Unsolved Math Problems
Unsolved Math Problems
The Three Types Of Unsolved Math Problem
[First box:] Weirdly Abstract [Ponytail stands in front of an equation.] Is the Euler Field Manifold Hypergroup Isomorphic to a Gödel-Klein Meta-Algebreic ε<0 Quasimonoid Conjection under Sondheim Calculus? Or is the question ill-formed? ⬙ℝ̇ℤ/Eℵ₅ [The Z is raised and underneath it is a double-ended arrow bent at a right angle. One points toward the R the other toward the Z. The ₅ is double-struck (𝟝) like the R and Z.]
Second: Weirdly Concrete [Cueball stands in front of a grid with 6 columns and 7 rows] If I walk randomly on a grid, never visiting any square twice, placing a marble every N steps, on average how many marbles will be in the longest line after N*K steps? Somehow the answer is important in like three unrelated fields. [The path starts in the 3rd row and 3rd column, a small circle indicates the start. It takes the path: North, East, North, East (a black dot representing the 1st marble is placed here, so N=4), South, East, South, South (2nd marble), West, South, West, North (3rd marble), West, South, South, South (4th Marble), West, North, West, West (this goes offgrid to the West. There is no visible line or marble outside the grid). The 1st, 3rd, and 4th marbles are colinear and there is a dotted line connecting them. The line's slope is 3.]
Third: Cursed [A Megan with unkempt hair stands next to a curve] What in God's name is going on with this curve? Is it even math? [The curve starts at the bottom of the screen, rises straight upward, begins to wobble left and right a little. It lists to the left and the left-right motion increases, then decreases. It begins a large counter-clockwise arc, spiraling inwards twice, then ends]

Math has many problems that remain "unsolved." This is not simply a matter of finding the correct numbers on both sides of an equal sign, but usually require proving or finding a counterexample to some conjecture, or explaining some property of some mathematical object. Sometimes this might involve extending an existing proof to a wider range of numbers like reals, complex numbers, or matrices.
A concrete problem is one that is very obviously connected to a real world process, while an abstract problem is one which seems unconnected to actual problems. In modern math, many problems tend to be very abstract, requiring complicated notation to adequately state the problem in the first place, like many of the millennium problems . On the other hand, many unsolved problems are very concrete; for example, there are very many problems related to packing objects into spaces that are very difficult to solve although quite easy to state, such as the Collatz conjecture . Finally, Randall describes a third category of "cursed problems," that have strange, seemingly random behavior, such as the behavior of turbulence or the distribution of prime numbers.
In the first panel, Ponytail describes a weird abstract problem. Her description seems to be a meaningless jumble of terms that are either mathematical or just sound mathematical. And the mathematical terms are from disparate branches of mathematics: group theory, topology, and calculus.
Finally she asks whether the problem statement is ill-formed; considering that it's mostly gibberish, this may be true.
Many real unsolved math problems appear similarly abstract. One example is the Hodge conjecture , a Millennium Prize problem. It states "Let X be a non-singular complex projective manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X." These words may appear nonsensical to a layperson. And even to an expert, the question is `abstract'. (Given a specific manifold, even an abelian fourfold, how on earth do you determine if a given 2,2 class is a cycle?)
In the second panel, Cueball describes a concrete random walk problem, and then mentions that this somehow has applications in three unrelated fields. This is actually not uncommon. The Wikipedia article says that "random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. Walking randomly on a grid never visiting any square twice is known as a self-avoiding walk ." This panel may have been inspired by some of the tricky unsolved problems about self-avoiding walks. Many of these problems have to do with rigorously proving properties of random walks that have been guessed by physics intuition, so these problems are connected to physics. The part about the maximum number of points in a line is reminiscent of problems in combinatorial geometry, which often involve counting points lying on different lines. Python code simulating this situation can be found here: [1] . C++ code simulating this situation can be found here: [2] .
In the final panel, Megan is looking at a strange curve that seems to have no consistent pattern. At the bottom it's mostly straight, with a few little wobbles. In the middle it looks like a wild, high-frequency wave that suddenly bursts and then dies down. And the top is a spiral that looks like a question mark or a Western-style Crosier . She wonders if this could even be mathematical. 
On one hand, considering the weird shapes that come from plotting some mathematical processes (e.g. the Mandelbrot set ), it could well be. For example the unsolved Riemann hypothesis , another Millennium Prize problem, concerns the properties of a weird and at-first-glance random curve .  In number theory, the term "cursed curve" has been used to describe the "split Cartan" modular curve of level 13, which resisted attempts for many years to compute its set of rational points .
On the other hand, the question if could even be mathematical suggests that this may indeed not be a mathematical symbol. The curve looks like the unalome symbol, which is a Buddhist symbol which represents the path taken in life, or the journey to enlightenment. It could be argued that this indeed represents an unsolved problem, although not a mathematical one - which might then be part of the humoristic meaning.
In the title text, the curve in the final panel is further explained based on the consensus of supposedly a group who has studied it and the procedure that generates it, commenting that "it's just like that" as their conclusion, which is really not an explanation at all.
The Three Types Of Unsolved Math Problem
[First box:] Weirdly Abstract [Ponytail stands in front of an equation.] Is the Euler Field Manifold Hypergroup Isomorphic to a Gödel-Klein Meta-Algebreic ε<0 Quasimonoid Conjection under Sondheim Calculus? Or is the question ill-formed? ⬙ℝ̇ℤ/Eℵ₅ [The Z is raised and underneath it is a double-ended arrow bent at a right angle. One points toward the R the other toward the Z. The ₅ is double-struck (𝟝) like the R and Z.]
Second: Weirdly Concrete [Cueball stands in front of a grid with 6 columns and 7 rows] If I walk randomly on a grid, never visiting any square twice, placing a marble every N steps, on average how many marbles will be in the longest line after N*K steps? Somehow the answer is important in like three unrelated fields. [The path starts in the 3rd row and 3rd column, a small circle indicates the start. It takes the path: North, East, North, East (a black dot representing the 1st marble is placed here, so N=4), South, East, South, South (2nd marble), West, South, West, North (3rd marble), West, South, South, South (4th Marble), West, North, West, West (this goes offgrid to the West. There is no visible line or marble outside the grid). The 1st, 3rd, and 4th marbles are colinear and there is a dotted line connecting them. The line's slope is 3.]
Third: Cursed [A Megan with unkempt hair stands next to a curve] What in God's name is going on with this curve? Is it even math? [The curve starts at the bottom of the screen, rises straight upward, begins to wobble left and right a little. It lists to the left and the left-right motion increases, then decreases. It begins a large counter-clockwise arc, spiraling inwards twice, then ends]
