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Try Solving This Riddle From a High School Math Competition

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Math olympiads feature some of the brightest minds to enter high school. The problems given to competitors are meant to be tricky, but there’s one question that’s so notorious it even has its own Wikipedia page.

The "Cheryl’s Birthday" riddle first appeared on a math olympiad test given to students in Singapore. Local television presenter Kenneth Kong posted a picture of the original text on Facebook in 2015. The clunky English makes for a riddle that's hard to decipher, so The New York Times published an edited version that reads as follows:

"Albert and Bernard just met Cheryl. 'When’s your birthday?' Albert asked Cheryl. Cheryl thought a second and said, 'I’m not going to tell you, but I’ll give you some clues.' She wrote down a list of 10 dates: May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17. 'My birthday is one of these,' she said. Then Cheryl whispered in Albert’s ear the month—and only the month—of her birthday. To Bernard, she whispered the day, and only the day. 'Can you figure it out now?' she asked Albert. Albert: I don’t know when your birthday is, but I know Bernard doesn’t know, either. Bernard: I didn’t know originally, but now I do. Albert: Well, now I know, too! When is Cheryl’s birthday?"

If you read through this quickly, it’s easy to get lost. How is it possible to guess someone’s birthday when you only have one half of the date? And how could Albert and Bernard help each other guess correctly without sharing their intel out loud? The secret lies in the possible dates Cheryl chooses to share.

In her list, every day is repeated once except for the 18th and the 19th—so if she had whispered “18” in Bernard’s ear he would immediately know her birthday was June 18, and if she’d whispered “19” he’d know it was May 19. Bernard confirms it couldn’t be either of those dates when he says he “didn’t know originally.” That means June and May are no longer options for Albert, which narrows it down to five possibilities: July 14, July 16, August 14, August 15, and August 17.

When Bernard says “I didn’t know[…]but now I do” that disqualifies 14, because in that case he would have two potential answers (July 14 and August 14) to choose from. These leaves three dates—July 16, August 15, and August 17—that could be correct. When Albert says, “Well, now I know too!” he eliminates July from the equation because that’s also a case where he’d be left with two options. July 16 is therefore the one true answer.

Does your head hurt yet? Unfortunately, math problems intended for grade-schoolers in Singapore don’t get much simpler. Check out this perplexing homework problem given to first-graders if you need convincing.

[h/t The New York Times]

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Man Buys Two Metric Tons of LEGO Bricks; Sorts Them Via Machine Learning
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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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Why Your iPhone Doesn't Always Show You the 'Decline Call' Button
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When you get an incoming call to your iPhone, the options that light up your screen aren't always the same. Sometimes you have the option to decline a call, and sometimes you only see a slider that allows you to answer, without an option to send the caller straight to voicemail. Why the difference?

A while back, Business Insider tracked down the answer to this conundrum of modern communication, and the answer turns out to be fairly simple.

If you get a call while your phone is locked, you’ll see the "slide to answer" button. In order to decline the call, you have to double-tap the power button on the top of the phone.

If your phone is unlocked, however, the screen that appears during an incoming call is different. You’ll see the two buttons, "accept" or "decline."

Either way, you get the options to set a reminder to call that person back or to immediately send them a text message. ("Dad, stop calling me at work, it’s 9 a.m.!")

[h/t Business Insider]

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