Can You Solve the Duplicator Lab Riddles?

“You have unwittingly stumbled into a duplicator lab…”

As you watch this brief, animated story about the duplicator lab – which includes 3 riddles – you may find yourself reaching for  a piece of paper, a pencil, and an entirely new kind of strategy.

How challenging will it be to solve the riddles from inside of the duplicator lab?

Watch the video, answer the riddles, and post any comments you have below.

ALSO:  I’m trying something brand new on my blog, which is to set up a way to deliver MY PERSONAL ANIMATED FILES which I’m using to tell the story directly to you.  Not only can you use the video just as it is in your class, but you can also have the PowerPoint files delivered to you, so YOU can tell the story yourself.



If you want to receive the PowerPoint files through email, just fill out this form.  You’ll receive the PowerPoint file I used and I’ll also send you new updates as they appear.


Send Me the Duplicator Lab PowerPoint!

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I’m going to send the answers separately so I don’t give them away at the end of the PowerPoint.  I want to preserve that vital space between the question and the answer.

When you are ready to see the answers, just fill out this form, and I’ll send them right over to you.  If you don’ see it right away, remember to check to see if your email has filtered it out of your main email box.  Enjoy!


I’m Ready To See The Answers!

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I’m learning as I go, and I appreciate your support!


Here are some still frames of the first 6 duplication events.  I recommend using the PowerPoint file, but this image might be a useful reference.


Also, I just wanted to send a note of appreciation to Jeremiah Ruesch, @mathkaveli who willingly dove into this challenge and provided me with feedback!  Thanks, Jeremiah!


All my best,



After reading this blog post, take a look at  THE 12 MOST POPULAR MATH STRATEGIES AND DOWNLOADS ON THIS BLOG.


P.S.  There are now 105 free Splat Lessons that you can download from my blog.


The Fraction Splat Series!

The Nested Splat Series!





Steve Wyborney Avatar

22 responses to “Can You Solve the Duplicator Lab Riddles?”

  1. Leonie Avatar


    Thanks for sharing. these are great and my year 5 students will love this. We are working on it tomorrow.

    1. Avatar

      Hi Leonie,
      Thank you for posting a comment here. I’d love to hear how it goes.

  2. Ellen Avatar

    Ok, just found the answer to my question. I guess it pays to read all the way to the end! I think I will still wait awhile before I even let them know there are answers to check against!

  3. Lisa Sanderfoot Avatar
    Lisa Sanderfoot

    We appreciated the definitive answers. The fifth-grade class working together finished correctly all 3 riddles in about 30 min. Now I owe them pizza. #mathfun

    1. Avatar

      What fun! I wonder what would happen if you sent 2 pizzas into the duplicator lab. Tell your class, “Great job!” I’m going to have to cook up a new challenge. Way to add to the fun!

  4. Aaron Bieniek Avatar
    Aaron Bieniek

    Hi Steve . Really interesting problem. I’m curious to see where you are going with this. In the mean time I’ve been trying to automate the process as much as I can. As of now, I can determine the “state” of Duplication Events 3n, 3n+1, and 3n+2 for any n, if given 3n, 3n+1, and 3n+2 for the previous n, with the n=0 initial states of (1,0,0), (1,1,0), and (1,1,1).

    What I’m curious about is how much more generalized I can make this. For example, I can only think in groups of 3 right now, so like I tried to express above, I can tell you about Duplication Events 24, 25, and 26 but only if I know the states of Duplication Events 21, 22, and 23.

    1. Avatar

      Hi, Aaron. I really appreciate your comment. Your insights and curiosities are very impressive. I just released the answers.

  5. Jim Doherty Avatar

    I will be sharing this question with all four subjects I teach in HS this week and will report back on the differences in comfort level. I am also curious about asking my Ss to retry the problem with a different starting point instead of 1 robot.

    1. Avatar

      I can’t wait to hear what you learn about the comfort levels. If you try with more robots you will be experiencing some of what appears in the sequel. 🙂

      I’ve posted the answers now:

  6. Barry Parker Avatar
    Barry Parker

    Enjoyed the puzzles. If my math is correct, that famous sequence suggests a neat predictive formula after several generations of robots, a formula which can be used to double check calculations.

    Thanks for the riddles,

    Barry Parker

    1. Avatar

      Thanks, Barry! I’ve posted the answers now, so you can see if it matches the sequence. 🙂

  7. Lisa Sanderfoot Avatar
    Lisa Sanderfoot

    My 4th and 5th graders are working on this, there are a few of them that are a bit obsessed. Thanks for the excellent math, hoping for a definitive answer at some point.

    1. Avatar

      Hi, Lisa. I added an answer download box on the blog so you’ll be able to find out now. Thank you for posting this comment!

  8. Alicia Hess Avatar
    Alicia Hess

    I had my 4th Graders working on this as teams this morning, and shared it with parents so that they can work on their own to try to solve the riddles at home if they want to. I posted a large piece of paper on the classroom door with a spot for each riddle, and sticky notes, for kids to post their answers if they think they’ve solved them. At the end of the week I’ll reveal the answers. The kids are totally engaged and during their team time they was debate, collaboration, math discourse and of course… healthy disequilibrium! Thanks (as always) for being so generous with your creations, Steve!

    1. Avatar

      Hi, Alicia! What wonderful strategies. I really like how you have taken the intrigue of the riddle and extended it across the week and to families. It will be fun to find out what the students post on the papers. The reveal at the end of the week should be really fun, too! Thank you for taking up this challenge and for sharing your experience.

    2. Kimberly McLean Avatar
      Kimberly McLean

      Great idea Alicia! I am going to try that in my 4th grade classroom too next week.

  9. Nate Schmolze Avatar

    I ended up trying to visualize it in a spreadsheet. Each station had a variant of the same color; red, blue, or green. Each shade (light to dark) was the labs over time.

    The one pattern that emerged was an unknown cell (red for example) could be figured out by a known previous cell + the previous red cell. One could use that to answer the riddles. I believe the 19th event would be the last one prior to the lab hitting 1000.

    1. Avatar

      Here is a link to the post – which now includes an option to download the answer. Good luck!

  10. Husky88 Avatar

    I also enjoyed this puzzle. I found a pattern while recording ways to keep track of my thinking. This allowed me to extend the pattern until I was able to answer the last riddle. Keep them coming!

    1. Avatar

      Thank you! I will keep them coming. I’ve also now posted an option to download the answers here:

  11. Tyler Auer Avatar


    I enjoyed this puzzle. The third riddle was, of course, the hardest. And, I admit to using OEIS in my search for an explicit formula — no luck.

    Like all good problems, I was surprised to discover a cool connection, specifically the solution’s similarity to another quite famous sequence. Hopefully my vagueness will not spoil this for others.

    1. Avatar


      I’m honored that you have read the post, solved, and commented. I’m also glad that you discovered that connection. I think your answer has exactly the right amount of vagueness for those who are reading our words here. This puzzle becomes more and more fascinating the more I look at it – and at variations. For example, what happens if there is already a robot in room 1 before the story begins. Or what if there is a robot in each room before it begins. Thank you for taking time to solve, comment, and share your insights here!

      All my best,

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