Emoji Democracy

Democracies can be attacked by altering the boundaries of voting regions. This has been done in many countries from Apartheid’s South Africa to Hong Kong under the Chinese Communist Party. Its English word is “gerrymandering.” In this puzzle, we are going to be the red emojis trying to subvert democracy.

Here we have failed because the majority (yellow happy faces) have won four regions and we (the red emojis) have only won one. Can we do better?

Before you try – we must first set some ground rules.

1) all regions must be rectangular.

2) each voter must be in exactly one rectangular region.

3) no region may contain twice (or more) as many votes as another region.

Here we failed again because we have won two and lost two regions. The last region is a tie. At least we didn’t lose! But we want to win! Try again!

This fails to get us a majority again. We have tied 3-3. This also makes it too obvious we are rigging the election because the smallest region is half the size of the biggest region (see rule 3.)

Take 20 seconds to try to destroy this election!

Here is a 1-2 loss…

Here is a 3-2 win.

Can we destroy this 5×5 election using fewer wizards? Try!

You can win…

This 4-3 win uses the fewest red wizards. Let’s try something bigger. I’m not sure if I’m using the fewest wizards, but that’s what I’m trying to do.

These 36 voters have 11 red emojis in their midst. Show how they can win the election.

There is a 5-4 victory!

Here are our results summarized in a table. Perhaps you would like to add a row for the percentage of red emojis that are needed to win. I’m not going to do it here. It just depends where your class is at. Do your students have a hypothesis? Can the fraction of evil wizards drop below 1/4? Can they drop below 1/5? Can they drop below 1/6? That’s just something to ponder.

Try destroying this 7×7 democratic vote.

Destroyed! We won 7-6 that time. I think that’s the fewest red emojis needed to win. (Rainer Bedrich showed that there are many ways to win with just 14 red emojis.)

Don’t think too long to destroy this 8×8 election.

Destroyed with 19 red emojis! Now destroy it with only 18 red emojis.

This arrangement of the 18 evil red emojis works. Destroy that election!

I tried 17, but could not manage to destroy the election with that few.

Our best results so far…

This was the best I could do for 9×9…

This 7-6 win made the red emojis crave more power. They decided to try 10×10… How many would they need?

With twenty-five red emojis the destruction was easy to find. Perhaps it could be done with 24, but I couldn’t find how.

8-7 victory!

I can’t easily go further…

If you do, I’d love to see the destruction you create. Send me an email: gord@mathpickle.com

Emoji Democracy

(MathPickle, 2020)

Emoji democracy is a perfect puzzle to get your students counting and thinking of the fraction one-half. This has the possibility of getting political, but gerrymandering has been used in many countries including in my own country, Canada. There is no need to get political unless you want to join together history and math classes 😉

Rainer Bedrich (below) has played with emoji democracy a lot! Look at his beautiful results here.

“Emoji democracy” used to be called “destroying democracy.” Here are two videos I made with this old theme shortly after the puzzle was created. I like the gentler new name. We need more light hearted puzzles even through the math might be hard! 

Rob Pratt has also explored the puzzle and has optimal results up to 12×12. See them here.  

Andrew Parkinson has written up an OEIS page on the problem. This will be where optimal solutions will be gathered going forward. Thank you to everyone who has found joy in playing with emoji democracy.

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

 
MP1 Toughen up!

Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP2 Think abstractly!

Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.

MP3 Work together!

Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. Try pairing up elementary students and getting older students to work in threes.

MP4 Model reality!

Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.

MP5 Use the right tools!

Students should use the right tools: 0-99 wall charts, graph paper, mathigon.org. etc.

MP6 Be precise!

Students learn to communicate using precise terminology. Students should not only use the precise terms of others but invent and rigorously define their own terms.

MP7 Be observant!

Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist! 😉

MP8 Be lazy!?!

Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?

(http://www.corestandards.org/Math/Practice/)

Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉 For a free poster of MathPickle's ideas on elementary math education go here.

Gordon Hamilton

(MMath, PhD)