Mathematics can
get pretty complicated. Fortunately, not all math problems need to be
inscrutable. Here are five current problems in the field of mathematics that
anyone can understand, but nobody has been able to solve.

##
__Collatz Conjecture__

__Collatz Conjecture__

Pick any number.
If that number is even, divide it by 2. If it's odd, multiply it by 3 and add
1. Now repeat the process with your new number. If you keep going, you'll
eventually end up at 1. Every time.

Mathematicians
have tried millions of numbers and they've never found a single one that didn't
end up at 1 eventually. The thing is, they've never been able to prove that
there isn't a special number out there that never leads to 1. It's possible
that there's some really big number that goes to infinity instead, or maybe a
number that gets stuck in a loop and never reaches 1. But no one has ever been
able to prove

**.***that for certain*##
__Moving Sofa
Problem__

__Moving Sofa Problem__

So you're moving
into your new apartment, and you're trying to bring your sofa. The problem is,
the hallway turns and you have to fit your sofa around a corner. If it's a
small sofa, that might not be a problem, but a really big sofa is sure to get
stuck. If you're a mathematician, you ask yourself: What's the largest sofa you
could possibly fit around the corner? It doesn't have to be a rectangular sofa
either, it can be any shape.

This is the
essence of the

**. Here are the specifics: the whole problem is in two dimensions, the corner is a 90-degree angle, and the width of the corridor is 1. What is the largest two-dimensional area that can fit around the corner?***moving sofa problem*
The largest area
that can fit around a corner is called—I kid you not—the sofa constant. Nobody
knows for sure how big it is, but we have some pretty big sofas that do work,
so we know it has to be at least as big as them. We also have some sofas that
don't work, so it has to be smaller than those. All together, we know the sofa
constant has to be between 2.2195 and 2.8284.

##
__Perfect Cuboid
Problem__

__Perfect Cuboid Problem__

Remember the
pythagorean theorem, A2 + B² = C²? The three letters correspond to the three
sides of a right triangle. In a Pythagorean triangle, and all three sides are
whole numbers. Let's extend this idea to three dimensions. In three dimensions,
there are four numbers. In the image above, they are A, B, C, and G. The first
three are the dimensions of a box, and G is the diagonal running from one of
the top corners to the opposite bottom corner.

*l is to find a box where A² + B² + C² = G², and where all four numbers are integers. Mathematicians have tried many different possibilities and have yet to find a single one that works. But they also haven't been able to prove that such a box doesn't exist, so the hunt is on for a perfect cuboid.*

**The goa**##
__Inscribed Square
Problem__

__Inscribed Square Problem__

Draw a closed
loop. The loop doesn't have to be a circle, it can be any shape you want, but
the beginning and the end have to meet and the loop can't cross itself. It
should be possible to draw a square inside the loop so that all four corners of
the square are touching the loop. According to the

*, every closed loop (specifically every plane simple closed curve) should have an inscribed square, a square where all four corners lie somewhere on the loop.***inscribed square hypothesis**
This has already
been solved for a number of other shapes, such as triangles and rectangles. But
squares are tricky, and so far a formal proof has eluded mathematicians.

##
__Happy Ending
Problem__

__Happy Ending Problem__

**problem is so named because it led to the marriage of two mathematicians who worked on it, George Szekeres and Esther Klein. Essentially, the problem works like this:**

*The happy ending*
Make five dots at
random places on a piece of paper. Assuming the dots aren't deliberately
arranged—say, in a line—you should always be able to connect four of them to
create a convex quadrilateral, which is a shape with four sides where all of
the corners are less than 180 degrees. The gist of this theorem is that you'll
always be able to create a complex quadrilateral with five random dots, regardless
of where those dots are positioned.

So that's how it
works for four sides. But for a pengaton, a five-sided shape, it turn out you
need nine dots. For a hexagon, it's 17 dots. But beyond that, we don't know.
It's a mystery how many dots is required to create a heptagon or any larger
shapes. More importantly, there should be a formula to tell us how many dots
are required for any shape. Mathematicians suspect the equation is M=1+2N-2,
where M is the number of dots and N is the number of sides in the shape. But as
yet, they've only been able to prove that the answer is at least as big as the
answer you get that wa

Err, the 'Perfect Cuboid' problem is, well, not a problem. There are an infinity of solutions. For example: 3^2 + 4^4 + 12^2 = 13^2 Perhaps you're thinking of the problem where the exponents are cubes? i.e. A^3 + B^3 = C^3? This problem is part of Fermat's Theorem, which was recently (finally) proven.

ReplyDeleteIn 'Perfect Cuboid' problem the goal is to make all face diagonals (d, e, f) to be integers as well.

DeleteThe formula for the "Happy Ending" problem should read 1+2^(2n-2).

ReplyDeleteAlmost. It is 1+2^(N-2).

DeleteIn the Collatz conjecture, you must start with a positive integer, not just "pick any number"

ReplyDeleteAn update on the happy ending problem

ReplyDeletehttps://www.quantamagazine.org/a-puzzle-of-clever-connections-nears-a-happy-end-20170530/