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Recursive Returns

Click here to try using Recursive Returns for yourself. Keep reading to learn more about the principles behind it and how to use it. Once you understand this, you can try the challenge and learn how problem solving can work.


A Recursive action is one that keeps occurring in a repetitive fashion.

In this interactive activity you can create a dazzling array of patterns by having a simple action repeated any number of times. This activity is like a simplified, visual version of the LOGO drawing language.


Below are some examples of what you can do in the Recursive Returns activity:


Change the values in the following activity to make different patterns. If you want to reset the activity to the original settings, hit the “Refresh” or “Reload” button in your Internet browser.

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Example (how the activity above starts)

When you start this activity in Normal mode, the following recursive process takes place:

  • Walk one pace then turn 90 degrees to the right;
  • Walk two paces then turn 90 degrees to the right;
  • Walk three paces then turn 90 degrees to the right;
  • Continue by repeating these three commands for four cycles in the same order.

Try changing the “Angle to Turn” to 89 degrees - it will help you see the example above.

Details of How this Activity Works

Normal Mode:

This activity uses the following recursive process:

Expert Mode:

Now, for each Step you can change the Angle and Number of Paces separately and repeat for a given number of Cycles.

The Challenge: Predictable Patterns

In the example above, after a total of 24 paces, you end up in exactly the same place as where you started! Can you discover the conditions under which you will end up in the same place as where you started? Test your theory by making some predictions and seeing if they work in the activity above.

When you get the hang of normal mode, try expert mode (click the tab in the top-right hand corner of the activity). Be sure to try changing all variable, including “Angle”. Does your theory above still hold true?

Problem Solving

The process you have used above is a good example of how to problem solve. In this case, you can think of problem solving as:

  • Understand the situation
  • Get familiar with the situation.
    • It can help to start with some existing examples.
    • Then try running through the situation on your own.
  • Understand the problem - now that you fully understand the situation, you are ready to understand the problem in that situation.
  • Experiment and Observe - try the situation under different conditions and observe how different changes can produce different results.
  • Develop a Theory - a way of explaining or answering the problem.
  • Use Your Theory to make predictions about how different changes can produce different results.
  • Test Your Predictions - using increasingly complex and extreme conditions. If your predictions still hold true, then there is a good chance your theory is true. If your predictions fail, then you will have to revise your theory.

However, remember that even though you can prove a theory wrong, you can never prove a theory right - there could always be a condition you did not think of that makes your predictions fail. Therefore, the best you can ever say is that your theory holds under a given set or a given range of conditions.


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