The Super Mario Quadratics

-Shreya Doshi

As kids, we have all fascinated playing the game of Super Mario. It is the game where an Italian plumber, with an iconic moustache and a red cap with a bold “M”, jumps around the world, collecting coins and defeating monsters coming from sewers. However here’s something that the super Mario fans did not know about.

Completing a game of super Mario is a mathematical equivalent of solving complex mathematical calculations. Navigating the world’s most famous plumber-Mario through one level can require the same level of hard work as solving a complex equation of math.

The most important component of the game is to jump to avoid treacherous holes and collect the coins scattered across the world. But did you know, when Mario jumps or in fact anyone jumps or throws anything into the air on earth in its uniform gravitational force, the path the person or object follows is the shape of a parabola and hence can be described via a quadratic function. Through this article, I will show you how the game has actually been coded and how, in every jump you take, you make a parabola.

Parabola is a mirror – symmetrical plane curve which is approximately U shaped.

When Mario jumps, he isn’t just going straight up and down on the y axis but rather creates a parabola to jump up and to come back down. The game would look quite silly if Mario would beam up and then beam across on the x axis and then straight down on y axis.

So, keeping these things in mind, John Rowe, a teacher, made this game called Super Mario Quadratics on a free online graphing tool called Desmos. This game is pretty indistinguishable to the real Super Mario game, but instead of a joystick, the players would have to construct and pen down various quadratics to collect coins and stars in a series of levels. I think this would clearly show you how the real Super Mario game is also coded using different permutations and combinations of parabolas and quadratics.

In the earlier rounds, a pre-made quadratic is given and that has to be edited to help Mario reach his goal. But as you progress, it gets more complex. And in the end, the players have to form a quadratic function from scratch to win the game and help Mario save Princess Peach.

Now the game may seem very challenging. But it isn’t if you understand quadratic functions and what kind of parabolas they form.

A few simple rules of parabolas-

1. If you want to move the vertex of the parabola along the y-axis you would change the number you add or subtract from the values being squared (So in y = x² +1, the vertex of the parabola would be at the y coordinate of 1)

1. If you want to move the vertex of the parabola along the x-axis you would change the number being added or subtracted from the x before it’s squared (So in y = [x-(-1)]² the vertex of the parabola would be at the x coordinate of -1)

1. If you want to make the parabola thinner, you would increase the number that x is being multiplied by. If the number is negative, then the parabola would become inverse.

Let’s start off easy by looking at some examples of various levels:

The parabola must be in a position where Mario can jump over the coin. So as we know, if we increased the number being added after x is squared, then the jump would get bigger in the y-axis. The coin is at the (10, 6), so if we increased the +1 to a +6 then Mario would be successful in getting the coin as the parabola passes through the 6.

Now, let’s get into a more interesting one: level 6-

Here, you need to create 2 different quadratics to complete the level.

The 1st quadratic needs to get Mario to the 3 coins in the air and he needs to land safely onto the platform. This can be done by multiplying -0.4 in the beginning, making the parabola of perfect thickness, and it also makes sure Mario gets from one platform to the other safely. Then subtracting 0.7 from x before squaring it would mean that the vertex would be at 0.7 on the x coordinate which gives the perfect trajectory for Mario to collect all the coins. After squaring, the values are subtracted by 0.7, to have the vertex high enough to reach all the coins. The quadratic would be: y = -0.4*(x-0.7)² -0.7

The 2^nd quadratic needs to be quite thin, so we would multiply x by a small number (-3). The star is at (5.5,0) so before squaring x it would be subtracted by 5.5 so the vertex of the parabola is at 5.5 on the x-axis. The quadratic would be: y = -3(x-5.5)²-0

There’s many more levels on the website that you can try out, and see if you remember what you learned after reading this. I just showed one way of solving these levels, but you can write many more quadratics for just one level. So, have fun with parabolas and help Mario save Princess Peach!

If you wish to solve more such equations, just click on the link below:

https://teacher.desmos.com/activitybuilder/custom/5c7614041509d870d4838bfd#preview/1f34b3ad-6952-4bbe-a538-1d94aa760a87

References:

https://www.forbes.com/sites/quora/2016/10/21/this-is-the-math-behind-super-mario/#598121b62154

https://www.kidzsearch.com/kidztube/watch.php?vid=28822067a