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Kerry’s random walk

By David, 9 July 2015 Activity

a six pointed star with each point numbered.

Draw a compass rose and number each line coming out of it.

Kerry is at the country show, but Kerry is lost. In this activity, we’ll explore the maths of random walks, and find out if Kerry can see all the show has to offer!

You will need

What to do

Trianglar grid paper with several spots marked out as amusements.

Mark some places of interest on your map.

Your sheet of graph paper is going to be a map of the show.

  1. First, we need to make a compass rose. Choose an out-of-the-way intersection on your graph paper, and number each of the six lines coming out of the intersection.
  2. Around the graph paper, mark out the areas that Kerry wants to visit. Some suggestions are farm animals, cake competition, showbags, sideshow alley, food stalls and the music stage.

    Someone drawing a line with a red pencil. There is a dice nearby.

    Roll the dice, and draw a line in the indicated direction.

  3. Order your pencils in a rainbow, and choose an intersection near the middle where Kerry starts from.
  4. Roll your dice and consult the compass rose to find out which direction Kerry has gone. Take a coloured pencil and draw a line, one triangle long in the indicated direction.
  5. Roll again to find a new direction, and continue Kerry’s journey, once again moving one triangle length.

    A colourful squiggly line weaves its way all over the triangular grid paper.

    Keep rolling and drawing until you visit all the attractions, or fall off the page.

  6. Sometimes you may go back over a line you already drew. That’s okay – just remember where Kerry is when you continue the line further.
  7. Keep rolling and charting Kerry’s progress. After 10 rolls, change colours so you can tell where Kerry has been recently and places Kerry visited a long time ago.
  8. Keep rolling and drawing and changing colours until Kerry visits all the areas, or Kerry falls off the edge of the map.

What’s happening?

The way Kerry moves is an example of a random walk. Random walks are not a great way to explore a country show, but mathematicians find them very interesting. They occur quite often in nature – the motion of tiny tea leaves in a hot cup of water, and the hunting patterns of silky sharks are both forms of random walks.

There are many interesting mathematical questions to do with random walks. For a long time, mathematicians wanted to know if every random walk would re-visit their starting points. After many years, they concluded that random walks on a flat surface would almost always return back to their starting point eventually. But walks in three dimensions, like a tea leaf in water, would only return about one third of the time.

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