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Twelve Envelopes Puzzle

I’m here today to share with you the Twelve Envelopes Puzzle.

twelve envelopes puzzle

I’ve finally reached a point in the summer where my brain has started thinking about ideas for the new school year. Actually, I’ve been thinking about next year since around February or March, but I’ve finally stopped *just* thinking about next year and started creating resources.

It’s been a number of months since I’ve had a chance to just sit down and create resources without any time pressure, and I’ve forgotten how rewarding the process can be.

Two days ago, I created a Google Classroom Poster. This past year was my first year in a 1:1 environment and my first year using Google Classroom. Every time I got a new student this past year or had a student switch from one period to another, I had to remember to go and look up the Google Classroom code so they could join the class to get assignments and announcements.

I can’t tell you how many times I had to drop everything and look up a code for a student at an inopportune time. This next year, I will be posting the codes on a poster so that I can point students to the poster whenever a classroom code is needed. 

Google Classroom Poster

Yesterday, I started working on my plans for the first day of school. Last year, I had my favorite first day ever with the Twos to Nines Challenges. I absolutely loved being able to engage my students in mathematical thinking and problem solving from Day 1. Listening to their conversations gave me great insight into my students’ attitudes toward math.

Twos to Nines Challenge

The activity went so well that I would be tempted to do the same activity again this year, but half of my classes will be made up of students who I taught this year. Next year, I will be teaching 3 sections of Trig/Pre-Calculus (students I taught this past year in Algebra 2) and 3 sections of Algebra 2 (students I have never taught before). This calls for a new first day of school math task.

Last week, I was feeling a bit bored from staying home so much this summer, so my husband and I spent the better part of a day visiting thrift stores as an excuse to get out of the house. At Goodwill, I picked up a new puzzle book, Giant Book of Hard-to-Solve Mind Puzzles, for the bargain price of $2.

It’s currently out of print, but you can still pick up a few used copies for relatively cheap on Amazon.

In this book, I ran across a puzzle involving twelve envelopes. I posed the problem to my husband, and we reasoned through it together. We arrived at a lovely, unique solution, and I decided that it would make the perfect puzzle for this year’s first day of school.

I appreciate the fact that this puzzle 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. 

Yesterday I typed up the twelve envelopes puzzle and sent out a quick tweet about it. I’ve been absolutely blown away by the number of likes and retweets, so I figured I should blog about the activity now instead of waiting until after school starts in August.

Tweet about finding twelve envelopes puzzle.
Twelve Envelopes Puzzle
Twelve Envelopes Puzzle
Twelve Envelopes Puzzle

Shaun and I ended up working through the twelve envelopes puzzle in the opposite way than the book’s original wording suggested, so I made a couple very slight changes to the wording of the puzzle to reflect this since we figured students would also benefit from taking this approach.

Basically, the book had the envelopes numbered 1 through 12 and the cards numbered 110-121. We reversed this.

The puzzle now reads: “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?”

I typed up the twelve envelopes puzzle, and Shaun taught me a new Publisher trick to add a white outline to my words to make them more readable over the image of the envelope in the background. I didn’t even know that was possible! 

Twelve Envelopes Puzzle

Then, I made 1-12 number cards to place inside each envelope.

Twelve Envelopes Puzzle

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.

Twelve Envelopes Puzzle

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.

Twelve Envelopes Puzzle

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.

Twelve Envelopes Puzzle

Similarly, 3 could be placed with 120. 3 could also be matched with 111, 114, or 117.

Twelve Envelopes Puzzle

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.

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.

divisibility rules printable chart pdf.

Digital Versions

Google Slides

Check out Cathy‘s Google Slides Version.

Twelve Envelopes Puzzle Google Slides.


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.

Twelve Envelopes Puzzle in Desmos.

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 with information about what you teach and where you teach. I will be happy to forward an answer key to you.


Friday 9th of July 2021

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"


Tuesday 14th of April 2020

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!

Sarah Carter (@mathequalslove)

Wednesday 24th of June 2020

Link to poster:


Tuesday 9th of July 2019

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.

Karen Wainright

Thursday 20th of June 2019

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?


Friday 14th of June 2019

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.

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