2529: Unsolved Math Problems - Revision history

FaviFake: i'm sorry but this is too much in the lead

2025-08-03T11:51:31Z

i'm sorry but this is too much in the lead

Kynde: /* Explanation */

2025-08-03T09:03:00Z

‎Explanation

Kynde: /* Explanation */

2025-08-03T09:02:10Z

‎Explanation

FaviFake: /* Explanation */

2025-07-31T14:47:17Z

‎Explanation

FaviFake: /* Transcript */

2025-07-30T09:35:52Z

‎Transcript

FaviFake: /* Transcript */

2025-07-30T09:35:37Z

‎Transcript

FaviFake: /* Weirdly Concrete */

2025-07-30T09:35:28Z

‎Weirdly Concrete

FaviFake at 09:35, 30 July 2025

2025-07-30T09:35:15Z

FaviFake: /* Explanation */

2025-07-30T09:32:54Z

‎Explanation

Kynde: Mentioning the category for this series

2025-07-16T06:51:54Z

Mentioning the category for this series

@@ Line 8: / Line 8: @@
 ==Explanation==
-This comic was recognized as the first in the [[:Category:Unsolved Problems|Unsolved Problems series]], when it was followed four years later by [[3115: Unsolved Physics Problems]]. That comics follows the exact same format as in this one with three similar categories. In the sequel, however, all the examples given are real phenomena as opposed to in this comic. Later it was then realized that [[2943: Unsolved Chemistry Problems]] that came out a year before the physics version was the second in the series. It doesn't follow the exact same format, however the idea (and title format) is the same.
+This comic is part of the [[:Category:Unsolved Problems|Unsolved Problems series]] and was followed by [[2943: Unsolved Chemistry Problems]] and [[3115: Unsolved Physics Problems]]. 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 {{w|millennium problems}}. On the other hand, many unsolved problems are very concrete, such as the {{w|Collatz conjecture}}. Additionally, there are the many problems related to {{w|Packing problem|packing objects into spaces}} which are often very difficult to solve though quite easy to state. 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.

@@ Line 8: / Line 8: @@
 ==Explanation==
-This comic was recognized as the first in the [[:Category:Unsolved Problems|Unsolved Problems series]], when it was followed four years later by [[3115: Unsolved Physics Problems]]. That comics follows the exact same format as in this one with three similar categories. In the sequel, however, all the examples given are real phenomena as opposed to in this comic. Later it was then realized that [[2943: Unsolved Chemistry Problems]] that came out a year before the physics version was the second in the series. But it doesn't follow the exact same format, but the idea is the same.
+This comic was recognized as the first in the [[:Category:Unsolved Problems|Unsolved Problems series]], when it was followed four years later by [[3115: Unsolved Physics Problems]]. That comics follows the exact same format as in this one with three similar categories. In the sequel, however, all the examples given are real phenomena as opposed to in this comic. Later it was then realized that [[2943: Unsolved Chemistry Problems]] that came out a year before the physics version was the second in the series. It doesn't follow the exact same format, however the idea (and title format) is the same.
 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.

@@ Line 8: / Line 8: @@
 ==Explanation==
-This comic became the first in the [[:Category:Unsolved Problems|Unsolved Problems series]], when it was followed four years later by [[3115: Unsolved Physics Problems]]. The comics follows the exact same format with three similar categories. In the sequel, however, all the examples given are real phenomena as opposed to in this comic. 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.
+This comic was recognized as the first in the [[:Category:Unsolved Problems|Unsolved Problems series]], when it was followed four years later by [[3115: Unsolved Physics Problems]]. That comics follows the exact same format as in this one with three similar categories. In the sequel, however, all the examples given are real phenomena as opposed to in this comic. Later it was then realized that [[2943: Unsolved Chemistry Problems]] that came out a year before the physics version was the second in the series. But it doesn't follow the exact same format, but the idea is the same.
 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 {{w|millennium problems}}. On the other hand, many unsolved problems are very concrete, such as the {{w|Collatz conjecture}}. Additionally, there are the many problems related to {{w|Packing problem|packing objects into spaces}} which are often very difficult to solve though quite easy to state. 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.

@@ Line 8: / Line 8: @@
 ==Explanation==
-This comic became the first in the [[:Category:Unsolved Problems|Unsolved Problems series]], when it was followed four years later by [[3115: Unsolved Physics Problems]]. The comics follows the exact same format with three similar categories. In the sequel, however, all the examples given are real phenomena as opposed to in this comic.
+This comic became the first in the [[:Category:Unsolved Problems|Unsolved Problems series]], when it was followed four years later by [[3115: Unsolved Physics Problems]]. The comics follows the exact same format with three similar categories. In the sequel, however, all the examples given are real phenomena as opposed to in this comic. 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 {{w|millennium problems}}. On the other hand, many unsolved problems are very concrete, such as the {{w|Collatz conjecture}}. Additionally, there are the many problems related to {{w|Packing problem|packing objects into spaces}} which are often very difficult to solve though quite easy to state. 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.

