5 More Mind Bending Mathematical Mysteries

There are a lot of unsolved mysteries in mathematics. Some of them, like the Millennium Prize Puzzles that we covered previously on this channel, are extremely complex problems with huge implications for advancing the fields of math and science. However, many of them pertain to recreational math.

Recreational math is exactly what it sounds like: math done for fun rather than research. It is often performed by amateurs, though not exclusively, and discoveries resulting from recreational math often can have important consequences as well; they just don’t tend to be as obvious.

What makes these unsolved mysteries so interesting is that they’re extremely simple to understand and even attempt on your own. The solutions may not even require any understanding of higher level math, and several partial and complete solutions have been found by amateurs that were inspired to play around with the problems thanks to YouTube videos. So who knows, you could make the next big breakthrough in any of these unsolved mysteries of mathematics.

The Magic Square of Squares

Magic squares have captivated humans for thousands of years, with the earliest examples dating back to China in 2800 BC. It was originally believed that these squares actually had magical or spiritual powers. Even without supernatural abilities, they do still have some amount of practical application in things like quantum physics and internet search algorithms.

For those unfamiliar, a magic square is a square of unique numbers arranged in such a way that each row, column, and diagonal sums to the same number. They can come in many sizes, but the most simple example is a 3x3 square using the numbers 1 to 9. Although you can rotate or flip the square, there is only a single solution: 5 goes in the center, even digits go in the corners, and odd digits go on the sides to make each line add up to 15.

There are infinitely many magic squares, and while 3x3 are the most commonly seen since they’re the easiest to construct, a square of any size is possible. The current record for the largest magic square contains over 3500 rows and columns. With so much known about these mathematical peculiarities, it’s hard to imagine that there would be any mystery here.

But then, in the 1700s, Swiss mathematician Leonhard Euler decided to make things a bit more complicated. Instead of using any old integers for the numbers, they had to be squares, like 2², 3², etc. In 1770, Euler constructed a 4x4 magic square of squares. Others then created 5x5, 6x6, and 7x7 squares. But it’s been over 250 years since Euler constructed his first square of squares, and nobody has been able to construct a 3x3 example.

And that brings us to the open question: is a 3x3 magic square using nine different square numbers even possible? In 1996, Martin Gardner offered $100 to anyone who could construct one. In 2005, Christian Boyer upped the ante to €1,000 and a bottle of champagne to anyone who could either construct the magic square or prove that it’s impossible.

Many attempts have been made, and people have gotten close. Gardner himself was able to construct a semi-magic square, a square in which the rows and columns all add up to the same value but the diagonals do not. Unfortunately getting close not only doesn’t count, but it doesn’t actually help people get closer to a proper solution.

As with most things, it’s only a matter of time before we have an answer. That answer may be that it’s impossible, but if you can prove that’s the case then the €1,000 and champagne would still be yours.

Sums of Three Cubes

From the magic square of squares we just covered to Fermat’s Last Theorem, it seems that adding exponents to simple tasks suddenly makes them immeasurably more complicated. Such is the case with the sum of three cubes problem.

The idea is pretty simple, as is often the case with these sorts of problems. The conjecture is that every number can be represented by infinitely many different sums of three cubes…except for the ones that can’t. Those ones that can’t are the numbers that equal 4 or 5 mod 9. That may sound confusing if you have no experience with programming or modular arithmetic, but some number mod 9 is just asking what the remainder is when you divide that number by 9.

For example, 13 divided by 9 has a remainder of 4, and as such could not be represented as the sum of three cubes.

That part is actually easy to prove. If the target number mod 9 is either 4 or 5, then the mod 9 of the three cubes need to sum to 4 or 5 as well. However, any cube mod 9 will either be 0, 1, or 8, and there is no way to add up three of those numbers that can result in 4 or 5. This also shows us that some numbers are more difficult to calculate than others, as needing the three cubes mod 9 to all equal 1 or all equal 8 is a severe restriction on what cubes we can use.

As an example of this, 29 is a pretty simple number to represent as three cubes. It can be written as 3³ + 1³ + 1³, or 27 plus 1 plus 1. But going to 30 is much more difficult, because 30 divided by 9 has a remainder of 3. That means that we can only use cubes that mod 9 give us 1 in order for those three cubes to have a total mod 9 of 3. Again, if the terminology sounds at all confusing, remember that we’re just talking about the remainder when a number is divided by 9.

While 29 was simple, to represent 30 as a sum of three cubes the smallest integer equation available is this thing that’s too long to even say.

