Comfy Math Thread

[Anonymous No.10420227]

Alright lads, post some comfy math problems to try out.
I'll start:
Prove that the greatest number of equal radius spheres which can fit around a central sphere of the same radius is 6.

[Anonymous No.10420230]

>>10420227
All spheres must be touching the central sphere*

[Anonymous No.10420977]

>>10420227
hexagons ennit

[Anonymous No.10421277]

Let [math] n [/math] be an odd integer [math] >2 [/math] and let [math] f(x)\in \mathbb{Q}[x] [/math] be an irreducible polynomial of degree [math] n [/math] such that the Galois group [math] Gal(f/\mathbb{Q}) [/math] is isomorphic to the dihedral group [math] D_n [/math] of order [math] 2n [/math]. Let [math] \alpha [/math] be a real root of [math] f(x) [/math]. Prove [math] \alpha [/math] can be expressed by real radicals if and only if every prime divisor of [math] n [/math] is a Fermat prime.

[Anonymous No.10421290]

>>10420227
>fit around
What did he mean by this?
Something like maximum number of spheres tangent to the original, probably.
[super]I'm also pretty sure it's false for spheres and it should be circle. Spheres feels like 8.[/duper]

[Anonymous No.10421296]

>>10421290
>Spheres feels like 8.
spheres are 12
take the hexagonal arrangement for a circle and set one of them touching the top half of the sphere and another touching the bottom

[Anonymous No.10422550]

>>10420227
>Alright lads, post some comfy math problems to try out.
>I'll start:
>Prove that the greatest number of equal radius spheres which can fit around a central sphere of the same radius is 6.

Something to do with there being 6 directions (2 for each axis), if they are of equal radius, then there will be no room for more spheres than 6 (pic related). Currently working on a formal proof.

[Anonymous No.10422589]

>>10422550
But where is the central sphere?

[Anonymous No.10422765]

>>10422589

In the middle of the hexagon in the picture, think of the radius of the central sphere being equal to half the distance between each sphere on the outside (therefore each sphere has the same radius)

[Anonymous No.10422803]

>>10422765
It's ok, this is a math thread. We forgive you for lack of reading comprehension.

[Anonymous No.10423121]

>>10420227
It's only 6 for 2D spheres anon, if you extend it to 3D you trivially get at least two spots to put new spheres on

[Anonymous No.10423161]

>>10422550
Wrong, you can fit at least 8

[Anonymous No.10423237]

>>10423161
Wrong, I managed 9 and I'm convinced it's the upper bound.

[Anonymous No.10423378]

>>10422589
>>10421290
>>10421296

OP here, I was thinking on a surface, so yeah, it should be circles