@@ Line 53: / Line 53: @@
 ==Transcript==
-The Three Types Of Unsolved Math Problem
+:The Three Types Of Unsolved Math Problem
 :[First box:] Weirdly abstract

@@ Line 55: / Line 55: @@
 The Three Types Of Unsolved Math Problem
-:[First box:] Weirdly Abstract
+:[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?
@@ Line 61: / Line 61: @@
 :⬙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
+: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?

@@ Line 41: / Line 41: @@
 Many real unsolved math problems appear similarly abstract. One example is the {{w|Hodge conjecture}}, a {{w|Millennium Prize Problems|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?)
-===Weirdly Concrete===
+===Weirdly concrete===
 In the second panel, [[Cueball]] describes a concrete {{w|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 {{w|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 [http://colab.research.google.com/drive/1nWrByCGBckwVdbAwow7tCYTOvqObYXyR?usp=sharing here]. C++ code simulating this situation can be found [http://github.com/AMindToThink/WeirdlyConcreteXKCD here].

@@ Line 14: / Line 14: @@
 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 {{w|millennium problems}}. On the other hand, many unsolved problems are very concrete, such as the {{w|Collatz conjecture}}. Additionally, there are the many problems related to {{w|Packing problem|packing objects into spaces}} which are often very difficult to solve though quite easy to state. 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.
@@ Line 40: / Line 41: @@
 Many real unsolved math problems appear similarly abstract. One example is the {{w|Hodge conjecture}}, a {{w|Millennium Prize Problems|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 {{w|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 {{w|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 [http://colab.research.google.com/drive/1nWrByCGBckwVdbAwow7tCYTOvqObYXyR?usp=sharing here]. C++ code simulating this situation can be found [http://github.com/AMindToThink/WeirdlyConcreteXKCD here].
 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 {{w|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 {{w|Mandelbrot set}}, or the {{w|bifurcation diagram}} of the {{w|logistic map}}), it could well be. For example the unsolved {{w|Riemann hypothesis}}, another Millennium Prize problem, concerns the properties of {{w|File:RiemannCriticalLine.svg|a weird and at-first-glance random curve}}.  In number theory, the term "cursed curve" [https://www.quantamagazine.org/mathematicians-crack-the-cursed-curve-20171207/ has been used] to describe the [https://annals.math.princeton.edu/wp-content/uploads/annals-v173-n1-p13-s.pdf "split Cartan" modular curve] of level 13, which resisted attempts for many years to compute its [https://www.jstor.org/stable/10.4007/annals.2019.189.3.6 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 {{w|Yantra_tattooing#Types_and_designs|unalome}}, 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 humorous meaning.
 The title text states that, despite decades of intensive study on the properties of the cursed curve, the best anyone's been able to come up with to explain its strangeness is "it's just like that." This lack of a satisfying explanation is commonplace with advanced math topics. As one famous example, the {{w|monster group}} ([https://youtu.be/mH0oCDa74tE explanation video]) is known to be the largest of a category of objects called {{w|sporadic groups}}. Similarly to the cursed curve in the comic, the monster group has a bizarre and complex structure which has, so far, managed to elude any logical explanation aside from "it's just like that."

