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Can You Solve the Three Gods Riddle?

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In this TED-Ed riddle, we explore a classic logic puzzle invented by logician Raymond Smullyan. As the video tells us, it has been called the hardest logic puzzle ever. They're not kidding.

So here's the situation. You have crash-landed on a mysterious planet. The only way to escape is to appease three alien overlords (they were "gods" in the original telling, hence its name).

You know that the three aliens are named Tee, Eff, and Arr. There are also three artifacts on the planet, each of which matches a single alien (for the sake of simplicity, let's assume they are labeled "Tee," "Eff," and "Arr"). To appease the aliens, you need to match up the artifacts with their aliens—but you don't know which of the aliens is which!

You are allowed to ask three yes-or-no questions, each addressed to any one alien. (You can address multiple questions to the same alien, but you don't have to.)

To further complicate things, each alien has a specific behavior with regard to telling the truth. Tee's answers are always true. Eff's answers are always false. Arr's answers are random.

Yet another problem is that while you know the alien words "ozo" and "ulu" somehow correspond to "yes" and "no," but you don't know which is which. (I know, this situation just keeps getting worse!) So while you can communicate enough of the alien language to ask questions, they will respond only with "ozo" or "ulu." You may ask the questions one at a time, building on each response if you wish—meaning you have time to think about the next question based on what you have learned.

So your task is to ask three yes-or-no questions, while not knowing which answering words correspond to "yes" and "no," of three aliens whose identity is unknown, but whose behavior is predictable...if you knew their identity. How can you figure out which alien is which, so you can hand the right objects to the aliens? Put your thinking caps on, then roll the video explanation (which includes the answer, after a suitable pause):

For more on this puzzle, check out this TED-Ed page. For some flavor on logician Raymond Smullyan, check out his interview with Johnny Carson.

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Math Symbols Might Look Complicated, But They Were Invented to Make Life Easier
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Numbers can be intimidating, especially for those of us who never quite mastered multiplication or tackled high-school trig. But the squiggly, straight, and angular symbols used in math have surprisingly basic origins.

For example, Robert Recorde, the 16th century Welsh mathematician who invented the “equal” sign, simply grew tired of constantly writing out the words “equal to.” To save time (and perhaps ease his writers’ cramp), he drew two parallel horizontal line segments, which he considered to be a pictorial representation of equality. Meanwhile, plenty of other symbols used in math are just Greek or Latin letters (instead of being some kind of secret code designed to torture students).

These symbols—and more—were all invented or adopted by academics who wanted to avoid redundancy or take a shortcut while tackling a math problem. Learn more about their history by watching TED-Ed’s video below.

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History
The Homemaker Who Helped Solve One of Geometry's Oldest Puzzles
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The next time you find yourself staring at your bathroom floor tiles, thank Marjorie Rice. The San Diego homemaker helped solve one of the oldest problems in geometry: figuring out which shapes could "tile the plane," or seamlessly cover a flat surface in an endless, repeating pattern. Rice's hand-drawn doodles in the 1970s led to major discoveries in the last few years, finally answering the puzzle that had stumped classical thinkers.

Ancient Greek mathematicians believed that certain shapes could tile the plane, without overlapping or leaving any gaps, in a pattern called a tessellation. They proved that all triangles and quadrilaterals, and some convex hexagons (six-sided shapes), could tile the plane. But for centuries, no one knew how many tiling convex pentagons (irregular five-sided shapes) were out there.

The hunt for tiling pentagons began in 1918 when German mathematician Karl Reinhardt described the first five types of tessellating pentagons. For 50 years it was believed that he had found them all, but in 1968, physicist R. B. Kershner discovered three more classes. Richard James, a computer scientist in California, found another in 1975, bringing the total to nine.

That year, Rice read a column by Martin Gardner in Scientific American about the research and began experimenting to find more tiling pentagons. "I became fascinated with the subject and wanted to understand what made each type unique," Rice wrote in an essay about M.C. Escher's use of repeating patterns. "Lacking a mathematical background, I developed my own notation system and in a few months discovered a new type which I sent to Martin Gardner. He sent it to Doris Schattschneider to determine if it truly was a new type, and indeed it was."

Schattschneider, a mathematics professor at Moravian College in Bethlehem, Pennsylvania, deciphered Rice's notation and realized she had found four new types—more than anyone other than Reinhardt. Schattschneider helped formally announce Rice's discoveries in 1977.

"My dad had no idea what my mom was doing and discovering," her daughter Kathy Rice told Quanta Magazine.

It took another eight years for the next type of tiling pentagon to be found, this time by University of Dortmund mathematician Rolf Stein. Then the trail went cold for 30 years.

In 2015, mathematicians Jennifer McLoud-Mann, Casey Mann, and David von Derau at the University of Washington, Bothell, found the 15th class of tessellating pentagon using a supercomputer. Then, in July 2017, French mathematician Michaël Rao completed the classification of all convex polygons, including pentagons, that can tile the plane. He confirmed that only the 15 known convex pentagons could tessellate [PDF].

The immense amount of research and the scale of the recent discoveries makes the achievements of Marjorie Rice all the more impressive. Though she lacked more than a high-school education and access to supercomputers, Rice remains the most prolific discoverer of tiling pentagons to emerge in the century since Reinhardt first attempted to crack the problem.

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