Twelve Envelopes Puzzle
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Put your knowledge of divisibility rules to the test with this fun twelve envelopes puzzle.
Every year, I try to change things up on the first day of school. This new puzzle is my attempt to engage students in mathematical thinking and problem solving on Day 1. Listening to their conversations gave me great insight into my students’ attitudes toward math.
Last year, we tackled the Twos to Nines Challenges on Day 1, and it was one of my most memorable first days ever.
It went so well that I would be tempted to do the same activity again, but half of this year’s classes will be made up of students I taught last year.
This year’s activity is especially delightful in the fact that it is solveable with only basic arithmetic.
It’s an interesting puzzle but not so complicated that it should take our entire 50 minute class period.
After all, I still need to leave myself time on the first day to take roll, go over basic syllabus information (particularly the supplies that students will need to purchase), answer student questions, and take care of the sort of housekeeping items that always seem to pop up during the first week of school.
Instructions
I have twelve envelopes numbered 110 through 121 and twelve cards numbered 1 through 12. Can you place one card inside each envelope so that the number on the envelope is divisible by the number in the envelope?
Printing and Prepping the Materials
I typed up the puzzle instructions on top of an image of an envelope. Shout-out to my husband for teaching me a new trick to add a white outline to my words to make them more readable over the background image. I didn’t even know that was possible!
Next, I prepared 1-12 number cards to place inside each envelope.
I actually considered using actual envelopes for this activity, but I ended up deciding against that since having to open every single envelope to check students’ work would be a hassle.
Ultimately, I decided to find a clip art image of an envelope and created my “envelopes” that way. This way, students could just lay the card on top of the “envelope” to make for easy checking.
Originally, I had planned on cutting apart the twelve envelopes, but then I realized that I could just leave them connected and laminate them that way.
This means the activity has fewer pieces to be cleaned up. Plus, it should make the checking process simpler since the envelopes will not be in a random order.
Completing the Activity
Students must match each card with an envelope that is evenly divisible by that card. For example, the 5 card could be matched with the 115 envelope since 115 is divisible by 5.
But, the 5 card could also be matched with 110 or 120. Students will need to use mathematical reasoning and the process of elimination to figure out exactly where to place the 5.
Similarly, 3 could be placed with 120. 3 could also be matched with 111, 114, or 117.
I haven’t used this twelve envelopes puzzle with students yet, so I have no idea how long it will take. I plan on making a set of cards/envelopes for each table group on the first day.
I will print each set of cards/instructions on a different color of paper to make it easier to return lost pieces to their home. It can’t just be my students who seem to lose the single random card to the floor without realizing it, right?
I’m planning on using the twelve envelopes puzzle as a first day of school activity at the high school level, but it could easily be used during a unit focusing on divisibility rules for younger grade levels.
I will be restricting my students from any calculator or technology usage during this activity. It is my hope that this will force students to use mathematical reasoning throughout the activity instead of simply plugging numbers in a calculator or attempting to google for an answer.
Original Source of Puzzle
This puzzle was originally published in the Giant Book of Hard-to-Solve Mind Puzzles which I was able to pick up for the bargain price of $2 at a local Goodwill.
Unfortunately, it’s currently out of print, but you can still pick up a few used copies for relatively cheap on Amazon.
Note: I did slightly modify the puzzle from the book to make the puzzle better fit my classroom. The book had the envelopes numbered 1 through 12 and the cards numbered 110-121. I chose to reverse this.
Another Divisibility Activity
Want another fun divisibility activity? Check out my divisibility puzzle!
Divisibility Rules Chart
Do your students need a reference chart to help them with their divisibility rules? I created a free printable divisibility rules chart for students to use.
Free Download of Twelve Envelopes Puzzle
Twelve Envelopes Puzzle (PDF) (4977 downloads )
Twelve Envelopes Puzzle (Editable Publisher File ZIP) (1662 downloads )
Digital Versions
Google Slides
Check out Cathy‘s Google Slides Version.
Desmos
Scott McEwen shared a Desmos version of the 12 Envelopes Puzzle with me as well. It’s modified from a version created by Nathaniel Highstein.
Puzzle Solutions
I intentionally do not make answers to the printable math puzzles I share on my blog available online because I strive to provide learning experiences for my students that are non-google-able. I would like other teachers to be able to use these puzzles in their classrooms as well without the solutions being easily found on the Internet.
However, I do recognize that us teachers are busy people and sometimes need to quickly reference an answer key to see if a student has solved a puzzle correctly or to see if they have interpreted the instructions properly.
If you are a teacher who is using these puzzles in your classroom, please send me an email at sarah@mathequalslove.net with information about what you teach and where you teach. I will be happy to forward an answer key to you.
Not a teacher? Go ahead and send me an email as well. Just let me know what you are using the puzzles for. I am continually in awe of how many people are using these puzzles with scouting groups, with senior adults battling dementia, as fun activities in their workplace, or as a birthday party escape room.
Thank you very much Sarah! I was pondering what new activity to use…so lovely to check a box knowing the activity will be a success!
This activity sounds awesome! I'm going to try it out the first week. Thank you so much for sharing your great ideas! I had the same problem my first year with Google classroom so the next year I had my students write it on the 1st page of their notebooks with some other class info. When a new student came into class I had them copy that info from their partner. It had them talking to their partner right away and I had to spend less time explaining some details of our classroom.
I love your Google Classroom poster! I have the same problem with students transferring into my class, and I usually forget to have them join my classroom until it is time to work in it. Then I have to look up the code, etc. What did you use to create your poster? How large will it be?
Thank you for sharing this activity! I am excited to use this on the first day of school. I will also be teaching the same students I taught last year and needed something new. I love the discourse they will have working this out.
I love your blog! Great resources! Is there a link to the google classroom code poster? I would love to use it if so. Thank you!
Link to poster: https://mathequalslove.net/google-classroom-poster/
Awesome stuff for first day at school. All the best !
Also there is a App in Google play store called Envelope Puzzle. It is based on the Two Envelope Paradox. Please give it a try and it would be fun to discuss the Two Envelope Paradox and possible solutions.
Google Play Store -> Search -> “Envelope Puzzle”