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YouTube // TED-Ed
YouTube // TED-Ed

Can You Solve the Prisoner Boxes Riddle?

YouTube // TED-Ed
YouTube // TED-Ed

Here's a riddle. Ten band members have had their musical instruments placed randomly in boxes. Each band member gets five shots at opening boxes, trying to find their own instrument. (Thus, a 50% chance of each individual finding the desired instrument.) They're not allowed to communicate about what they find. If the entire band fails to find their instruments, they're all fired...and the odds of them all finding their instruments via random guessing is 1 in 1,024. But the drummer has an idea that will radically increase their odds of success. What's the big idea?

This puzzle is based on the 100 Prisoners Problem, though it has been simplified just a bit, and lightened up to be about a band (in the original 100 Prisoners Problem scenario, the punishment is death rather than loss of a job). Here's what the TED-Ed video gives us as the rules:

1. Instruments have been randomly placed in 10 boxes.

2. The pictures on the boxes don't necessarily correspond to the instruments inside.

3. Each musician can open up to 5 boxes. They have to close all of the boxes they open.

4. All 10 musicians must find their own instruments.

5. The musicians can't in any way communicate to each other what they find.

Given all this, it seems rather hopeless. But think on it. What could you do if you talked it through beforehand, as this puzzle allows? Watch this video for the setup, and then pause it when it tells you to, in order to have a good long think. I'd like to think I could come up with a solution like this, were my band (or band of prisoners) in a similar situation. Enjoy:

For more on the original 100 Prisoners Problem, check out this nice writeup. There's also more info from TED-Ed (check the "Dig Deeper" bit for useful links).

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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
V via Flickr // CC BY-NC 2.0
V via Flickr // CC BY-NC 2.0

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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