League of Steiner Trees

Though Pokemon once again stood out as a prime example of what a Steiner Tree looks like in the video game world (more specifically Kanto and Johto’s regions), I already made a post about that, so instead I focused on yet another video game I have seemingly been addicted to these days- League of Legends.

The above picture shows a place known as the Summoner’s Rift, where “summoners” channel these magical crystals and summon powerful characters to do battle against one another on teams of 5 v 5. The different vertices on this map signify the positions (except for blue, that’s the shop.) These also signify which “lane you are in.

The goal of the game is to destroy the other team’s base, run down your opponents and in some cases be as rude as possible while doing so! This being said, most professional players of the game actually take into account timing objectives (slaying the dragon, killing towers, etc.) and must have each player rotate back and forth from shop to upgrade their items and from lane to lane, capturing objectives and fighting the enemy. For the sake of our class, I thought it would be cool to showcase some trees within the rift!

In any case, the “jungler” character, who traverses the jungle defeating monsters, often must put this into practice the most. During the first fifteen minutes of the game, one or two characters are matched up against one another to collect as much gold from each others’ minions as possible.

The jungler’s job is to plan out their routes efficiently to both capture various jungle monsters and help their teammates in lane defeat enemy characters and capture turrets. This is much harder than it sounds, League is like a game of chess; every move you make has its benefits and drawbacks.

I just thought it would be cool to showcase some trees within yet another one of my favorite video games! If you’re more interested in what the game really is, and had a hard time understanding my vague description, feel free to visit https://play.na.leagueoflegends.com/en_US or simply search the game on YouTube, tons of people stream it!

Source for Photo: http://greatamericanroadtrip.us/league-of-legends-maps/league-of-legends-maps-how-to-play-league-of-legends-beginners-guide-with-600-x-448/

More Doesn’t Always Mean More

In regards to the Four Color Theorem, I found it extremely interesting and, honestly, really cool that, in this case, adding more pieces to a puzzle doesn’t always mean that there will be more required to solve it. In fact, in most cases that we did in class, I found that the more pieces we added to the puzzle, the less colors we needed to put in. In the end, the best way to require the most colors was to think our way through the puzzles and put the pieces in certain arrangements, not adding in more.

Playing Around with the Four Color Theorem

First of all, I wanna give a quick shoutout to this dope snow storm that made me leave my house and go to Barnes and Noble just to have WiFi to post this. You rock!!!!!!

Anyways, the only thing I wanted to do today was play around with this four color theorem.  So… I researched some different aspects about it to widen my horizons.  We all know that the four color theorem is the idea that you can color in any reasonable map on a plane with just four distinct colors.  But how was this discovered and what happened in order for us to come to that conclusion?

In 1852, the theorem was brought up as a possible answer to this question of whether or not this was actually possible, but was not proven until over a CENTURY later in 1976 when it was the first major theorem to be proved using a computer.  How cool is that?

At this point, the people that were proving this theorem asked themselves if they could go any lower than four colors, so they tried three.  Many maps they experimented with showed success with this, since they started out just using maps of regions of the world… until they got to Europe.  If you look at the countries surrounding Austria, you’ll see that it is surrounded by a ring of seven countries, which clearly poses a problem because it’s not an even number, as shown in the picture below.

However, with the entire map of Europe, the four color theorem still holds its own.

In conclusion, my heart is still with the four color theorem.  Hope you all stay warm in this storm and hopefully not as butthurt as I am about having to leave my house!! If you’d like to read more about this theorem, please check out the link below:


Four Color Theorem

So you guys might remember the Four Color Theorem from class, which states that any planar map needs no more than four colors to be colored in so that no bordering regions are the same color. This got me thinking, does this really only work for a flat map, or can it work on something that is also 3D?

In this example, a 3D figure was completely colored in using only 4 colors, confirming the theory that it can, in fact, be done.

In this 3D object, you will notice that 2 of the blocks are not filled in (the ones that are a yellow color and look like they are caving in), but can easily be filled in with a green one on the top and a blue one on the bottom.

Other 3D objects can be colored with four colors if they are flattened first. This does not work for all objects though, as most maps on a torus require 7 colors.

Below, I have posted a video of Torus Earth, to show how the flattening process has created the map.

Here is a link to read more about the sphere:


And here is the link to the 3D four color theorem:




As you can see above, I posted a picture of the T system (Subway) in Boston. I grew up in Boston, so this has become very easy for me to navigate, but if you are not from the area it could be an absolute nightmare. There is “inbound” and “outbound” as well as different color lines, that will reach specific routes and destinations.

