Math Tricks, Negative Space, and Simple Beauty

Once again we start with two of my favorite things: soccer and math. I’ve talked about them both at length, for example in my “geometry in soccer” post from March 2009. Both are related by a similar underlying, structured framework. Both have rules, methods, and strategies for finding success, whether that’s solving a problem or winning a game.

What most non-players don’t understand is that despite the rules that govern both math and soccer, there are tricks to the game as well. These are the visions and insights that exist not within the simple rules and methods of an operation or a play, but rather in the negative space – the non-obvious space surrounding the operations and plays. You may find, more often than not, that recognizing these tricks in all aspects of life can provide the competing advantage necessary for happiness and success.

The soccer tricks will have to wait until after some knee surgery, so for now, I’ll stick with the math. There are thousands of known tricks in math, and probably an infinitesimal amount of unknown tricks waiting for an epiphany of recognition. Here’s an example:

Squaring Any Number Ending in “5”

Although this works for any number that ends in 5, it’s probably most practical for two digit numbers when no calculator is present. Let’s use 65 as an example, where we try to quickly compute 65 squared, or 65^2.

All you have to do is look at the number to the left of the “5” in the ones place. For our example, we have a “6”. Multiply this number by the number that follows it sequentially, which is “7” for our example. We get 6*7=42. To find our final answer of 65^2, all we have to do is take the result of our multiplication and append a “25” to the end of it, recognizing that the last two digits of the square of any number ending in “5” will always be “25”.

So for our example, we have “42” + “25” which gives us 4,225. The square of 65 is 4,225. Pretty neat, huh? Try it with some others…

25^2:     2*3=6,             “6” + “25”        = 625
95^2:     9*10=90,         “90” + “25”     = 9,025
475^2:   47*48=2,256, “2256” + “25” = 225,625

For a proof, I’ve looked to Dr. Math at the MathForum.org website. Here goes:

Let’s generalize a two-digit number ending in “5” by the representation X5, where X could be 1, 2, .., 8, or 9. Essentially, X5 is really a shorthand notation for the integer represented by

10*X + 5

Let’s go ahead and square X5:

(X5)^2 = (10*X + 5)^2 = (10*X + 5)*(10*X + 5) = 100*X^2 + 100*X + 25

Now factor our the 100 and an X from the first two terms:

= 100(X^2 + X) + 25 = 100*X*(X+1) + 25

Looking at this closely, you can see that this is exactly the product of X and the next sequential integer (X+1) with “25” appended to the end. Pretty cool, huh?

Notice that this trick works for squaring any integer that ends in “5”, not just two-digit numbers. Dr. Math shows us that for the the larger proof would have to be modified a bit (since all integers that end in “5” cannot be represented by 10*X + 5).

Seemingly Complex, But Beautifully Simple

Although the rules and structure of math may at times seem complex and chaotic, in the negative space of math we can find a beautiful simplicity through which things can fall in place. The same can be true for soccer, language, love, astronomy, cooking, and all aspects of life. Sometimes we’ve defined a framework (or have had it defined for us) of rules and methods to follow. But if we take a step back, look between the numbers and think outside the box, maybe we’ll find a simpler route to happiness and success.

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Visualizing the U.S. Men’s National Soccer Team Roster

As the World Cup approaches, countries begin to solidify their rosters, trying to optimize their squad to give the best chance of taking home the FIFA trophy. As the Bob Bradley recently announced the United States’ 30-man roster, we wonder where these players come from and how can their stats be visualized?

Obviously, it would be most valuable to visualize comprehensive stat sheets of the U.S. team players against all other teams and their players, especially the others in Group C (England, Algeria, Slovenia). Unfortunately, I don’t have that much time! So, elementary as these may be, here are some quick visualizations, given the data provided on the US Men’s National Team (USMNT) website.

1. Full 30-Man Roster, with Hometowns, Club Teams, and Total USMNT Goals (by Position)

2. Player Experience (Age vs Total Caps, by Position)

3. Player Size (Height vs Weight, by Position)

Some things to note, although I have not determined an international baseline from which these conclusions can be definitively made, is that our goalkeepers are old and our midfielders are relatively small, young and inexperienced. But I bet you didn’t need me to tell you that!

Regardless, the World Cup is surely a global spectacle and I’m very much looking forward to it. Hopefully the US squad can take Group C and show some true grit and determination on the international stage. Four weeks to go…

geometry in soccer

Soccer is a beautiful game of people and sciences: psychology (see post: mind-bend it like beckham), biology, and among others, mathematics. In particular, geometry can be found in almost all aspects of the game. To the standard viewer, these applications may not seem obvious. It probably looks like a bunch of people kicking a ball towards a common goal trying to keep another team from scoring on their own. That’s true, but it’s the science and language of movements that make one team beat another.

First, the simple applications of geometry and topology in soccer. The field is rectangular in shape with other shapes creating boundaries, halves, penalty boxes, corners, and goals. The field is geometrically symmetrical. The ball is semi-spherical adding it’s own element in shot curvature, reflections, etc. The shot itself must take on a very calculated velocity (angle + speed). Passing with a give-and-go, long ball, or off the wall indoors are all derivatives of geometric shapes. Using a wide-angled pass is less probable to be intercepted than a “square” pass. The wall aims to accomplish much of what a goalie does against a breakaway: take away the angle. Setting up a wall at 1o yrds or a goalie coming off a line reduce the target for a shooter. The cross must be a calculated trajectory, usually bending in or out to give more advantage to the goal-scoring team. Bending out allows for a more forward angle versus a completely perpendicular cross which is much tougher to head towards a desired target. These are some simple components of soccer in which geometry “takes shape”.

Let’s also think about team formation and spatial distribution around the field. Formation is not random, and it’s known that having the optimal formation at any one time greatly increases the chance of achieving an objective, whether that’s scoring a goal, playing stiff defense, maintaining possession, or slowing the game pace.

Connectedness: How many neighbor nodes does the person with the ball have? How passing options are there currently? The more interconnected the 11 nodes might be, the more optimized the network.

Compactness: The team should be like an accordion that can open and close, adapt and react, become dense and spread out. Localized compactness assists in defense, spreading out opens up for offense.

In passing, triangular formation creates multiple opportunities to give-and-go. Moving without the ball to create shapes builds the passing support network that creates scoring chances. Thinking ahead of reflected or return passes based on complementary angles off your teammate will get you the ball back in stride. Combine this geometric intuition with a good mental game is key.

In all, having a locally-connected space and visualizing complementary angles is advantageous for any team. The name of the game is adaptation and proaction. Thinking 3 passes ahead, using all sciences in your toolbox both mentally and physically, and anticipating the faults of your opponent will surely increase the probability of a win. Having a kick ass shot doesn’t hurt either.

mind-bend it like beckham

I’ve played soccer for pretty much my entire life. According to some quick calculations, I’ve probably spent almost 10,000 hours around soccer (practices, camps, indoors, outdoors, games, high school, middle school, intramurals, adult leagues, premier teams, district teams, travel teams, tournaments, refereeing, etc.). That’s equivalent to over 400 full days – over a year if I played all day/night (which sometimes I think I did).

Now considering I’m 24, that means about 1/24th of my life has been on the soccer field. To put that in perspective, I probably eat for 1.5 hours a day which would be 1/16th of my day. Therefore, eating and soccer combined have taken more than 1/10th of my life. Subtract sleeping, school, and work, and how much free time did I really have for a bat and turkey catching club (P.A.B.A.T.)?

The point is that I love the game. One must be physical but sensitive, dynamic yet passive, and logical yet imaginative in order to be a complete player. This balance of attributes while being prepared mentally will always make someone an asset to their team.

I think most of all it’s the pre-game mental preparation and in-game mind-reading that make it so much fun (and I’m talking not just of your opponents, but of your own players too). Studying the other team means identifying stand-out players or leaders and recognizing weaknesses in formation. Studying your team means knowing players’ strengths, weaknesses, and most of all, tendencies. You should be able to move as a cohesive unit and almost play blindly – the best teams I’ve been on have been where I can pass the ball without looking, knowing someone will be there who will know what I am expecting them to do with it.

While in the game, it’s about reading minds of players. Anticipation and probabilistic expectation play huge roles in gaining an advantage on the other team. A quick analysis of 2 connected passes should lead you to forecast subsequent moves based on the prior movements of nearby players.

It’s true, soccer is math, and really I mean it. Optimize your position and forecast movements based on prior states – all from looking at someone else’s eyes, hearing their communication, and most of all, intuition based on thousands of hours of experience. It’s pretty simple.

“The scoreboard never lies but it rarely tells the whole truth.”