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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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History
When Math Discoveries Led to Banned Numbers
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The literature world has seen more than its share of controversy. The best stories tend to provoke the strongest reactions—both positive and negative—in readers, which is why so many classic books have been banned at one point or another. But even a more objective field like math isn’t immune to conflict. In its new video, TED-Ed rounds up the numbers that caused such a stir when they were introduced that they were banned in math circles.

One of the earliest examples comes from ancient Greece. A mathematician named Hippasus was having trouble solving certain equations with fractions and whole numbers alone, so he came up with irrational numbers to make these values easier to express. The ruling school of thought at the time dictated that everything in nature could be explained elegantly with the numbers that already existed. Threatened by Hippasus’s new notion, his fellow mathematicians rejected the irrational numbers and had him exiled.

Other numbers have been banned for legal reasons. When Arab traders brought their positional number system, which included zero, to Italy in the Middle Ages, Florence banned it from record-keeping fearing that they would be easier to forge than Roman numerals. The Arabic way of counting also led to the rise of negative numbers, which were regarded with disdain by many experts into the 19th century. For more banned numbers, including some that are prohibited today, check out the full story below.

[h/t TED-Ed]

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Euclid
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Live Smarter
An Ex-Google Engineer Just Reinvented the Measuring Cup
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Euclid

Recognizing a problem most people didn’t even know they had, former Google and Facebook software engineer Joshua Redstone has made a bold claim for his recent Kickstarter venture: He’s developed a better measuring cup.

According to the Boston Business Journal, Redstone spent four years tinkering with a solution to something that had long annoyed him as an amateur chef: Traditional measuring cups, which are stocky and not very well tapered, don’t do a great job of accurately measuring their own contents. Redstone believes the shape of a cup determines its success, particularly when a cook overfills a liquid or solid by a tiny amount. The smaller the volume, the more the problem is magnified.

Euclid

Redstone’s cup, Euclid, resolves the issue. According to the Kickstarter page: “With traditional measuring cups, the smaller the amount, the harder it is to measure accurately. The culprit? The shape. Straight sides magnify errors when measuring lower down in the cup. Some have tried to solve this problem with conical measuring cups, but their results fall short of Euclid’s by up to 60 percent. Euclid is the only measuring cup with a mathematically optimal, tapered design for consistent accuracy across amounts.”

Euclid is just about ready to overshoot its $30,000 Kickstarter goal. Backers can pay $24 for the cup now, or wait until it’s available at retail for a slightly higher price to be determined. The cup is scheduled for release in May 2018.

[h/t Boston Business Journal]

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