Until we learned about this in math, it never clicked as to why there are different color lines that run. Since the train actually can’t turn around, it has its own path ways and essentially goes back and fourth on the path, but crosses a mutual point with other colors. They all intertwine, but branch out to an end point because it really is a tree format. It does not create a circuit.

An example of a circuit in the real world that I haven’t thought of – until now, would be a rollercoaster. The tracks go up, down and in different directions depending on how it is set up, and then comes back around to meet, which allows the carts to go around and around. While riding the rollercoaster, you always stay going “forward”, but in the subway you would essentially be rolling “backwards” at points, to go back in a straight shot the way you came in. It doesn’t loop around because it doesn’t have a circuit.

It is very interesting how all the points made in class are actually applied to absolutely everything we do. I have enjoyed learning about it this far.

“Crowdsensing” and Trees

I have personally only experienced the hustle and bustle of the subway station a handful of times in my life. With this being said, it it quite apparent that the system is messy and often times chaotic.

When I heard about UrbanEngines, a company that uses math, specifically the trees we have beendiscussing this past week, to infer the real-time state of a transit system, I wanted to read more. It was founded by led former Googler Shiva Shivakumar and Balaji Prabhakar, a Stanford computer science professor.

Users of the subway simply enter and exit the system, and the company then produces a digital replica of the city’s transportation network.

By tapping at the station when a commuter arrives and tapping out upon exiting…

“An individual’s commuting history has limited information,” Prabhakar said. “But if you have the trips of everybody, you can now reconstruct.” It just takes some math: algorithms that can compare all those trips and figure out how long it took for the buses to run or the trains to arrive. It is no surprise that Prabhakar’s background is in traffic routing for computer networks. I’m going to give the algorithm category a name: we call it crowdsensing.

This system doesn’t rely on seriously complex data- they just need to know where people tapped in and out. The really interesting thing is, if they bundle enough of the trips together, they can piece together what’s happening in the system into a real-time visualizations with astute accuracy.

“Urban Engines’ work offers potentially revolutionary solutions for addressing the complex issue of commuter congestion through incentives and data­-driven insights,” said Shomik Mehndiratta, the World Bank’s Lead Transport Specialist for Latin America.

It’s a pretty fascinating way to simply solve such a monumental problem-congestion especially in big cities- and all with a couple of “taps”.

To learn more:



Dark Souls and Trees

Brad Wall


Blog Post

Today in class we learned about trees and how they are used in things like transportation. I spent most of the day today thinking about how I can relate this to my own experiences in a way that is different than others. I finally found what I needed. I took a break from homework and brainstorming today to play some video games with some friends. When I sat back down to continue this post I realized what I could talk about.

Recently my friends and I have been playing a video game called Dark Souls 3. This video game is known for being one of the hardest games to ever play. It takes place in a world where humans are starting to decay and turn evil. Essentially you are the last champion to undergo the task of reigniting the flame of life. If you do not darkness will take over. To reignite the flame you must kill all the Lords of Cinder. By killing them you gain the materials you need to spark the flame. There are four Lords of Cinder and they each reside in a different place on the map. Each lord is a boss and each boss is protected by a vast world of decaying monsters that are trying to kill you. To make this even harder of a task at any point in the game another real life human can come into your game and kill you with their champion and take your money.

Like most video game communities there is a subdivision that is devoted to beating the game as fast as possible. These people are known as speedrunners. This is where I see the first part of math used in this game.  The map that they use resembles the tree that we had discussed in class.

Each box is an area in the game and the arrows show how they connect.

This tree is seen in another aspect of the game as well though. Throughout the game, the only way to save your progress is to find a bonfire in the area that you are in. Some are obvious while others are not. The community has made this map to show the connections to each bonfire and when the player will be able to access them.

Overall the game does not have an actual map that displays the land and where the player needs to travel. In response, the community used the tree method to display this and have made it easier for people to find their way around the extremely dangerous map.

Traveling Salesman Problem vs. Star Wars

Wow look who’s finally remembering to post!

OK so if you couldn’t figure it out I love Star Wars and kinda detest math. However, since we could bring in Star Wars, I decided to better one thing that’s always bothered me in movies.

You know how characters zip all over the place in movies with no problems? No traffic, no stopping to refuel, no checking maps, no figuring out the best method to get from place to place?

