Mathematical Connections to Wrestling: Pt. 2
This will be the fifth and last installment in my "connections" series, because, frankly, I have run out of ideas after this.
Mathematics can be linked directly to wrestling, like I have shown before, as not only are there mechanical connections within wrestling itself, but other processes within the business are also heavily influenced by mathematics.
In this article, I will aim to consider and solve two frequently discussed issues within pro wresting and the WWE.
Landing in the middle of the ring
The mathematics behind this section is pretty sound, and although a bypass of the meaning of the key equation is omitted in the middle, it is a fairly accurate proof of my statement.
It is known by many in the wrestling world that the middle of the ring is the best place to land, but why?
The first concept that needs to be addressed in understanding this problem is how a wrestling ring works in general.
The way that a fall is “cushioned” is to increase the time that an opposing force is applied to a wrestler’s body. A given momentum has to be stopped by applying a specific total force, so if you apply it over a longer time the force at any given moment can be reduced.
Consider how a bungee rope works. If you were to jump off a bridge with a bungee cord attached you would begin to feel the rope pull at you from only a few seconds into the jump, right up until the moment you stopped.
No harm would be done as a small force would be applied over a long period of time.
Whereas if you had no ropes attached you would hit the ground. The same force could be applied by the ground to stop you, but it would all occur within a fraction of a second.
A large force would be applied to the body, and it is easy to imagine what the result would be.
So with that concept in mind, why does the middle of the ring provide a better cushion than the edge?
One mechanism that applies aptly in this instance is the workings of a spring.
A spring applies an opposing force on an object proportional to the distance of the displacement from its equilibrium position, and to an individual spring constant.
In vector form F = - k X (where underlining indicates a vector, F is the opposing force, k is the spring constant, and X is the displacement on the mat by the wrestler).
In this instance the spring constant in any given wrestling ring should be roughly equal across the entire mat.
As wrestlers only really apply force downwards when falling, I will be considering only the upward force produced by the mat.
This equation may look confusing, but all it is basically saying is that the bigger the force you apply to the mat, the harder it will push back.
The question I want to solve, using this equation, is while considering a specific displacement to the mat, where in the ring will the least force be felt?
The area that provides the smallest force for a given displacement will be also be the area that will allow the most time in contact with the mat. From my explanation above, this will be the best place to land in the ring.
I will consider movements of a distance “x” away from the centre of the ring for a given displacement downwards “a”, and try to determine if this increases or decreases the overall force applied.
After many hours spent doing calculations, re-ordering equations, and re-examining the problem several times, I came up with a solution.
Using the equation for a spring above, and some use of trigonometry, this is the formula I derived for the force experienced for a given displacement:
This equation is quite complex, and I don’t expect anyone to have an idea what it means, but I can show you graphically what happens.
The picture for my article shows how the force varies with x. The horizontal axis represents position away from the centre, and the vertical axis shows the force experienced.
If you find it difficult to see what is happening in the picture, then follow the instructions at the end of the article.
You can see that the force is minimal at 0, i.e. the centre of the ring, and the further out you get the higher it becomes.
The areas close to the centre provide roughly the same the centre itself, so landing close to the middle is relatively safe too.
It is only when you approach the edge of the ring that the force applied back to your body becomes larger, as you can see from the graph.
I hope the way in which I have shown this was understandable, regardless of the complexity of the equation. It was hard to go about due to the nature of the mathematics, but I hope you can all see the implications.
Although some assumptions have been made, this strictly proves that the best place for a wrestler to land is the middle of the ring.
Does the draft have limitations?
This section relies heavily on some key assumptions, which are not necessarily accurate. However, what they imply is still applicable, and the figures used will be representative of the true nature of the draft.
One of the comments that I frequently hear from the wrestling community is “I wish the WWE would just pick completely new rosters instead of the draft,” and as a mathematician I have often wondered what the limitations of the draft are compared to picking completely new rosters.
If new rosters were picked there would be no restrictions. As long as the numbers on each roster are kept vaguely the same anyone could essentially be put anywhere.
The difference between this and the draft is that during the draft certain wrestlers are picked to move from an already established roster into another. So the belief is that as there is only a certain number of draft picks, the new rosters will not be as altered as they could be if new ones were picked.
The problem is thus to determine how many different rosters can be composed without restriction, and to compare that with how many different rosters can be made by using draft picks on an already established set of rosters.
The method with the most different possibilities will be the best at “shaking things up.”
To analyze this problem I first need to make some assumptions.
First, I want to make it clear that I will only be analyzing the possibilities in terms of male wrestlers, just to make things easier. Similar results can be said of the divas so it makes little difference.
