Here’s a one-minute video that introduces the problem from The 1-to-10 Card Investigation I wrote about in a blog about a year and a half ago.
And here’s a comment that was posted just recently. Actually, it was a request.I used to have the order written down and I haven’t used this for years. I can’t find the card order and have been working on it for about 20 minutes. Can someone give me the card order? Thanks.
I don’t have the card order written down either, but I got a deck of cards and fished out ten cards, Ace through 10. I’ve solved this problem many times over many years so it really isn’t a problem for me any longer. I was able to figure out the card order fairly quickly.
Then I thought about how to respond to the request for the solution. I began a conversation with myself. Providing the answer would be quick and easy now that I’ve solved it again, but I wondered about what additional information I might also include. Actually, I had quite a long conversation with myself about how to respond.
The conversation in my head went something like this:
— Hmm, in the classroom, I resist giving students answers. Should I do it here?
— Well, why not? She obviously really wants to know and why should I withhold information that I have?
— But, then again, I worry that giving the answer betrays my educational belief that productive struggle has value. And what I especially love about this problem is that you know that you’ve solved it when you can deal the cards the way I showed on the video. No answer book is needed.
— But if a problem is frustrating, then struggle may not be productive. Maybe she doesn’t want to spend any more time on trying to figure it out. Also, if someone really wants the answer, maybe that will help spark some thinking.
— A confession: There are times, most recently when I was tutoring my granddaughter in high school math, when I check the answer in the answer key at the back of the textbook and then work backward from that to figuring out a solution strategy.
— So if the answer can be useful, why didn’t I just put the answer above in the original blog?
— Well, I worry that the solution would either be a spoiler for people who were curious or a so-what for people who weren’t interested anyway. In this instance, I think the answer would most likely close down thinking.
— About the problem itself: Why did I select it for the blog and make that hokey video at my kitchen table? The problem really isn’t a biggie, not up there on the top ten problems I love. (Well, maybe it’s close.) But if you never came across this problem, or never figured out a solution, it’s no big deal. You’ve gotten this far in life without knowing how to arrange ten cards so they come up in order in the very specific way I modeled of dealing and putting cards underneath. How important is that?
— Well, then, that supports my not giving the solution. I don’t think that knowing the answer will be a huge enhancement to anyone’s life, mathematical or otherwise. And not knowing probably won’t be much of a detriment.
— Back to the classroom: When teaching, I want to give students challenges that ask them to make sense of problems, encourage them to persevere when solving them, push them to express their solution strategies with precision, have them communicate their ideas to their classmates, talk with them about patterns and structure that can help them solve related problems, and more. All of these ideas directly relate to the Standards for Mathematical Practice, which describe important ideas for doing mathematics.
— Well, where’s the mathematics payoff in this problem? The numbered cards seem to imply that the problem has something to do with math, but I could just have easily labeled the cards with letters from the alphabet―a-b-c-d-e-f-g-h-i-j―and presented the same challenge. Would that be a problem I would use in math class?
— Yes, because I think it has the potential to engage students with the Standards for Mathematical Practice, as I mentioned above, in a way that’s playful, low stakes, and hopefully engaging.
— OK, should I give the answer and be done with it?
— I wouldn’t do that in class.
— But I’m not in class. I’m at home in front of my computer reading a comment from a person who is interested and stuck. She worked on the problem for 20 minutes and hasn’t felt successful. She made the effort to reach out on my blog site. I want to encourage her to continue to engage with the ideas I express in my blog posts. I want to be helpful.
— Hmm, what’s this about working on a problem for 20 minutes? How much time should someone spend trying to solve a problem. I googled “how long to work on a math problem.” My search returned over 92 million hits. (This conversation with myself was now becoming a time suck.) One site caught my attention and seemed promising. It’s from the Mathematics Stack Exchange, which describes itself as a question and answer site for people studying math at any level and professionals in related fields. It had a post titled How much time is too much (to put into a single problem)?. See what you think.
Now this is really turning into way too much of a time suck. It’s time to get back to life.
Thanks, Jo, for your comment. It got me thinking and for that I’m appreciative.
ALERT: THE SOLUTION JO REQUESTED IS AT THE END OF THIS POST. IF IT WILL BE A SPOILER FOR YOU, HERE’S MY ADVICE: STOP READING, CLOSE YOUR BROWSER, AND GET SOME CARDS.
A Last Comment
I’ve posted the answer below, Jo. For a challenge, here’s an extension: Solve the problem again, but this time after turning over a card, instead of dealing just one card to the bottom of the deck, deal two cards. Can you arrange the ten cards so they’ll come up in the correct order?
I’ve solved that problem before but I don’t have the solution. I’d have to work it out again. Let me know if you get stuck.
Oh, and the solution: 1-6-2-10-3-7-4-9-5-8.
This is such a tricky question. I offer up my approach for criticism, not as a model:
On my blog, I almost never give answers when I pose questions. I have started using spoiler buttons to let readers choose whether to see an answer or not. I’d not that spoiler buttons can be wonky to implement.
In other cases, it is context dependent. If one of my children (the biological ones that live at home with me) asked for a solution or an answer, I would almost never give them a solution. The exception is if they wanted help on an intermediate part of a problem/exploration and it was counterproductive to struggle with that particular step.