I was inspired by the Car 54  problem and decided to create my own for y’all to solve. This map (granted, not from an official Star Wars site) shows all the trips taken in the various SW movies, which is super cool!

Me being the nerdy lady I am, I decided to start a hop on/hop off flight tour for the most interesting (in my humble opinion) places in the SW universe and find the quickest way to do that.

Here’s the map without lines: Here’s MY map with my tour locations indicated with red dots. Included (in no order) are: Hoth, Tattoine, Alderaan (with a yellow dot because RIP), Courasant, Naboo, Dagobah, Endor, and Mustafar.

Here’s the much simpler map with the points represented by their initial and indicated by pushpins:

Since a lightyear is approximately 5.88 trillion miles and Jupiter is 0.000082 of a lightyear away from earth, I’m going to put the average planet distance at 0.02 lightyears for the fictional SW map as their galaxy is far larger. (Ooo math!)

Here’s the map with the “weights” added:


  • Corrusant – Alderaan: .02
  • Alderaan – Tattooine: 1.05
  • Tattooine – Naboo: .04
  • Naboo – Dagobah: .06
  • Dagobah – Mustafar: .03
  • Mustafar – Hoth: .02
  • Hoth – Endor: .07
  • Endor – Corrusant: 1.15

This to me looks like the shortest Traveling Salesman distance – I’m wondering if anyone in the comments wants to try finding another way to travel? You have to begin at Corrusant, go to Alderaan, and then continue.

I actually had a lot of fun working on this – from a 90% creative, 10% logical mind like mine, I get to create a route and then find it’s efficiency.

Have fun playing around with this!

P.S. I thought this anecdote from Business Insider about the TSP was pretty good and another reminder of how much math there is: “As we can already see with these small numbers of cities, the number of paths grows extremely quickly as we add more cities. While it’s still easy to take a given path and find its length, the sheer number of possible paths makes our brute-force approach untenable. By the time we have 30 cities, the number of possible paths is about a 9 followed by 30 zeros. A computer that could check a trillion paths per second would take about 280 billion years to check every path, about 20 times the current age of the universe.”  http://www.businessinsider.com/p-vs-np-millennium-prize-problems-2014-9

Icosian Game

As we learned in class, the goal of a traveling salesman problem (TSP) is that a salesman should travel to every single city in an area, visiting each city only once. The salesman also needs to end up in the same city where he starts his journey.

We also learned about Hamilton’s path and circuit. Similar to Euler’s path, a Hamilton path is one that passes through each vertex exactly once. A Hamilton circuit is actually just a Hamilton path, but is a cycle.

In 1857, William Rowan Hamilton developed a mathematical game called the Icosian game.  The game’s object is finding a path along the edges of a dodecahedron such that every vertex is visited exactly once, and the ending point is the same as the starting point. Hamilton’s game was eventually turned into an actual board game. And, if we think about it, even the Icosian game is really just another variation on the traveling salesman problem! Just as the dodecahedron has 20 vertices, we could imagine Hamilton’s game as a map of 20 cities. Using this analogy, our traveling salesman would need to find a way through 20 cities, visiting each city once, and ending up at the starting point.

Solution To The Icosian Game in 2D and 3D:


Please check out the link below for more information:


Traveling Salesperson Problem

I never find much use in math where I learn something that I’m going to ever actually use again in my life, but this is something that I know I will quite frequently. I actually use it a lot in real life and have for a long time without knowing it. Whether it’s going to town to do errands I figure what is the best route to stop at all the places I need to go, in the grocery store what isles I should go to in what order for the things I need, and even when I’m getting ready for the day trying to figure out the quickest most efficient way to do everything in the least amount of time.

What I connected this week to was a trip I’m taking this year. It’s for a 4 day trip with a group of friends to go car racing. To get to Florida using American Airlines directly it would be about $380 dollars. To stop at just one place along the way from Boston to Virginia would be $231 and then Virginia to where I will be flying into would be closer to $130. Of course these prices can and obviously will change, but for now I plan to use the theorem from the TSP to gauge although it could be a little longer time wise that it’ll be more cost effective to make a stop somewhere along the way.

Now that we do know these I feel like I am going to surprisingly find myself using this way more often in real life, especially now that I know it’s a real thing and not just something unnamed that everyone uses but no one actually has a real name for it. I’m looking forward to finding out some more useful things from class that I will actually use in my life, not just learn for a test, regurgitate the information, and forget it days later to never use again.