Original image

The IRS's Favorite Mathematical Law

Original image

When it comes to catching tax cheats, the IRS has more than just federal law on its side. The agency’s arsenal also includes a mathematical truth known as Benford’s law. Armed with this law, the IRS can sniff out falsified returns just by looking at the first digit of numbers on taxpayers’ forms.

While most Americans wouldn’t put it past the IRS to use black magic, the truth behind Benford’s law is far from mystical. In 1938, GE physicist Frank Benford undertook a comprehensive study of numbers and how they occur. His findings mirrored the discovery of American astronomer Simon Newcomb, who had undertaken similar research in 1881. Benford found that when it came to naturally or socially generated data, the distribution of the first digit in a series of numbers is not uniform.

In analyzing 20,000 sets of numbers from a variety of sources—numbers from a newspaper, population figures, American League baseball stats—Benford found that a whopping 30 percent of the numbers in his sample had one as their first digit. The numeral two turned up in the first position 18 percent of the time, and three occurred 12 percent.

There’s a simple explanation for what Benford observed. In the number set 0 to 99, 11 percent of the numbers start with 1, and 11 percent start with each digit from 2 to 9. In the number set 0 to 199, over half of the numbers start with 1, and less than 6 percent start with 2 to 9. In the number set 0 to 299, 37 percent start with 1 and 37 percent start with 2, and the numbers 3 through 9 start 3.7 percent each. This situation goes on forever, so over a large enough data set, the distribution of leading digits follows a predictable pattern. The bigger the integer, the less likely it is to be the first digit in a data set.

How does the IRS use this distribution? Many folks who happen to fudge a bit on their tax returns or expense reports call attention to their creativity by using too many dollar amounts that start with an eight or nine (the least common integers found in the first position) and not enough that start with numeral one. Savvy CPAs know what to look for, and many computer systems that tabulate figures are also programmed to catch any suspicious strings of numbers.

Original image
iStock // Ekaterina Minaeva
Man Buys Two Metric Tons of LEGO Bricks; Sorts Them Via Machine Learning
May 21, 2017
Original image
iStock // Ekaterina Minaeva

Jacques Mattheij made a small, but awesome, mistake. He went on eBay one evening and bid on a bunch of bulk LEGO brick auctions, then went to sleep. Upon waking, he discovered that he was the high bidder on many, and was now the proud owner of two tons of LEGO bricks. (This is about 4400 pounds.) He wrote, "[L]esson 1: if you win almost all bids you are bidding too high."

Mattheij had noticed that bulk, unsorted bricks sell for something like €10/kilogram, whereas sets are roughly €40/kg and rare parts go for up to €100/kg. Much of the value of the bricks is in their sorting. If he could reduce the entropy of these bins of unsorted bricks, he could make a tidy profit. While many people do this work by hand, the problem is enormous—just the kind of challenge for a computer. Mattheij writes:

There are 38000+ shapes and there are 100+ possible shades of color (you can roughly tell how old someone is by asking them what lego colors they remember from their youth).

In the following months, Mattheij built a proof-of-concept sorting system using, of course, LEGO. He broke the problem down into a series of sub-problems (including "feeding LEGO reliably from a hopper is surprisingly hard," one of those facts of nature that will stymie even the best system design). After tinkering with the prototype at length, he expanded the system to a surprisingly complex system of conveyer belts (powered by a home treadmill), various pieces of cabinetry, and "copious quantities of crazy glue."

Here's a video showing the current system running at low speed:

The key part of the system was running the bricks past a camera paired with a computer running a neural net-based image classifier. That allows the computer (when sufficiently trained on brick images) to recognize bricks and thus categorize them by color, shape, or other parameters. Remember that as bricks pass by, they can be in any orientation, can be dirty, can even be stuck to other pieces. So having a flexible software system is key to recognizing—in a fraction of a second—what a given brick is, in order to sort it out. When a match is found, a jet of compressed air pops the piece off the conveyer belt and into a waiting bin.

After much experimentation, Mattheij rewrote the software (several times in fact) to accomplish a variety of basic tasks. At its core, the system takes images from a webcam and feeds them to a neural network to do the classification. Of course, the neural net needs to be "trained" by showing it lots of images, and telling it what those images represent. Mattheij's breakthrough was allowing the machine to effectively train itself, with guidance: Running pieces through allows the system to take its own photos, make a guess, and build on that guess. As long as Mattheij corrects the incorrect guesses, he ends up with a decent (and self-reinforcing) corpus of training data. As the machine continues running, it can rack up more training, allowing it to recognize a broad variety of pieces on the fly.

Here's another video, focusing on how the pieces move on conveyer belts (running at slow speed so puny humans can follow). You can also see the air jets in action:

In an email interview, Mattheij told Mental Floss that the system currently sorts LEGO bricks into more than 50 categories. It can also be run in a color-sorting mode to bin the parts across 12 color groups. (Thus at present you'd likely do a two-pass sort on the bricks: once for shape, then a separate pass for color.) He continues to refine the system, with a focus on making its recognition abilities faster. At some point down the line, he plans to make the software portion open source. You're on your own as far as building conveyer belts, bins, and so forth.

Check out Mattheij's writeup in two parts for more information. It starts with an overview of the story, followed up with a deep dive on the software. He's also tweeting about the project (among other things). And if you look around a bit, you'll find bulk LEGO brick auctions online—it's definitely a thing!

Original image
Name the Author Based on the Character
May 23, 2017
Original image