These moves must not take into account tag teams, stables, or other similar limitations. I am simply focusing on the mechanics of the draft.
In terms of numbers I will say that RAW and SmackDown each have 25 wrestlers and ECW has 12 (which isn’t too far from the truth). Also, let’s assume there will be 30 draft picks in total that involve male wrestlers, with each roster having the same proportion of their wrestlers drafted.
For those who didn’t read my last mathematics article, I discussed the “nCr” function, or “choose” function, which has uses in combinatorial counting. In this instance it will tell me how many distinctive ways I can pick a certain number of wrestlers from an individual roster.
I will first find how many different rosters are possible given no limitations.
To solve the problem I will first find how many different rosters can be made for the RAW brand, given a free choice of any WWE star. The number of ways of picking 25 wrestlers from the total group of 72 is 72C25 (roughly 1.52x10^19, or 152 quadrillion).
Then for each of these possibilities we must determine how many different SmackDown rosters can be picked. This is the number of ways to choose 25 wrestlers from the remaining 37. The numeric value of this is 37C25 (1,852,482,996, close to two billion).
The ECW roster will then automatically be decided, as there are only 12 left to pick from.
The order in which I pick wrestlers for each roster does not matter, due to the symmetric nature of the choose function.
The total number of ways is then the product of the numbers above (62C25 x 37C25), which yields roughly 2.73x10^26, which is just over 273 thousand quadrillion possibilities.
That’s quite the number to get your head round.
Onto the number of different rosters the draft can produce.
From the 30 drafts picks, I will assign 12 each to SmackDown and RAW, and six to ECW, as this fairly represents their roster sizes.
The number of unique ways of picking 12 wrestlers to be drafted from 25 on either SmackDown or RAW is 25C12 (5,200,300).
Similarly for ECW, we must determine how many ways there are of selecting six people from 12, which is 12C6 (924).
Once these wrestlers have been decided, each will move to either of the two other rosters.
As there are 30 men in total that will have this choice, the number of different unique ways in which they can all move is 2 to the power of 30 (1,073,741,824, around a billion).
Using the product of the figures above (25C12 x 25C12 x 12C6 x 2^30), I calculate that the number of different rosters possible after the draft is 2.68x10^25, which is roughly 27 thousand quadrillion.
Comparing the first and second numbers of possibilities (the first with no limitations, the second from the draft) we see that the draft limits the numbers of possible rosters by a factor of 10.
This means that for every possible roster after the draft, a further nine could be created given no limitations.
This certainly would seem to support the theory that the draft has limitations.
However, I do not believe this is a problem.
We have to remember that the WWE will always do what they feel is necessary for their business. If they want to move certain wrestlers to certain brands then they will do it.
This year’s draft saw an increase of eight draft picks on last year’s, which goes to show that the WWE are flexible in who and how many they move.
My reasoning also does not consider the fact that some of the roster possibilities in the first instance would not be possible. For example, a roster made up entirely of mid-carders would not be feasible.
The same could be said about the draft, but it will be to a much lesser extent. As there are already established rosters for each brand, it becomes less likely that all of one group of wrestlers (jobbers, mid-carders etc.) will end up on one show.
In a similar respect, my model also does not account for the fact that an equal number of main eventers/mid-carders/jobbers are moved to and from a brand. For example, if RAW lost John Cena in the draft they might receive Triple H.
Maybe the draft does have some limitations, but it’s not the be all and end all. Many wrestlers have moved brands outside the draft, so don’t despair if things didn’t go how you wanted following a draft.
All these aspects would mean that the draft could in theory “mix things up” just as easily as picking from scratch, as a more refined model could reduce the factor of 10 I deduced quite dramatically.
I hope this article has lain to rest any fallacies surrounding the issue.
Well, that is the final installment in my series.
I have had a blast writing it, as it has certainly taken me some places I never thought I would go, and discovered things about wrestling I can only imagine no-one has ever considered.
I implore you to read this article’s predecessors if you have not already. Specifically those regarding mathematics, a field in which I am most proficient.
I thank you all for reading, and I hope that you have enjoyed my tangential views of the world of professional wrestling.
If you follow the link below, and insert the formula that’s inside the quotation marks into the function box, along with the parameters I have given, you can view the graph used as my article picture for yourself: http://www.walterzorn.com/grapher/grapher_e.htm; “-2((sqrt(((10-x)^2)-0.01)-(10-x))*sin(atan(0.1/(10-x)))+(sqrt(((10+x)^2)-0.01)-(10+x))*sin(atan(0.1/(10+x)))).” Enter in the following parameters: x min = -10, x max = 10, y min = 0, y max = 0.01.
Thanks must go to that web site for providing such a useful tool.
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