which deserves attention. It is a very clever concept, though it takes some levels before you approach a level which cannot be done very easily by trial and error.

As with most such games that are purely designed for entertainment but which have educational problem-solving potential, the number of tries again is too great. Fewer moves would mean that players would need to develop more of a strategy and to think of multi-step solutions. However, we can’t have everything. Well worth a look.

By the way, if you are in any doubt as to the problem solving potential, please try to work through to Stage 11, which is an absolutely lovely puzzle! I did it… eventually… and I have to admit that I used squared paper to help me work it out!

ReMaze is a small Flash game which has some fascinating puzzles to stimulate problem-solving, and which is really quite engaging. This is well worth a look.

My only concern about a game such as this for education is that too often while I was dimly aware of the sort of strategy I should employ to solve the puzzles, I was too often relying on trial and error to solve the problems. To convert this into a thorough educational game would require careful thought about how to make the problem-solving more explicit.

Apparently there are 21 levels!

Gravity Pods is a 50-level game which requires a great deal of problem solving and creativity.

It is a good example of a game which is educational in the sense that it requires students to problem solve and explore the structure of the game. It is certainly not intended to be educational, and has some elements that are not perhaps suitable for an educational scenario (the name of the website for example).

Nowhere on the curriculum could you pinpoint an aspect of mathematics or any other subject which this game addresses. However, if that is a necessary criterion for you as an educationalist, you are in the wrong place! I do not believe that what is learned in mathematics classrooms can be meaningfully described in terms of the criteria prescribed by the national curriculum.

Oh, and I was unable to do the 50th level. It is really, really hard.

At University I was introduced to a really interesting CD called Problem Pictures. It had some fascinating images that all had some mathematical significance. I printed out about fifty and they made the walls of my classroom bright and appealing. It also had questions related to each photo, but I found that I rarely used them, preferring to create my own questions and ideas.

Enter Flickr, a great online photo sharing tool. Though you will not get questions related to each photo, the database is huge. Typing into the Flickr search engine ‘geometry’ turns up 10,810 photos at the time of writing. Most of them are really relevant; some of them are utterly spectacular. Here is a random sample: