To see part 1, click here.
The professor had spent some time examining collections of 10 reportedly random points on 10-by-10 grids, and noticed that these collections often times contained clusters, with a majority of the points lying in a considerably smaller rectangular subset of the large grid. He made no effort to share this with his class—rather, he suggested that we pick a Saturday, and as a class, spend some time rolling 10-side dice and populating as many of these 10-by-10 grids as we could, in exchange for some pizza and compensation a little bit better than minimum wage. He then posited, “and for anyone who can write a computer program to do it—I’ll pay you a lot more.”
The invitation rang as a silent bell to me, and for the next week or so, I thought nothing of it, except that I’d get free lunch and some spending money some Saturday in the near future. After all, I was two semesters away from graduation and had no desire to put forth any more than the minimum amount of effort required to escape the system that I had spent so much time in (perhaps a different story for a different day).
Within two weeks, two other students in the class, with whom I shared a statistics course during the class period immediately following College Geometry, were actively discussing and pursuing the apparent money-making computer program that would pay for their next party or Rubik’s cube event (note that I have no derogatory intent toward those students, and don’t necessarily think they would have used the money for either of those things). During one of these conversations my interest spiked, as I rudely eavesdropped on their discussion. The problem was a simple one: create random sets of points, feed those points into some computer program that would plot them, and hit print. And, if they could do it for one grid, they could do it for 1 million.
I have spent a decent amount of professional development time over the last 5 years becoming intimately familiar with Microsoft Excel—training seminars, help files, internet message boards—and it’s something that I use on a daily basis. My reaction to their conversation was simple as well (and I think I even said it out loud): “I think I can get Excel to do that.”
Operating independently of those two students, I enlisted the help of my brother. He also was a student at BW at the time, and I (not so randomly) had plans with him for dinner that day. So, over dinner, I introduced the problem to him and asked his input on how I might get Excel to create the desired output (it is noteworthy here to mention that, although my brother was not as well versed as I am in all things Microsoft Excel, he did have a good mind for thinking objectively through a problem and theorizing on how to solve it—and his unofficial expertise in that arena proved very useful).
His response was immediate: “Haven’t you done something like that before?” What he was referring to was the Super Bowl Square randomizer that I created for a coworker who regularly orchestrated fundraisers for his son’s travel baseball league. The coworker needed a quick and easy way to split a list of several hundred names randomly into (none other than) a series of 10-by-10 grids. And, while I don’t want to give away the secret too quickly, I will point out that the worksheet that served as our starting point did not in any way resemble the output which we eventually produced. In essence, our starting point was the random number generator built into Microsoft Excel, and it provided what turned into 1200 grids that I was able to submit to the professor for his perusal and analysis. I will devote some energy later into describing the process behind creating these grids.
Upon submitting the grids to the professor, I inquired about his interest in the grids and how he intended to use them. His response was not surprisingly cryptic: he thought he noticed a pattern that he wanted to explore further. And for a week, the conversation was over, as was my random encounter with The Grid (how I will from this point forward refer to the 10-by-10 grids each with 10 random points selected). But, curiosity (and a bit of arrogance, perhaps) got the best of me, and the following week I returned, thinking that I could again use Excel to analyze the data to his liking—if only I knew the pattern that he thought existed.
The base conjecture for The Grid is this: given ten randomly selected points on a 10-by-10 grid, with each coordinate ranging from zero to nine, fifty percent or more of the points would be contained in a rectangular area of twenty-five percent or less of the total area of the grid—and this would happen at least eighty percent of the time.
The analysis is seemingly simple, and something that I thought I would no doubt be able to replicate with Excel—but I would soon discover that my not-so-limited knowledge of the program did not leave me well-enough equipped to develop a program to analyze the grids I had created. And, after perhaps three weeks of tweaking, I sadly reported my findings (or lack thereof) to the professor, bruised ego and all. He countered with an offer to spend the next semester devoting a few hours each week studying The Grid—and randomness in general. I practically jumped at the opportunity. Another part of the semester would be devoted to a review of the book The Drunkard’s Walk: How Randomness Rules Our Lives by Leonard Mlodinow. To some degree, it was hoped that these two puzzle pieces would help prepare him to deliver a brief lecture on the perception of randomness to a section of the Annual Meeting and Conference of the National Council of Teachers of Mathematics in April 2012. His intended lecture would be a follow up and extension of his presentation of the 2010 Annual Meeting of NCTM.
More to come. Sorry for the lack of updates recently–work has been a bit crazy. Be well and stay tuned.
inspiredbyrandom 