[3ACTS] Toothpicks

I enjoy tasks that exhaust a finite supply of things in order to make some kind of interesting structure. Here, a finite supply of toothpicks (250 of them) are exhausted to make a pyramid. (Or consider the finite fencing around Pixel Pattern.)

At some point I’d like to test out the hypothesis that removing the finiteness would make the video a lot less perplexing on the whole. In other words, we wouldn’t be as perplexed by a guy plugging away at a pyramid with an inexhaustible supply of toothpicks. We’re perplexed because we know, at the end of the video, that he’s done and we want to know what the pyramid looks like.

Here’s the task page.

Here are several interesting questions that popped up at 101questions:

  • Alison: How many rows in the end?
  • Hope Gerson: How many toothpicks does it take to make the next sized equilateral triangle?
  • Douglas Moore: How many triangles?
  • Matthew Clark: What’s the perimeter of the final triangle?
  • Scott Westwell: How many small (3 toothpick) triangles can be made?
  • Jeff de Verona: How many “total” triangles will be created (any size)?
  • Gregory Taylor: How long did it actually take to go through the entire stack?
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. FWIW, I tend more often to find myself with an image in mind and wanting to know if I will have enough toothpicks (or whatever) to create it. This may be a case where maximizing “realistic” (in the sense of lived experience) and maximizing “perplexing” aren’t the same. I hypothesize that it’s more perplexing to imagine the person with the plan and wonder what they got, even if it’s more realistic to be a person with a plan and have to figure out if it will work. The first task requires some imagination and math, the second will always be solvable by tedious trial and error (and that’s obvious from the get-go and therefore un-perplexes me).

    Subtle things make a big difference in what makes a problem feel engaging and worth doing, thanks for continually identifying and making it easy to get data around those subtle things.

  2. Now knowing your affinity for finite things, I ask you how long would your paperclip chain have been if you would have exhausted your supply and not been up against a certain time?

  3. Jennifer Eaton

    January 10, 2013 - 3:42 pm -

    Thanks Dan, another fun one. I just used this TODAY! I have a new class of students who have failed Algebra I first semester. Counseling is still transferring students into the class, so the first 2 days have been 3 Acts (We did Taco Truck yesterday).

    It’s been very interesting. They’ve been engaged (mostly) and are showing me how smart they really are!!
    Now, the harder question….how do I help them become proficient in Algebra (and remediate basic math skills)? We are in a computer lab, so if anyone has any analysis of the merits of any software or online tools, or recommendations for curricula, I’m open to anything!! Thanks!!!

  4. Thanks for the feedback, Jennifer. We’ve been having conversations in another thread about the merits of ST Math, which is one of the better tools available, from my limited exposure. Dreambox is good for elementary and DragonBox is an interesting solving equations app for the iPad.

  5. Great stuff as always, Dan.

    Regarding the role that finiteness plays, if we view 3-act lessons as an exercise in students posing (or discovering) intriguing questions, this just seems so much more difficult without some sort of constraint. For this particular video, if you were to cut to a graphic of shelves covered with an unnamed number of toothpicks, or some screenshot showing that the supply of toothpicks was unlimited (or unknown), then cut back to you creating triangles and end with an ellipsis, what question would I ask about this task? Those that come to mind are:

    1. How does the ratio of rows:toothpicks relate to the number of rows?
    2. What is the number of toothpicks it would take before there is no more space in this room to continue the pattern?

    You’ll notice that, for #2, I picked my own constraints.

    I could be that I’m just not very imaginative, but these are the best I can come up with. To me, these pale in comparison to the questions already generated with the number of toothpicks given.

    I think there is a sweet spot where just the right number of constraints are provided: too many and you fall into the textbook trap where it’s all right there; too few and the problem just isn’t all that intriguing.

  6. Hi, Dan. I would like to ask you something and I hope that this is the right place. I tried to solve the task by using arithmetic sequence. I noticed that on each level the number of toothpicks increases by 3. The sum of all the toothpicks is 250, d=3, A1=3, An=A1+(n-1)d and using the formula for the sum of arithmetic sequence we end up with a quadratic equation with only 1 solution n=12,4196(which is ok because we don’t have only 12 levels). Furthermore 12,4196-12=0,4196. Multiplied by 36(the value of A12) is equal 15,106(here I tried to get the exact number of the toothpicks left). The actual count is 15. Where do you think my mistake is? In 0,4196*36?

  7. Unless n=12.4196 was the exact solution, I’d anticipate some error from rounding when you subtracted the theoretical answer from the experimental answer.

  8. Well I had some of my kiddos play with this one yesterday. After some thinking, they got the number of rows AND could tell me how many toothpicks he had leftover for 250 and 500 toothpicks. Coming up with the formula stumped them but I DID have one group come up with this. t=3[n + (n-1) + (n-2) + (n-3) +… ] Can’t really expect more than that when they haven’t studied series! :) It was a good day

  9. The real magic happened when students shared their ideas at the end about how they solved for the number of triangles and rows. The comment that stuck out to me was one student mentioned that he started to draw out the triangles but then he stopped. As soon as he saw a pattern emerging he could switch to more efficient process of using numbers.

    I was able to quite succinctly go on to describe how we use numbers to describe patterns but that algebra gives us the power to describe patterns for any given number, in this case, of toothpicks.

    Next week we start the unit on ‘What is a function?’. I hope this has given me enough to build upon.

  10. I used this as a starting activity before we jumped into sequences and series in my advanced precalculus class today. It was super fun to do. I brought toothpicks for them to use and gave them giant whiteboards, and had them in groups of 3 and 4. Just gads of fun to listen to them work through the puzzle. I barely said anything once they were off working. And I heard some awesome observations.

    Their work for over the weekend?

    1) For 12,594 toothpicks, what will the final picture look like?
    2) How many vertices are there in the final picture?

    Thanks for a fun class!