Destroying 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 evil red wizards trying to subvert democracy.

Here we have failed because the majority (yellow happy faces) have won four regions and we (the evil red wizards) 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 5x5 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 evil wizards 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 evil wizards 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 7x7 democratic vote.

Destroyed! We won 7-6 that time. I think that's the fewest red wizards needed to win.

Don't think too long to destroy this 8x8 election.

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

This arrangement of the 18 evil red wizards 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 9x9...

This 7-6 win made the evil wizards crave more power. They decided to try 10x10... How many would they need?

With twenty-five evil red wizards 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

Destroying 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 evil red wizards trying to subvert democracy.

Here we have failed because the majority (yellow happy faces) have won four regions and we (the evil red wizards) 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 5x5 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 evil wizards 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 evil wizards 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 7x7 democratic vote.

Destroyed! We won 7-6 that time. I think that's the fewest red wizards needed to win.

Don't think too long to destroy this 8x8 election.

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

This arrangement of the 18 evil red wizards 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 9x9...

This 7-6 win made the evil wizards crave more power. They decided to try 10x10... How many would they need?

With twenty-five evil red wizards 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

Fractured Fractions

(MathPickle, 2012)
Grade 5 students tackle Fractured Fractions. Excellent engagement of the full spectrum of student ability is typical with beautiful and interesting curricular puzzles.

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!

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

MP3 Work together!

This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

 
MP6 Be precise!

This is where our 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!

One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist.

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

Please use MathPickle in your classrooms. If you have improvements to make, please contact us. We'll give you credit 😉

Gordon Hamilton

(MMath, PhD)

 

Lora Saarnio

(CEO)