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Can You Solve the Passcode Riddle?

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Here's a nice riddle. Three members of a team have been captured. On their way into the prison (guarded by ravenous mutant salamanders), they pass a series of numbered doorways, each with a keypad featuring the numbers one through nine. Each keypad opens with a code...but you have no idea what that code might be. One member will be allowed to try to escape by facing a challenge. Can the remaining two listen in on that challenge, figure out the correct hallway, and figure out the passcode to open it? With some basic math, they can succeed.

Some more details: Zara, the team member participating in the challenge, has a one-way audio transmitter that allows the other two team members to listen. As Zara is led to the challenge through one of the hallways, she is informed that her challenge is to guess the passcode for her hallway based on rules. Zara is told that the passcode will contain three positive whole numbers, in ascending order (like 1, 2, 3—the second number is greater than or equal to the first, the third likewise to the second).

Zara is told that she may ask up for up to three clues about the code—but she can't say anything else, or else she too will be fed to the mutant salamanders! Through this process of requesting clues and thinking through the problem, Zara implicitly passes information to her compatriots about what the answer is.

Zara asks for the first clue, and is told that the product of the three numbers in the code (x * y * z) is 36. Zara asks for the second clue, and is told that the sum of the numbers in the code (x + y + z) is the same as the number of the hallway she entered. There is a long silence. Then she asks for the third clue, and is told that the largest (greatest) number appears only once in the combination. Shortly after, Zara punches in the code and escapes.

Given that information, can you figure out the passcode? The video below walks through the puzzle and its solution. Here's the text of the puzzle in its simplest form, transcribed from the video (it helpfully asks you to pause before explaining the solution!):

Find three numbers in ascending order!

1. The product of the three numbers is 36.

2. The sum of the three numbers is the same as Zara's hallway number, which you don't know but she does.

3. The largest number must be unique.

4. Zara, a perfect logician, needed clues 1-3 to escape.

Start your figuring!

Then tune in to the video to test your solution:

For a bit more on this puzzle, check out this TED-Ed page, especially the "Dig Deeper" section which contains links to various math resources that you may find useful. (It also includes a link to a puzzle variant that may be interesting after you solve this one.)

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

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Name the Author Based on the Character
May 23, 2017
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