Ultimate Toothpick Pattern

Meeting Summary

We met to talk about a very simple, but really interesting, new pattern that can be created in the following way.

  1. Start with a single toothpick
  2. Add another toothpick at a right angle to each of the pointy-ends of the pattern
  3. 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:

toothpicks_50

Bonus Question:

Which stage of the pattern is shown in the picture?

Related Links

A Few Photos from the Meeting