We met to talk about a very simple, but really interesting, new pattern that can be created in the following way.
- Start with a single toothpick
- Add another toothpick at a right angle to each of the pointy-ends of the pattern
- If two toothpicks, meet at their pointy-ends, don’t add toothpicks to the place they meet.
We started with actual toothpicks, and eventually started to see a neat pattern emerging. And then, things got all mathematical!
Questions We Asked
- How many toothpicks are needed at each stage of the pattern?
- How many NEW toothpicks are needed at each stage of the pattern?
- How many squares get made in the pattern?
- How many rectangles get made? What are their dimensions?
- Will the pattern ever end?
By carefully keeping track of the toothpicks at each stage, we figured out a way to generate all the numbers in the ‘toothpick sequence’ using previously calculated numbers. This means, we found a recursive pattern in the sequence of numbers.
One picture we used as a reference:
Which stage of the pattern is shown in the picture?
- Overview on Wikipedia
- An applet for animating the pattern
- Research on this sequence
- Known information about the sequence