It took a long time to find this solution, and not even every number up to 1,000 has been solved yet. In 2019, a solution to 42 was finally found, completing every number up to 100. But the solution included three numbers that were 17 digits long, so you can only imagine how difficult finding some of these answers is. The only numbers under 1,000 that still need a solution are 114, 390, 627, 633, 732, 921, and 975, and you are more than welcome to try to tackle any of those numbers yourselves.

But even if answers are found for those specific cases, this will still be an open problem. We can demonstrate by example that every number from 1 to 1000 that mod 9 is not 4 or 5 has a solution, but it doesn’t prove that every number can be written as the sum of three cubes, let alone that there are infinite sets of cubes that sum to each number. We’re confident that it’s true even if finding specific examples can take millions of hours of computing, yet we have no way to actually prove it.

The Hadwiger-Nelson Problem

This problem is related to the four colour theorem, a theorem which has now been proven. The four colour theorem was first proposed in 1852 by mathematician Francis Guthrie as a result of pure happenstance. He was colouring a map of the counties of England and realized that he only needed four colours to do so. Guthrie then speculated this would be true for all maps.

It took over 120 years, but this theorem was proven in the 1970s thanks to some help from computers. You can divide a map into absolutely any regions you want, real or imaginary, and it will always be possible to colour the map using only four colours such that no two regions of the same colour share an edge.

Named after mathematicians Hugo Hadwiger and Edward Nelson, the Hadwiger-Nelson Problem was first formulated in 1950 and published in 1960. Though similar, it is a more complex problem than the four colour theorem.

Imagine you are standing on an infinitely large, flat plane, and that plane is painted different colours. Starting from any point, you want to be able to take a step in any direction and land on a different colour. For the purposes of this question every step must be the same length, so we’ll just call it a meter though the actual unit size isn’t important. The Hadwiger-Nelson problem asks what the minimum number of colours needed is such that you could walk 1 meter in any direction from any point and always land on a different colour.

Intuitively, you’d probably think that we would need a lot of different colours for this to be possible. At the very least we would need more than the four colours used in a map. If this was your intuition, you would be correct.

It’s easy to prove that it cannot be done with two colours, because there’s no rule that you can’t visit the same point twice. If you were to walk one meter, turn 60 degrees, and repeat this two more times you would have made a triangle. If your starting position was blue, the next point on the triangle would have to be a different colour, let’s say red. The third point would then have to be blue, but now there’s a problem.

Stepping back to the first point on the triangle would be going from blue to blue, so this doesn’t work.

There are similar disproofs for three and four colours, though the paths you would have to take are a bit more complicated. We can also prove that it is possible to design such a plane with only seven colours pretty easily by using a bunch of hexagons in a repeating pattern.

You need to be able to step from the very edge of one hexagon all the way across it and onto a different one, so they need to be slightly smaller than one meter but not so small that you could step from one hexagon to another one of the same colour. If they’re .9 meters each for example, then this configuration will allow you to step in any direction and always land on a different colour than what you started on.

Of course, the problem still isn’t solved. We know that such a plane is possible with seven colours, but can we do better? We can’t go as low as four, but what about five or six? We simply don’t know yet. The fewest colours that can be used to construct such a plane is either five, six, or seven, but we aren’t sure which.

The possibility of using 4 colours was disproven in 2018 by anti-aging scientist Aubrey de Grey who worked on it for fun in his spare time. So you don’t need to be a professional mathematician if you want to be the person to prove or disprove 5 or 6 colours, though having a computer science degree probably helped de Grey and his computer assisted proof a little bit.

The Moving Sofa Problem

If you’ve ever had to move furniture, you know what a giant pain in the ass it can be. Moving things like sofas or tables around corners and through doors can be an absolute nightmare, usually resulting in it being lifted up and turned in all sorts of directions to squeeze it through seemingly impossible spaces.

But what if you couldn’t lift it? Furniture can be extremely heavy, and maybe you are only able to move this piece by sliding it along the floor. What is the largest, two-dimensional sofa that could be maneuvered through a 90 degree turn in a hallway? Naturally we can assume for this that the width of the hallway is the same both before and after the turn, which will just call 1 meter again.

There are some pretty simple answers for what types of sofa you could fit through this hallway. The most basic is just a 1 meter square. This could be pushed flush against the corner, and then pulled down the other side of the hallway. But we can do a lot better than that.

