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  • Sometimes math can get a bad rep for being confusing and difficult.

  • But ask a mathematician, and they'll tell you it's anything but!

  • There are lots of unsolved mysteries in the world of math,

  • and many of them start off with a deceptively simple premise.

  • So here's a few of the weirdest and most interesting

  • unsolved math problems inspired by the everyday and beyond.

  • If you've ever experienced the joy of trying to pivot a sofa around a narrow corridor,

  • you'll understand the struggle of the moving sofa problem.

  • You're trying to move out of your apartment

  • and you turn your sofa around a 90-degree corner

  • And depending on the sofa, it might get stuck.

  • So here's the question: assuming you have to turn around the corner,

  • what's the biggest sofa, area-wise of any shape,

  • that you can slide around it like this without lifting it?

  • This problem was first proposed back in 1966 by Leo Moser.

  • It might sound like a simple one, but to this day,

  • no one's been able to come up with the biggest possible sofa,

  • but some have come close!

  • In 1992, mathematician Joseph Gerver came up with the biggest sofa

  • so far by stitching together a series of optimized curves

  • which you might recognize as a telephone receiver.

  • And this is just one of the infinite sofa shapes you can come up with.

  • But proving you've got the biggest one, that's a lot harder.

  • A proof in math makes new truths from already-known ones.

  • It has to be a totally airtight, completely logical argument,

  • not just an idea with lots of evidence in its favor.

  • In this case, a proof would demonstrate the sofa fits

  • and is the biggest and that there are no other possibilities that beat it.

  • But going through every possible sofa is a daunting task,

  • so scientists wrote some computer code in 2017.

  • The more equations the code solved, the smaller the upper bound got,

  • a specific number which the sofa area couldn't surpass.

  • They concluded that either Gerver's sofa could be the biggest possible sofa,

  • or it's only a few percent smaller than the biggest possible sofa.

  • So the solution might be near, but it requires all sofas to be tested

  • so the quest continues today.

  • But the sofa problem wasn't the only problem Moser proposed,

  • he also came up with a worm to go with it!

  • We now call it Moser's Worm or sometimes lovingly Mother Worm's Blanket.

  • It involves a worm that wants to make a blanket

  • that fully covers a sleeping baby worm.

  • The question, then, is this: what's the smallest blanket

  • that will cover the baby worm no matter what position it's in?

  • We'll say Baby Worm doesn't move in their sleep, to make things easier.

  • Clearly, a square blanket where each side is twice as long as Baby Worm

  • would work, but you can go way smaller.

  • Clipping the square corners to make a circle gives you a blanket that still works,

  • but that blanket is only 79% of the square blanket's area.

  • How much more can you cut before it stops working?

  • Turns out quite a bit.

  • Like with the sofa problem, finding a shape isn't the issue,

  • proving you've got the smallest one is.

  • It's hard to check every shape one-by-one,

  • and even harder to prove it's the smallest one.

  • So most of the research focuses on clever arguments to find bounds,

  • or limits for the smallest blanket's size by saying, well,

  • the blanket can't be much bigger or smaller than the worm.

  • And researchers have actually come up with a bunch of

  • clever proofs along those lines and narrowed down the limits quite a lot.

  • Instead of needing a square blanket where each side is twice the worm's length,

  • a blanket about six percent as big can

  • still cover the worm no matter what position they're in.

  • To be precise, it's now known that the smallest possible blanket

  • is between 6% and 6.5% of the area of the original square blanket.

  • And while researchers still don't know the actual size

  • of the smallest possible blanket, whatever that shape ends up being,

  • we'll be saving mother worm a lot of knitting work!

  • The worm problem turns out to have a surprising connection

  • to another problem that's a bit moresurvivalist-oriented.

  • Imagine you're a hiker lost in a forest.

  • You have a map with the size and shape of the forest, but no landmarks,

  • and on top of that you don't know which part you're in, or which way you're facing.

  • So, how can you find the shortest path out of the forest?

  • This is the Lost-In-A-Forest Problem, which was first proposed in 1955.

  • In 2002 one researcher called it a “million-buckproblem,

  • implying that it's secretly one of the most important unsolved problems in math.

  • He thought the techniques you would need to invent to solve the problem

  • would be hugely useful in other areas of math, as well.

  • Sadly, mathematicians are just as lost as the hiker when it comes to this problem.

