Twelve Envelopes Puzzle

I’m here today to share with you the 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. 

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.

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.

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! 

Then, I made 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.

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.

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.

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.