Jacob H. from Austin, Texas wrote a few computer programs to perform the numerical recycling procedure from Challenge 13.

In his response video, Jacob demonstrates how his programs work and describes his results. He also poses a fantastic new question about the numerical recycling procedure that emerged as he studied the original challenge.

What other thoughts do you have about Challenge 13? Do Jacob's results give you any ideas? How would you approach the new question Jacob poses? Get in touch and share your thoughts!

We're taking a break this month, but there are still problems to be solved! Check out our recent challenges:

Make a mathematical mess in Challenge 08: The Confetti Problem. Or, be the first to take a stab at a number that's nearly 6000 digits long in Challenge 09: The Awe of Large Numbers.

We'll be back again in April with Challenge 10, so stay tuned!

Challenge 07 from Collaborative Mathematics is here! Watch the video, try out the probability experiment, and send in your data. Or, tackle it theoretically!

Plus: we've received a great response to Challenge 06 from Dani, and a fantastic new approach to Challenge 03 from Paul.

There are several very cool blogs out there -- among them Math Mistakes and Bridging the Gap -- that highlight the value of thinking deeply about student responses, the mistakes they make, and the explanations they give for doing things the way they do.

One of the ideas behind Collaborative Mathematics has always been that participants could learn something from exploring other people's solutions, in addition to exploring the challenges. Some of the response videos in the archive contain mistakes, and I think that's fantastic. Rather than being an answer key of tidy solutions, they represent a library of arguments and ideas that require additional critical thinking.

One aspect of this is that I avoid posting my own answers to the challenges. I don't want my way of solving a problem to be interpreted as "the right way" just because I'm the one who posed the challenge in the first place. Over the next few posts, though, I would like to share my thoughts on a particular solution as a way to talk about my own mathematical curiosity, and what I've learned from a really creative approach to one of the video challenges.

But before I start going on and on about what I think, I want to give you a chance to explore for yourselves. Here's what I suggest:

• Take a few minutes to visit the archive page for Challenge 03: Finger Counting.
• Solve the challenge on your own, if you haven't already.
• Watch a few of the response videos.
  
• In particular, watch Maile's solution and see what makes you curious.

Next time I'll share how I've come to think about the mathematics of Maile's approach.

I recently received response videos to a number of earlier challenges, so I'll summarize the updates here. Click the links below to visit the archive page for each challenge, where the new videos are highlighted. Plus, you can watch the original challenge video and the other responses, and take on the follow-up challenges.

• A new solution to Challenge 01, in which our problem solvers go back to the beginning and hope to count the even "Ermer Numbers".
  
• A new solution to Challenge 03, in which our problem solvers take on the most popular challenge from the spring, "Finger Counting".
  
• A new solution to Challenge 04, in which our problem solver sets about to explain how we know we have found the minimum number of button presses on the less-than-useful "Two-key Calculator".
  
• A new solution to Challenge 05, in which our problem solvers bust out the rulers for a practical approach to "International Paper".
  

The first response videos have arrived! Check them out, and explore some follow-up challenges, by visiting the Challenge 05 page in the archive!

None of the videos (so far) have taken up the challenge of describing a new piece of paper for us to explore, so I'd like to encourage you to put on your problem-posing shoes!

• What if the paper had been a square? A golden rectangle?
• Describe a piece of paper such that exactly one-third of the paper is inside the largest circle that can be drawn. Can you find a real piece of paper that fits your description?
  
• What other questions could we pose?