Gerrymander!

Logic Level 1

This grid of voters (colored by party affiliation) has to be split such that:

There are 5 regions of 5 voters each
The regions have to be "continuous", in the sense that each square must share an edge with another square in its region
The Blue party gets to choose the regions

Can the Blue party win the election by winning a majority of the regions, despite only having $\dfrac{9}{25} = 36\%$ of the voters?

Yes, it's possible No, it's impossible

6 solutions

Michael Mendrin
Jun 28, 2016

Relevant wiki: Mathematics of Voting

Fun puzzle to work out one way to do it. Maybe I should go work for the Republicans.

Well, you know the old saying: "when the U.S. sneezes, Canada catches a cold." I've followed U.S. politics since a B-film actor beat a peanut farmer for President; Canadian politics is boring by comparison. Besides, I have one niece studying south of the border and another who might want to but won't pursue that option if Trump wins, so my interest is quite personal as well.

So I guess SCOTUS will hear that case in the Fall session. I suppose they could use the "If it walks like a duck and quacks like a duck..." test like they did with the Texas abortion bill case. If Justice Kennedy continues on his leftward trajectory the Republicans should be put in their place on this one as well.

Brian Charlesworth - 4 years, 11 months ago

Even if Clinton wins and Democrats control both Senate and House, and 3 more liberal justices are appointed to SCOTUS, it sill will not be easy to curb gerrymandering. What are we, if not mathematicians? We should develop and offer rational guidelines on how to curb gerrymandering that doesn't depend on the "If it walks like a duck..." rule. Do you want to start with a proposal? You know, this should be similar to the Fair division problem, which is already pretty well studied.

Michael Mendrin - 4 years, 11 months ago

It appears that there are some proposals already out there, but they could make the problem worse, in some cases. Perhaps, as is stated in the concluding paragraph, some consideration should be given to moving away from winner-take-all, geographic districting, but the question is whether or not there is the political will to do so. We have had cases of "creative" districting up here too but generally there aren't too many complaints; it's a far more pervasive problem in the U.S..

Brian Charlesworth - 4 years, 11 months ago

@Brian Charlesworth – It's more of a problem in US because 1) it's predominantly a 2-party country, and 2) it gives whoever is in majority control over state governments in census years too much power--and that sets up a potential feedback. That is the basic flaw. We need to consider "fair division" proposals instead of letting foxes manage the chicken coops. I think the more people understand what gerrymandering is and how it works, there will be pressure for reform. The upcoming SCOTUS review of the notoriously bad cases of gerrymandering in North Carolina should be very interesting.

Michael Mendrin - 4 years, 11 months ago

Although it's the Democrats that benefit from this particular "rigged system" :P Whatever it takes to help the Dems win and avoid a Trump-tastrophe, I'm all for it.

Brian Charlesworth - 4 years, 11 months ago

Brian---but aren't you a Canadian?

I hear that SCOTUS is about to have a hearing on gerrymandering in North Carolina---by Republicans, of course. It's a problem as old as the Republic, because any legal recourse against is fundamentally ill-defined. How can it be effectively prevented by any set of rules, other than to say, "gerrymandering shall be unlawful"? Maybe that's Phase II of this math problem.

I suspect that once Trump learns of the word "gerrymander", he'll use it in one of his twitters to insult somebody with it, without a clue as to what it means.

Michael Mendrin - 4 years, 11 months ago

That somebody must've been of Germanic descent.

Saya Suka - 6 months, 4 weeks ago

Hung Woei Neoh
Jun 30, 2016

Here's a third possible combination:

Sorry for the wriggly lines, it's hard to draw straight lines with Paint and a mouse

Not really, no. Use the line drawing tools with maximal thickness.

Jack Lam - 4 years, 10 months ago

This is the maximum thickness available. Paint in Windows XP and Paint in Windows 8 are totally different. I preferred the old Paint

Hung Woei Neoh - 4 years, 10 months ago

Gabe Smith
Jun 28, 2016

Here's one way, but there is at least one more -- can you find others?

Gabe, you need to fix your solution. One region has 6, another has 4. Move the line over.

Michael Mendrin - 4 years, 11 months ago

Thanks to both of you! Just a little drawing typo :)

Gabe Smith - 4 years, 11 months ago

Good , I see you corrected it now .But I am little troubled because , where is my comment in which anyway I made the observation that your drawing again is wrong and has to be corrected again sot say anyway ?

A A - 4 years, 11 months ago

Great question! A (really difficult) extension would be to find the number of arrangements of 9 Blue, 16 Red squares which can be gerrymandered so that the Blues can win. (Also for (B,R) = (10,15), (11,14), (12,13).)

Brian Charlesworth - 4 years, 11 months ago

That's a really interesting idea! I have some ideas for how to tackle that with a program, but that wouldn't help much for larger grids. Did you have any approach in mind?

Gabe Smith - 4 years, 11 months ago

What would make it so difficult ? You can start suppose from the some available possible configurations and derive the others anyway.

A A - 4 years, 11 months ago

You made a little mistake in your drawing. Section 589 has 6 squares the way you have draw it but can be easily corrected.

One way to check if there are more possible solutions is to consider , at least for this case since it is smaller which squares belong in the same section. Considering this , with a little bit of organizing the understanding you can come up with all possible solutions easily I think anyway.

A A - 4 years, 11 months ago

Hasmik Garyaka
Sep 22, 2018

Manuel Martins
Sep 15, 2018

Here is my solution. The problem should have been included the note for winner take all rule, though, which is implied but requires knowledge of the US voting system.

A A
Jun 28, 2016

I suppose firstly it is wise to check if theoretically (without any construction already) it can be done or not anyway. Therefore firstly observe that for the blue party to win they must have at least the minimum number of majority possible from the number of regions with the minimum number of majority of voters in those regions. For the case of the problem this means that the Blue party should have at least 3 regions where there are 3 blue voters. As such making some calculations , since the blue voters have 9/25 they could have that minimum number.

Secondly , after showing that theoretically is possible , considering the disposal of blue and red squares it should be verified if the regions can be constructed in a concrete way. One construction , which I found by considering it anyway , despite anyway the fact that I am not understanding completely the method of how to arrive at it and therefore grasp it's validity in principle to assure me it is or not the only possible solution , can be done but anyway I will let the readers find it just for fun anyway.

Yes, 9/25 is the minimum number of votes such that a winning gerrymander is even possible. But you haven't really shown how you could divide it up and satisfy things. In the following numbering, what are the 3 groups of 3 you think can go in the districts together, and how can you connect them?

Gabe Smith - 4 years, 11 months ago

AA, Please learn the meaning and way to use the word "anyway"

Tran Hieu - 4 years, 11 months ago

Does it bother you that much that I use the word "anyway" in an improper manner Tran Hieu ?

I suppose you still understand what I meant and it's not all that unpleasant to read it but I don't know if it's appropriate to discuss this here anyway.

A A - 4 years, 11 months ago

Ok , I will show my solution in the comments sections as it is now hidden. I didn't showed it because I want everyone think at it.

The section which I found are 612 , 589 , and 437. The 612 connected almost obviously with another of the red squares in downwards from 6.

437 are connected by making a square with 43 and the red squares downwards and then choosing 7 anyway.

589 are connected , say , by taking 5 and 9 in a line and then selecting 8 and this anyway is my entire solution.

A A - 4 years, 11 months ago

Gerrymander!

6 solutions

0 pending reports