Another possible solution is to use a sofa in the shape of a semicircle that has a radius of 1 meter. This again can be pushed directly into the corner where it could then be pivoted at an angle to navigate it through the turn. It’s a solid improvement, and results in a sofa that’s about 1.57 square meters instead of just 1 square meter.

By using irregular shapes, we can do even better. The largest sofa thus far designed came from Joseph Gerver in 1992. It’s just over 2.2 square meters, and is shaped like the receiver for a corded telephone. It’s a big improvement, but is it the best? Probably not. Even though nobody has found a larger sofa that would work, we can’t just check every possible shape and design to see what would fit around the corner as there could always be another shape we hadn’t thought of.

Some work has been done to determine an upper limit for what the largest possible sofa could be, the largest possible area that could theoretically be navigated through the hallway. For example, it should be pretty clear that a sofa that was 50 square meters couldn’t possibly fit, but what might? Currently it has been determined that the absolute upper bound for how large the sofa could be is 2.37 square meters.

This was proven in 2017, lowering the previous upper bound of 2.82. The proof itself is more complicated than we need to get into, but we can now definitively rule out any shape with an area greater than that upper bound.

So is Gerver’s telephone shaped sofa actually the optimal design, or is there an alteration that could be made to increase it to the theoretical upper limit? It’s been almost 60 years since the moving sofa problem was published and longer since it was conceptualized, but we still don’t have an answer for what seems like a rather simple problem.

Odd Perfect Numbers

A perfect number is a number that is equal to all of its positive divisors, excluding itself. For example, 6 is divisible by 1, 2, and 3 which add up to 6. Likewise, 28 is divisible by 1, 2, 4, 7, and 14 which sum to 28. We could go on, but the next two perfect numbers are 496 and 8,128, so hopefully you got the idea.

The idea of perfect numbers was laid out by Euclid over 2,000 years ago. He even identified the first four such numbers as well as created a formula that would identify them, though it was unclear whether or not this formula could find every perfect number. There are formulas that can identify prime numbers, but no such formula can be used to calculate every prime number, just ones that meet a specific criteria.

Ever since perfect numbers became a matter of discussion and number theory all those years ago, one question has persisted: are there any odd perfect numbers? Euclid’s formula was only able to produce even perfect numbers, but there was no guarantee that his formula would find every perfect number.

In 1496, French theologian Jacques Lefèvre claimed that Euclid’s formula would give all perfect numbers, and thus no odd perfect numbers existed. Unfortunately when it comes to math, people have a tendency to claim lots of things whether they can prove them or not. It’s a great way to get your name attached to the history of something in case someone actually is able to prove it hundreds of years later.

But thus far, nobody has been able to prove that odd perfect numbers don’t exist, though nobody has found one either. A lot of interesting and useful math has been discovered as the result of the search for odd perfect numbers, though at this point the restrictions imposed on odd numbers are starting to get a little silly. Through extensive research, it has been determined that if an odd perfect number exists, all of the following must be true:

• The number is at least 1,500 digits long.
• It is not divisible by 105.
• Its largest prime factor must be greater than 100 million.
• It must have at least 101 prime factors.
• The number mod 12 must equal 1, or mod 468 must equal 117, or mod 324 must equal 81.

There are actually a lot more restrictions that get much more complicated.

Euclid’s formula for even perfect numbers stated that if P was prime and 2p-1 was prime, then 2p-1(2p -1) was a perfect number. It was an extremely simple formula, even if finding prime numbers that actually fit wasn’t a simple task. It’s hard to believe that the formula for even perfect numbers could be so simple while any formula for odd perfect numbers would be so ludicrously convoluted, and most mathematicians don’t believe odd perfect numbers exist.

But believing something and proving something are not at all the same. Until somebody can either find an odd perfect number or definitively prove that none exist, it will remain an open question. Mathematicians have only spent about 2,300 years on this problem thus far, so with a little more effort perhaps they’ll finally make that big breakthrough.

Key Takeaways

• Recreational math problems are simple to understand and can be solved by amateurs.
• The Magic Square of Squares problem involves creating a 3x3 square using unique square numbers.
• The Sums of Three Cubes problem seeks to represent every number as a sum of three cubes.
• The Hadwiger-Nelson Problem asks for the minimum number of colors needed to color a plane.
• The Moving Sofa Problem aims to find the largest sofa that can be maneuvered through a 90-degree turn.

Sources

• Original Side Projects video: 5 More Mind Bending Mathematical Mysteries

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