@@ Line 16: / Line 16: @@
 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.
-*'''Euler field:''' An Euler vector field represents a space where every point is rotating with its own speed and direction. The name "Euler field", however, is something like "John Smith" - fields are very common algebraic structures, and {{w|Leonard Euler}} was a prolific Swiss mathematician who influenced {{w|List_of_things_named_after_Leonhard_Euler|so many areas of study}} that some of his discoveries are named after whoever wrote about them next, just to avoid naming everything after him.
+; Euler field: An Euler vector field represents a space where every point is rotating with its own speed and direction. The name "Euler field", however, is something like "John Smith" - fields are very common algebraic structures, and {{w|Leonard Euler}} was a prolific Swiss mathematician who influenced {{w|List_of_things_named_after_Leonhard_Euler|so many areas of study}} that some of his discoveries are named after whoever wrote about them next, just to avoid naming everything after him.
-*'''{{w|Manifold}}:''' A manifold is a topological space which is locally Euclidean - the shortest distance between two points is a straight line, the ratio between a circle's circumference and diameter is always pi, parallel lines are always the same distance apart, everything generally behaves the way you'd expect. A globe is a two-dimensional manifold, because a small-enough area is indistinguishable from a flat map. Using manifolds as an example of impenetrably occult maths may be a nod to the Tom Lehrer song "Lobachevsky", which makes a similar joke about "the analytical algebraic topology of locally Euclidean metrizations of infinitely differentiable Riemannian manifolds (Bozhe moi!)".
-*'''{{w|Hypergroup}}:''' An ''algebraic structure'', like arithmetic, is a set of well-defined operations (addition, subtraction, multiplication, division) mapping inputs to outputs over a domain of elements (the real numbers). A ''hyperstructure'' is an algebraic structure including an operation that maps a single input to multiple outputs - the simplest example is the square root, which maps a positive number like 4 to both positive and negative 2. A ''hypergroup'' is a hyperstructure with an operation that takes a pair of input elements, and, depending on which pair, can output every element or combination of elements in its domain... but also preserves association (1 + 2 + 3 = 6 whether you start by adding 1 + 2 or 2 + 3) and reproduction (if either input is "the entire domain", then the output will still be the entire domain). It's a decent indication of how abstract a hypergroup is that it takes at least three to five sub-definitions to make it remotely understandable.
+; {{w|Manifold}}: A manifold is a topological space which is locally Euclidean - the shortest distance between two points is a straight line, the ratio between a circle's circumference and diameter is always pi, parallel lines are always the same distance apart, everything generally behaves the way you'd expect. A globe is a two-dimensional manifold, because a small-enough area is indistinguishable from a flat map. Using manifolds as an example of impenetrably occult maths may be a nod to the Tom Lehrer song "Lobachevsky", which makes a similar joke about "the analytical algebraic topology of locally Euclidean metrizations of infinitely differentiable Riemannian manifolds (Bozhe moi!)".
-*'''Isomorphic:''' {{w|Isomorphism}} describes whether all the attributes of one structure can be mapped to properties of another structure. The structures usually have to be of the same type; it is unclear how a hypergroup would map to a "conjection".
-*'''Gödel-Klein:''' {{w|Kurt Gödel}} was a famous 20th-century mathematician who studied logic and philosophy (he's most well known for {{w|Gödel's incompleteness theorems}}) and {{w|Felix Klein}} was a famous 19th century mathematician who studied group theory and geometry; the two probably never collaborated.
+; {{w|Hypergroup}}: An ''algebraic structure'', like arithmetic, is a set of well-defined operations (addition, subtraction, multiplication, division) mapping inputs to outputs over a domain of elements (the real numbers). A ''hyperstructure'' is an algebraic structure including an operation that maps a single input to multiple outputs - the simplest example is the square root, which maps a positive number like 4 to both positive and negative 2. A ''hypergroup'' is a hyperstructure with an operation that takes a pair of input elements, and, depending on which pair, can output every element or combination of elements in its domain... but also preserves association (1 + 2 + 3 = 6 whether you start by adding 1 + 2 or 2 + 3) and reproduction (if either input is "the entire domain", then the output will still be the entire domain). It's a decent indication of how abstract a hypergroup is that it takes at least three to five sub-definitions to make it remotely understandable.
-*'''Meta-algebra:''' Not a real term, though derived from the real term {{w|Metamathematics}}. Also, 'algebraic' is normally spelled with two a's, not two e's.
-*'''ϵ<0:''' Another joke term. In analysis, {{w|ϵ}} is usually defined to be an arbitrarily small ''positive'' number.
+; Isomorphic: {{w|Isomorphism}} describes whether all the attributes of one structure can be mapped to properties of another structure. The structures usually have to be of the same type; it is unclear how a hypergroup would map to a "conjection".
-*'''Quasimonoid:''' A quasimonoid is a type of algebraic object, namely a non-associative {{w|monoid}}. Here it may also be meant to evoke the character {{w|Quasimodo}} from ''The Hunchback of Notre-Dame''.
-*'''Sondheim Calculus:''' This refers to {{w|Stephen Sondheim}}, one of the most successful composers and lyricists of American musical theatre -- the producer of his musical "Into the Woods" once [https://www.indiewire.com/2015/01/watch-singing-sondheim-is-like-calculus-in-into-the-woods-behind-the-scenes-video-exclusive-189507/ remarked] that "Singing Stephen Sondheim is like calculus for singers and actors."
+; Gödel-Klein: {{w|Kurt Gödel}} was a famous 20th-century mathematician who studied logic and philosophy (he's most well known for {{w|Gödel's incompleteness theorems}}) and {{w|Felix Klein}} was a famous 19th century mathematician who studied group theory and geometry; the two probably never collaborated.
-*'''Conjection:''' This may combine conjecture and conjunction, or be a joke on pros and cons plus projection.
 Finally she asks whether the problem statement is ill-formed; considering that it's mostly gibberish, this is probably true.