Activity 1-3: Coincidences

What is the strangest coincidence that you have ever come across? Coincidence is an idea that can sometimes be treated mathematically. That has ever happened to you?

For example, take the Birthday Problem. You discover that two of the people on a football pitch during a match (including the referee) have the same birthday; does this count as a coincidence? Given two people, what is the chance no pair has the same birthday? Given three people, what is the chance no pair has the same birthday? and so on...

This product is getting smaller and smaller; how quickly does it get to a half? Birthday Problem Spreadsheet So it turns out that the chance that some pair from 23 people have the same birthday is better than a half. So given two football teams and the referee; not really that much of a coincidence. Check the formula for Column B here. Task: can you construct an Excel spreadsheet to answer this? carom/carom-files/carom-1-3.xls

Two people A and B on Facebook are chosen at random. If A contacts all their friends, and they all contact their friends, and so on… How many messages will there be on average (assuming that there is a path from A to B) before B is reached? Facebook calculated this recently, and the answer was The world is getting smaller!

Paul Erdos was mathematician who gave his colleagues an Erdos number. If they had written a paper with him, their Erdos number was one. If they had written a paper with someone who had written a paper with him, then their Erdos number was two, and so on. We can use this idea in all sorts of ways...

Imagine a chain of people such that every person in the chain has shaken the hand of the people on either side of them. The Handshake-Number for two people is the number of people in the shortest possible such chain. How could you reduce your average Handshake-Number dramatically? Might there be two people with no Handshake-Number?

Beware; it is easy to manufacture coincidence if you try. If you are determined to find strange numerical significance in all the measurements of the Blackpool Tower, you can do!

But is there a force in nature heading in the direction of coincidence? The psychotherapist Jung developed the idea of synchronicity; that events have a tendency to occur in apparently coincidental ways. Twins are said to know if their partner dies, even if they are on opposite sides of the world at the time.

Coincidence can happen in mathematics: although sometimes the apparent coincidence can be explained. For example, pick six numbers a, b, c, d, e and f in arithmetic progression. Now solve the simultaneous equations ax + by = c, dx + ey = f. What results for x and y do you get? Try this a few times; is this coincidence? Many mathematical theorems try to give the reason behind an apparent coincidence.

The east coast of Latin America and the west coast of Africa appear to fit together. Is this a coincidence? You can find the number e on your calculator. e = Is the repetition of 1828 a coincidence? A coincidence? A total solar eclipse is remarkable because the moon almost exactly covers the sun. This is because the sun is about 400 times wider than the moon, and also about 400 times further away from us.

Carom is written by Jonny Griffiths,