For my students, I would provide escalating suggestions for solution strategies. For example, in this card puzzle, I would suggest they try:
(1) use fewer cards (maybe A to 4) and see if they can arrange those to match the desired dealing order
(2) arrange the cards in order A to 10, then deal them out and record the order the cards appeared when they follow the deal. What does that information tell them?
(3) work backwards
If an adult asked me for an answer, I would probably just tell them and discuss the solution strategy with them. That’s especially the case if I could follow-up with: “here are some other, similar puzzles, you might like.”
In this case, take a look at this page Mathematical card tricks. This is another Peter Liljedahl resource. I promise, I’m not a paid plant by the Simon Fraser University Faculty of Education, just flagging some great, related, resources!
Thanks, Joshua for your comment, and for another link to Peter Liljedahl. My typical go-to when students are stuck on the 1-10 problem is the first one you mention — using fewer cards. Also, it’s pretty common for students to keep trying without ever writing down their ideas so they have no record of their success, even if partial. And then I respect that the problem isn’t for everyone. I tried it with our four grandsons at dinner one night. Two dug in and worked until they solved it, one observed the other two and later tried it, one found something else to do. I’ve learned at home not to push, but in the classroom, I’m more insistent.
I used the activity with my 5th grade STEM after school club as well as at a training with teachers. The students enjoyed working on the solution. What I like best is that when you get the solution you know it, there is not answer key. Great activity for them to encourage a “growth mindset” , persistence and patience. Sharing with teachers was fun as well. As with students some are interested, some not, and some really interested. I had one teacher who wanted to explore the why and how. The teachers expanded the problem to include more than 10 cards. One interesting strategy that I had no thought of was working backwards. Marilyn thanks for sharing.
Thanks, Marilyn, for NOT giving the solution–this was super fun! It took me every bit of that 20 minutes to figure this out, and I ended up working backwards and drawing lots of arrows. It took me a while to figure out the last bit of the pattern, even after I stumbled onto the solution by some trial and error.
Then–I did see it! I tested it out with 20 cards, red and black, and it worked. 🙂 Now thinking of how to present this to 5th graders so they are challenged but not frustrated…I like the “start with fewer cards” idea.
Please keep sharing! I just bought “42 Elephants” as well…
When some of my students have become frustrated at not arriving at the solution, I have found that there is at least some value in drawing out this strategy as shown below, starting with the vertical stack of cards and labeling each one that is placed under until it rises back to the top. It takes four sequential (left to right) vertical columns to finish the ten cards. A similar approach could be used to reveal solutions to variations on the challenge. This strategy discovers the solution without even using the cards. Surely, there must be something for the frustrated student to learn from this logical approach:
Stack of Cards
_____ 1
_____ under 6
_____ 2
_____ under under under 10
_____ 3
_____ under 7
_____ 4
_____ under under 9
_____ 5
_____ under 8
As a teacher of 30 years, I have read your books since the 90s and followed your recipes. However, when I was working on my master’s degree, I took statistics.
The professor would give us 5 problems for homework and tell us the answers. At first, I thought…hummmm
BUT, he realized something I had to learn….
The process of getting to the answer was the challenge. He challenged us to use the method in the book and then find another. I worked every evening for approx. 4 hours on these problems. Sometimes, it took me two days to find that solution, sometimes more, sometimes, NEVER! Did I become frustrated…YES ! YES ! And did I say…YES!
But, after about a month, I realized that I had become pretty good at alternative solutions and how my brain now thinks differently when approaching problems. I started this process with my 3rd graders….. AMAZING RESULTS!
Sooooo, again I am with you in that sometimes it is ok to give the answer because of the process.
Keep thinking Marilyn. We love you!
Veronica, I love your idea of the answer given and two ways to get it … what kinds of problems do you give your third graders that they can find two ways to solve?
I teach sixth and they have no perseverance, but I think this might help – give them the answer – they think they won’t have work so hard.
Thank you Marilyn, for the repeat of the card “trick”. They really liked that!!
Hi. I was delighted to learn of this problem. It inspired me to create a lesson for my programming students, starting with a spreadsheet to solve it intelligently and then creating a Python computer program to solve it with brute force, after trying 32,551 permutations. If anyone is interested, my video lesson is here.
I plan to pose this problem to one of my programming groups tomorrow. Thanks!
This seems like a great activity and I am going to try it tomorrow in my 6th grade class. You mentioned in your blog that this is not one of your top ten problems, but it is close. Can you tell us what you top ten problems are? I love your blog! Thank you!
I don’t really have a top-ten list, but the qualities that I look for in problems are their potential to help uncover some important mathematics for students and their accessibility to students with a variety of interests and abilities. Then I comb resources for ideas. Years ago I wrote a book titled 50 Problem-Solving Lessons which drew from ten years of Math Solutions newsletters (before blogging existed). Here’s a link to Acrobats, Grandmas, and Ivan, the first one in the book, still one of my favorites: Acrobats, Grandmas, and Ivan
Is this an appropriate problem for the second week of 4th grade. I love the idea but it seems that it’s a lot to ask from someone who just starting to get the basics of math as my daughter. I am not sure if it stimulating or counterproductive in her case as she has learning disabilities. Thoughts?
I’m not sure. What’s important is the understanding that students have had enough prerequisite learning so they can engage and tackle.