  • There aren't even many proposed solutions, let alone proven ones.

  • The tough thing about this problem is that different shapes of forest

  • turn out to be separate problems that need solving in different ways.

  • For circles, as well as squares, pentagons, and all regular polygons with more sides

  • than a triangle, it's been proven that a simple straight line is the best path.

  • That means that no matter where you are in those kinds of forests,

  • walking in a straight line is always the fastest way to escape.

  • But a straight line isn't always the best option; it depends on the shape.

  • For equilateral triangles, it's known that

  • a certain kind of zigzag pattern is the best path to try.

  • That's because it's been proven that there's a zigzag line that

  • doesn't 'fit' inside the triangle, no matter how you try to slot it in,

  • and that zigzag is shorter than the triangle's sides.

  • So working on the problem can be tough.

  • But progress in this problem can also be used in the worm problem,

  • because the two are kind of inverses of each other.

  • If you find the smallest worm shape that doesn't fit inside a certain blanket shape,

  • then that's the same as finding the shortest path shape

  • that gets you out of that shape of forest.

  • Basically, lines that don't work for the worm problem

  • are the ones that do work for the forest one.

  • Some recent work has hinted that maybe there's a simpler way to check

  • if a certain type of path is the shortest possible for a particular shape of forest.

  • And making paths easier to check means that researchers might be able to

  • use computers to check a bunch of paths really quickly.

  • So researchers may not be lost in the woods for too much longer!

  • Moving away from geometry, another neat unsolved mystery is about numbers,

  • and it's actually a problem you can try to solve right now.

  • It involves magic squares, which are grids of different numbers where every row,

  • column, and diagonal adds up to the same number.

  • This is not a new problem, it has been played with for thousands of years,

  • from ancient China to medieval India to 18th-century Europe.

  • The unsolved problem is about magic squares of squares,

  • magic squares made only of square numbers,

  • which are numbers you get from multiplying a number by itself.

  • The most famous and unsolved one is the 3-by-3 magic square of squares

  • made famous in 1996 by legendary puzzle-maker Martin Gardner

  • in one of his Scientific American columns.

  • Again, this puzzle sounds simple, right?

  • All you need to do is write down the correct combination of nine numbers,

  • and you've solved it, like an extreme sudoku.

  • There have been a lot of near-misses: magic squares where two of the numbers

  • aren't squares, or where one of the columns doesn't work right,

  • or some numbers repeat.

  • But nobody's been able to find one that fully works,

  • and it's not clear that looking at near-misses gets you any closer to finding one.

  • And maybe there simply isn't one that fully works.

  • But if that's the casewhy?

  • Why does it work for all those other examples, but not here?

  • It's possible to make 3-by-3 magic squares by solving a big bunch of equations,

  • so maybe there's just no square-number solutions to those equations.

  • Though no one's been able to prove that.

  • That feels a bit unsatisfying as an answer, so many people, mostly hobbyists,

  • still hope there's a solution out there, and spend their time searching for it

  • until somebody proves it can't be done.

  • In fact, one person has offered a prize of a thousand Euros

  • and a bottle of champagne to anyone who finds

  • a 3-by-3 magic square of nine unique square numbers.

  • Not quite a million dollars, but not bad, either!

  • And the really cool part is that you don't need a math degree

  • to look for a square that works.

  • Anyone can just go out there and start playing around with numbers,

  • making this a neat introduction to the joy of recreational math.

  • In general, the beauty and fun of math is

  • precisely in asking and answering those kinds of puzzles.

  • Often the answers to random problems like these end up being

  • surprisingly useful in the 'real world'.

  • But it's just as worthwhile to play around

  • with these weird problems for their own sake.

  • If you like these math problems, then I bet you'll like a podcast full of tangents!

  • SciShow Tangents is a podcast is where the fun people of SciShow

  • get together for a lightly competitive knowledge showcase.

  • Every episode, they rack up points for teaching the others,

  • and everyone listening at home,

  • the most mind-blowing science facts related to the week's theme.

  • If you love science, laughing, and lighthearted, nerdy competitions,

  • you should check it out!

  • You can find SciShow Tangents anywhere you get your podcasts.

  • [♪ OUTRO]

Sometimes math can get a bad rep for being confusing and difficult.

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4 Weird Unsolved Mysteries of Math

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    joey joey に公開 2021 年 06 月 28 日
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