Before our myths of democracy were inked – Cleisthenes tossed and turned in bed: “All Men Are Created Equal?” “One Man, One Vote?” “Virtuous Man, Multiple Votes!” “That’s it!” he thought. And here is how it would work in practice. Let’s say there are five men ranging from a debauched character (virtue 1) through to a man of exemplary character (virtue 5). The five men cast one vote for each level of virtue they have.

An election was called. Each of the five men need to vote for either Yellow or Orange. After the votes were counted three things were evident:

  1. The yellow party won more votes.
  2. The yellow party was voted for by a majority of the men.
  3. If any of the men voting Yellow were to change their allegiance to Orange – then Orange would have won.

Here is one solution. Man 1, 3 and 4 vote Yellow. Find the other two solutions.

Here is one of the other solutions: Man 2, 3 and 4 vote Yellow. Yellow is getting 9 votes – Orange is getting 6.

Here is the other solution.

Before going further maybe take the time to tell a bit about Athenian Democracy being only for men. The suffragette movement was millenia in the future…

Try to solve this same problem with 1-6 and 1-7 and 1-8 men. One of these is impossible to solve. Use the same constraints…

  • The yellow party won more votes.
  • The yellow party was voted for by a majority of the men.
  • If any of the men voting Yellow were to change their allegiance to Orange – then Orange would have won.

Spoiler alert… try these now before progressing…

 

There is a unique solution to 1-6.

 

There is a unique solution to 1-7, but this is not it.

  1. The yellow party won more votes.
  2. The yellow party was voted for by a majority of the men.
  3. If any of the men voting Yellow were to change their allegiance to Orange – then Orange would have won.

Notice that 1) Yellow is winning, 2) there are 4 yellow-voting men (more than the number of orange-voting men.) Unfortunately, if you transfer the 2 from Yellow to Orange, The Orange team does not win (it would be a tie.)

There is no solution to 1-8 under these constraints:

  1. The yellow party won more votes.
  2. The yellow party was voted for by a majority of the men.
  3. If any of the men voting Yellow were to change their allegiance to Orange – then Orange would have won.

Why is this so hard? Your students might want to conjecture why and try out their prediction on 1-3, 1-4, 1-9, and 1-10. They also might want to change exactly one of the above constraints.

Have a three color election with men of  virtuous level 1-12 under these constraints:

  1. The Yellow party won more votes. The Orange party got the second most. The violet party got the least votes.
  2. The Yellow party was voted for by more men than the Orange party. The Orange party was voted for by more men than the Violet Party.
  3. If any of the men voting Yellow were to change their allegiance to the Violet party – then Violet would have won.

Is this possible. Yes. But let’s just write these constraints in short hand to make them easier to see…

Here are the constraints as you might write them on the white board:

  1. #Votes   Yellow > Orange > Violet
  2. #Voters  Yellow > Orange > Violet
  3. Any Yellow -> Violet         Violet wins!

Here is one solution. I have not checked if there are others. Notice – it satisfies all the constraints:

  1. #Votes   Yellow > Orange > Violet
  2. #Voters  Yellow > Orange > Violet
  3. Any Yellow -> Violet         Violet wins!

 

1-14 failed with the following constraints:

  1. #Votes   Yellow > Orange > Violet > Indigo
  2. #Voters  Yellow > Orange > Violet > Indigo
  3. Any Yellow -> Indigo         Indigo wins!

Does 1-13 work? 1-15? 1-16? 1-17? 1-18? 1-19? 1-20?

1-15 failed with the following constraints:

  1. #Votes   Yellow > Orange > Violet > Indigo
  2. #Voters  Yellow > Orange > Violet > Indigo
  3. Any Yellow -> Indigo         Indigo wins!

Does 1-16? 1-17? 1-18? 1-19? 1-20?

1-16 failed with the following constraints:

  1. #Votes   Yellow > Orange > Violet > Indigo
  2. #Voters  Yellow > Orange > Violet > Indigo
  3. Any Yellow -> Indigo         Indigo wins!

Does 1-17? 1-18? 1-19? 1-20?

1-18 worked with the following constraints:

  1. #Votes   Yellow > Orange > Violet > Indigo
  2. #Voters  Yellow > Orange > Violet > Indigo
  3. Any Yellow -> Indigo         Indigo wins!

 

Let’s change some constraints and add a geometric one to 1-19.

  1. #Votes   Yellow > Orange = Violet
  2. #Voters  Yellow > Orange = Violet
  3. Any Yellow -> Violet         Violet wins!
  4. All Yellow are connected. All Orange are connected. All Violet are connected.

 

This satisfies the constraints. I don’t know if the answer is unique.

  1. #Votes   Yellow > Orange = Violet
  2. #Voters  Yellow > Orange = Violet
  3. Any Yellow -> Violet         Violet wins!
  4. All Yellow are connected. All Orange are connected. All Violet are connected.

 

Students should create their own constraints. If they are too harsh there will be no solution. Here is one alternative set of constraints for 1-19:

  1. #Votes   Yellow > Orange > Violet
  2. #Voters  Orange > Yellow > Violet
  3. Any Yellow -> Violet         Violet wins!
  4. All Yellow are connected. All Orange are connected. All Violet are connected.
  5. Orange and Violet do not touch.

 

A solution to 1-19:

  1. #Votes   Yellow > Orange > Violet
  2. #Voters  Orange > Yellow > Violet
  3. Any Yellow -> Violet         Violet wins!
  4. All Yellow are connected. All Orange are connected. All Violet are connected.
  5. Orange and Violet do not touch.

 

Virtuous Democracy

(Gordon Hamilton, 2005)

Many top quality creative mathematics classrooms get all students to make conjectures about their mathematics explorations. True conjectures are easy to get excited about, but the mark of a strong mathematics classroom is that all students feel comfortable enough to make a false conjectures. Alison Hansel’s (twitter: @ms_hansel) class is particularly excellent.

We can think of conjectures as commentary on the output of a puzzle. The complement of conjectures are constraints. Sometimes students should explore adding new constraints to existing puzzles. If the constraints are too harsh, the puzzle may have no solution. That’s good for students to discover.

Not all constraints are created equal. Creative students should try to develop an aesthetic sense about what makes a beautiful constraint. There is no strong rule for this, but I’ll give you two examples and let you draw your own conclusions…

Example of a constraint which sounds ugly: “Numbers 2, 13, 14 and 17 must be Yellow.”

Example of a constraint which sounds beautiful: “All the yellow numbers must be consecutive.”

I invite you to explore virtuous democracy – both through making conjectures and adding your own constraints.

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. MathPickle recommends pairing up students for all its puzzles.

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 Know the tools.

Students master the tools at their fingertips - whether it's a pencil or an online app. 

MP6 Be precise!

Students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to 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)