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There are lots of unsolved mysteries in the world of math, and many of them start off with a deceptively simple premise, like: What's the biggest couch you can slide around a 90-degree corner?

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Moving sofa

Moser’s Worm


Magic Squares of Squares

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 more… survivalist-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-buck” problem, 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 case… why?

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].