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1500 questions
96
votes
7 answers
Can we cover the unit square by these rectangles?
The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1.
It is easy to show that
$$\sum_{1 \leq k } \left(\frac{1}{k} \times \frac{1}{k+1}\right) = …
Kaveh
- 5,362
96
votes
4 answers
A curious relation between angles and lengths of edges of a tetrahedron
Consider a Euclidean tetrahedron with lengths of edges
$$
l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34}
$$
and dihedral angles
$$
\alpha_{12}, \alpha_{13}, \alpha_{14},
\alpha_{23}, \alpha_{24}, \alpha_{34}.
$$
Consider solid…
Daniil Rudenko
- 4,296
95
votes
7 answers
How Much Work Does it Take to be a Successful Mathematician?
Hi Everyone,
Famous anecdotes of G.H. Hardy relay that his work habits consisted of working no more than four hours a day in the morning and then reserving the rest of the day for cricket and tennis. Apparently his best ideas came to him when he…
Justin Curry
- 2,684
95
votes
11 answers
Is it possible to capture a sphere in a knot?
You and I decide to play a game:
To start off with, I provide you with a frictionless, perfectly spherical sphere, along with a frictionless, unstretchable, infinitely thin magical rope. This rope has the magical property that if you ever touch its…
zeb
- 8,533
95
votes
36 answers
The concept of duality
I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come.
Wikipedia has a good page on several forms of "duality"…
Suvrit
- 28,363
95
votes
11 answers
Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?
Is there an infinite field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only finitely many points $y \in k$ do not lie in the image…
Philipp Lampe
- 2,580
95
votes
16 answers
Why is it a good idea to study a ring by studying its modules?
This is related to another question of mine. Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module. You tell him that modules generalize ideals and quotients, but he remains…
Qiaochu Yuan
- 114,941
95
votes
4 answers
Which manifolds are homeomorphic to simplicial complexes?
This question is only motivated by curiosity; I don't know a lot about manifold topology.
Suppose $M$ is a compact topological manifold of dimension $n$. I'll assume $n$ is large, say $n\geq 4$. The question is: Does there exist a simplicial…
Charles Rezk
- 26,634
95
votes
6 answers
Peer review 2.0
I have an idea for a website that could improve some well-known difficulties around peer review system and "hidden knowledge" in mathematics. It seems like a low hanging fruit that many people must've thought about before. My question is…
user2718
- 137
95
votes
5 answers
Is there a database for tracking the dependencies of mathematical theorems?
Given a proof for a result, one could denote the proof as a node on a graph, and then draw arrows to the node from axioms and previous results that the proof uses, and then draw arrows from the node to results that the result is used to prove. The…
Chill2Macht
- 2,622
95
votes
16 answers
Most 'unintuitive' application of the Axiom of Choice?
It is well-known that the axiom of choice is equivalent to many other assumptions, such as the well-ordering principle, Tychonoff's theorem, and the fact that every vector space has a basis. Even though all these formulations are equivalent, I have…
Tony Huynh
- 31,500
95
votes
8 answers
Mistakes in mathematics, false illusions about conjectures
Since long time ago I have been thinking in two problems that I have not been able to solve. It seems that one of them was recently solved. I have been thinking a lot about the motivation and its consequences. Mostly because people used to motivate…
user39115
- 1,785
95
votes
2 answers
Perfectly centered break of a perfectly aligned pool ball rack
Imagine the beginning of a game of pool, you have 16 balls, 15 of them in a triangle <| and 1 of them being the cue ball off to the left of that triangle. Imagine that the rack (the 15 balls in a triangle) has every ball equally spaced apart and all…
Phedg1
- 989
94
votes
4 answers
How can there be topological 4-manifolds with no differentiable structure?
This is a very naive question, and I'm hoping that it will be matched by a correspondingly elementary answer. It is well known that not every topological 4-manifold admits a smooth structure. So what's wrong with the following very sketchy proof…
gowers
- 28,729
94
votes
37 answers
What's a great christmas present for someone with a PhD in Mathematics?
Christmas is just around the corner and I haven't bought all the gifts for my family yet ( yeah, )
My Dad has a PhD in Mathematics, he works in Graph theory and his thesis was about Quasiperiodic tilings.
What do you think would make a good gift for…
Antenor Briareo
- 131