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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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B1 US sofa worm blanket math problem square

# 4 Weird Unsolved Mysteries of Math

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joey joey posted on 2021/06/28
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