Solving murder mysteries have police making connections between underworld characters. Strings are pinned up between existing suspects. New subjects are pinned up.  It all looks like a big mess. But never fear! We are here to bring order to this chaos! Hopefully, our ideas can turn the jumbled mess of these crime boards to something that’s quick-to-understand.

There is no one right way to bring order to this chaos, so this “puzzle” is as much “art” as “science.” For starters, we’ve focused just on the faces…

How could you make the connections clearer? Think about the overlapping lines. 

We’ve untangled the mess of interconnecting connector lines. From now on – no lines can cross over one another. Like it? It definitely makes it easier to figure out what’s going on at a glance. 

Instead of the lines being pinned anywhere on each criminal’s photo… let’s standardize…

You might not like it, but instead of connecting a peripheral part of the photo – we’re going to connect the centers… Let’s keep going. Let’s standardize the size and shape of the photos. What shape to choose?

Let’s make them all circles. Ideas for further improvement? Discuss the lines. Are those red lines good?

I don’t like the red lines. Let’s make the lines a more neutral color and put them to the back so they don’t cover the faces. Keep going! Discuss how to add order to this chaos by considering the length of each line.

Let’s standardize the lengths. Instead of just any length – let’s make the length equal to the radius of the circular criminal photos. How else can we standardize these pin boards? Consider the angles.

All angles around each criminal must be equal…

72 degrees for the clump of connected criminals on the right…

180 degrees for the connectors around this criminal…

All angles around each criminal are equal.

Follow the same rules to untangle this mess. Don’t worry about drawing the criminals.

Go back to the previous slide if you doubt that this is correct…

Untangle this mess…

Go back to the previous slide if you doubt that this is correct… you can see how untangling can provide a much clearer perspective of a criminal organization.

Untangle this group of criminals…

We have run into a problem. The set of rules we have developed does not work for this network of crime. Obviously, you cannot allow one photo to be pasted on top of another. Discuss how to solve this problem before going to the next slide.

One way to solve the problem is to increase the edge length until the circles do not overlap. Here the edge length has been increased to 10r (ten times a radius.) 

Maybe you have a different solution that is less wasteful of bulletin board space.

This is another possibility… instead of keeping all the edge lengths the same – you allow some to be length 2r. 

Untangle this web of villainy…

That was easy.

When edge length = r there is no overlap, but somehow the close proximity of two pairs of circles makes it appear as they are connected.

Design a rule that fixes this in 30 seconds…

We can increase edge length… but how far to go?

Let’s say that two criminal circles which are not connected must be at a distance greater than r away from each other. 3r is the minimum length which solves this.

PS. There is nothing correct or incorrect about our choices. You could choose an entirely different set of rules.

Our new rule also works here. All unconnected circles are distance >r away from each other.

Untangle this…

Not looking good at edge length r.

Even at edge length 3r this doesn’t work.

At edge length 4r the circles are separated but they still are closer

It is not until edge length 5r that the unconnected circles are all separated by length >r. This solution wastes a lot of pin board space. Use multiple edge lengths to find a more efficient solution. I’ll leave that with you.

For the rest of this challenge, we will only consider crime networks with no loops.

Is there such a crime network with no loops that is impossible to pin up using only one edge length. If so – find an example with fewest criminals.

This was my first attempt. Untangle it.

Is it possible to fix this with a single edge length?

No. 

So what do you think of this solution? Well before we fix the edge lengths you’ll notice the beautiful symmetry of the solution. Let’s always try to optimize symmetry as this structure makes the network easier to understand in a glance.

But this arrangement only has two lines of symmetry. Let’s make this a new rule: we will always optimize symmetry. We might want to define exactly what that means in the future, but here it’s obvious…

This arrangement is better because it has more symmetry.

Now we need to fix the edge lengths…

Done – the edge lengths are r and 4r.

This was my first attempt at finding the smallest number of criminals (17 here) which force a network to adopt more than one edge length. We can get fewer without doing much thinking…

This solution has only 14 criminals. There is no way to solve it using one edge length.

Untangle this…

With one edge length, it doesn’t work – we have two pairs of criminals stuck right on top of one another.

This presentation of the network looks good, but it does not optimize the symmetry.

This presentation optimizes symmetry. Does it preserve space as well as possible? Can you come up with a solution that maintains this symmetry, but uses less space?

This is more compact. So our rules are:

1) Equal angles around each criminal circle.
2) Edge lengths are all multiples of the criminal circle radius.
3a) As compact as possible (option A)
3b) Use as few different edge lengths as possible (option B)
4) The solution’s symmetry must be optimized.

But I’m not happy about that last solution. It seems to me that this less compact form reveals the structure better. Why? Because there is more small-scale symmetry in this solution. That’s difficult to define… but you might want to try 😉

Find a network with the fewest number of criminals that cannot be represented with less than three edge lengths.

This was my first attempt at finding a criminal network that requires more than two edge lengths. Of course, we can remove some criminals, but is there a bigger problem?
Yes – there is a bigger problem. This network only requires two edge lengths!
Try again!
These 161 criminals do require three different edge lengths, but we can get rid of many of them…
The 161 criminals have dropped to 122. Let’s drop it further…
The 161 criminals have dropped to 122 which have dropped to 110. That’s the best I could do.
The discussion about the best way to present a criminal network has a long way to go. We have not even begun to explore networks with loops. We have not even begun to consider that some criminal networks may be more readily understood by presenting the criminal circles in different sizes. Next time 😉

Pin Board Murders

(MathPickle, 2019)

This is a discussion rich puzzle in which students are trying to bring order to the chaos of those pin boards that detectives and obsessed undercover sleuths are constantly using in the movies. I’m sure it has a basis in reality 😉

Gord!

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)