## Random Walks through a-MAZE-ing Triangle

The main activity for last weeks lesson with the first and second graders was adapted from the book Math Circles for Elementary School Students by Natasha Rozhkovskaya (and it will likely be the first of many).

The kids were split into groups of two or three and each group was given a copy of the Random Maze.

This maze has exactly one entrance and seven exits, labeled 0 through 6.  Anyone with a coin can go through the maze as long as they follow this rule: At every crossroad you flip the coin.  If it lands on heads you go to the left and if it lands on tails you go to the right.

The kids in each group took turns going through the maze, using a stone to keep track of their positions.  They then recorded the exits they came out of in a table.  To motivate this I stole the story from Rozhkovskaya’s book about an ice cream vendor that wants to put his cart near the exit where the most visitors to the maze come out.

The kids had a ton of fun flipping the coins, and I was amazed that only on a few occasions did the coin land on the floor or all the way across the table.  Here are some pictures of the process:

About half way through the process, I overhead one girl saying, “Now the stone definitely won’t come out of exit 6.”  I looked over and saw that indeed, the coin had moved one step to the left!  I drew everyone’s attention to the observation and several minutes later I was hearing many of the groups discussing which exits were still possible from the various positions that their stones were in.  This was very exciting!

At the end of class, we compared the tables of results for all the groups.  Exit 4 was by far the most popular (so their throwing was somewhat biased, but not too terribly).  We then discussed that some exits had more paths leading to them than others and that we’d expect more people to come out of those exits.  Some kids noted that exits 0 and 6 had only one path leading to them (and in fact, no one ever came out of that exits in all of their experiments) and all other exits had more.

I then asked them which exit they thought had the most paths leading to it.  After a brief pause, one kid said that he thought it was Exit 3, and almost everyone agreed.  They were not swayed by the experimental data!  I am not sure how to explain that, but I find it very interesting.  I then asked them why they thought that Exit 4 was the most popular among their results.  No one seemed to have any thoughts on the matter, but it didn’t seem to bother them very much.

Next time I plan on showing them how to actually compute the number of paths leading to each exit and then play around with Pascal’s triangle.