43
votes
In how many different ways can a 9-panel comic grid be used?
Suppose we create a layout by deleting borders between the panels of a
$3\times 3$ grid, as has been suggested at various times.
But let's put some restrictions on which borders we can delete so that ...
- 100k
31
votes
In how many different ways can a 9-panel comic grid be used?
While this is not so much of a mathematical solution as a software one, I'm going to add it anyway, if only for the nice image at the end.
The process used to find all of the grids is as follows.
...
- 411
24
votes
In how many different ways can a 9-panel comic grid be used?
Unfortunately these kinds of geometrical questions rarely have a neat satisfying answer, like a formula for any sized grid. This is because a "divide and conquer" strategy does not work - as soon as ...
- 13.8k
18
votes
Accepted
What is the exact value of the radius in the Six Disks Problem?
The minimal polynomial for $r(6)$ is
$$\color{red}{7841367r^{18}-33449976r^{16}+62607492r^{14}-63156942r^{12}+41451480r^{10}-19376280r^8+5156603r^6-746832r^4+54016r^2+3072}$$
I obtained this result in ...
- 104k
14
votes
Tiling the plane with consecutive squares
Here is a solution with $n=7$. Edit. I added some comments expressing my belief that $n=7$ may well be the largest solution. (Well, I keep editing my answer, but at the end I came up with a solution ...
- 13.5k
13
votes
Accepted
Prove that the shortest side of one triangle is the longest side of another, given 3 pairs of points.
Maybe there's a more general combinatorial method to this interesting problem.
I was so scared by the length of your answer that concentrated all my mind power for an insight about such a method and ...
- 93.3k
12
votes
Accepted
How many circles do 11 points define?
It doesn't take two points to define a circle, because either of those two points could be the centre. Instead, the solution for part (a) is simply $\binom{11}3=165$ because three non-collinear points ...
- 104k
11
votes
Number of paths in a grid below a diagonal
We know that there are $\binom{2n}{n}$ ways of going $(n,n)$ from $(0,0)$ when there is no restriction. So the answer should be $\binom{2n}{n}-B$ where $B$ is the number of "bad paths", that is, ...
- 13.3k
11
votes
Accepted
Radius of inscribed sphere of n-simplex.
A $n$-simplex can be embedded in $\mathbb{R}^{n+1}$ by choosing $(1,0,0,\ldots),(0,1,0,\ldots),\ldots$ as vertices.
By this way the centroid of the simplex lies at $\left(\frac{1}{n+1},\frac{1}{n+1},\...
- 355k
11
votes
Accepted
British Mathematical Olympiad - December 2001 - Round 1 - Question 4
Number the people around the table $1,\ldots,12$.
As an example, the number of ways that persons $1,\ldots,12$ can engage in handshakes with no arms crossing given that person $1$ is shaking hands ...
11
votes
Accepted
Points in plane with every pair having at least two equidistant points?
A picture is worth a thousand words:
... but I will write a few words, nevertheless. This configuration has many symmetries and interesting properties, some of which I will describe here; however, I ...
- 346
10
votes
A largest subset of a cubic lattice with unique distances between its points
For each $r \in [1, \ldots, 3 n^2]$ let $E_r$ be the set of unordered pairs $(p,q)$ of points in $S_n$ such that $\|p-q\|^2 = r$. We want a largest possible subset $X$ of $S_n$ such that $X$ contains ...
- 455k
10
votes
In how many different ways can a 9-panel comic grid be used?
For square grids, the "Number of ways of dividing an $n\times n$ square into rectangles of integer side lengths" is OEIS sequence A182275. There are no formulas, but for $n\le12$ the following numbers ...
- 79k
10
votes
Accepted
Venn diagram with rectangles for $n > 3$
Solving the given problem I tried different approaches, obtaining better and better bounds. Also the last bound is the best, I leave the other too, because they can help to solve similar problems.
...
- 93.3k
10
votes
Accepted
Several rectangles cover the unit square. Can I find a disjoint set of them whose area is at least $1/4$?
The question linked by @AlonAmit in the comments answers exactly this question, and shows that the answer (at least with the constant $1/4$) is no. For a concrete demonstration, start with a $6\times ...
- 42k
10
votes
Tiling the plane with consecutive squares
$n=3$, $n=4$, $n=5$ all tile the plane:
Each of these 'symmetrically bitten rectangle' shapes tiles the plane by translation (e.g., attach them along opposite long sides to form diagonal bands, then ...
- 52.4k
10
votes
Accepted
Draw 7 lines on the plane in an arbitrary manner. Prove that for any such configuration, 2 of the those 7 lines form an angle less than 26◦
Big hint, with seven lines differing maximally in angle:
Because you are interested just in the relative angles, you can arbitrarily shift each line to go through the same point (the origin).
Do you ...
- 30k
9
votes
Divide the chessboard
Big hint
The centers of the chessboard form an $8$ times $8$ grid of dots. There are $28$ line segments on the border of this grid which connect two adjacent border dots. Namely, there are $7$ ...
- 78.3k
9
votes
British Mathematical Olympiad - December 2001 - Round 1 - Question 4
Let $S_{2n}$ = ways for for 2n people to shake n pairs, without crossing.
Example, for 4 people ABCD, we have 2 ways: (AB,CD),(AD,BC)
$S_2 = 1$
$S_4 = 2S_2 = 2$
$S_6 = 2S_4 + S_2 S_2 = 4+1 = 5$
$...
- 2,124
8
votes
A largest subset of a cubic lattice with unique distances between its points
Promoting the comments to an answer, as the upper bound seems to be surprisingly good early on.
We want to avoid repetitions of squared distances. The squared distance $d$ between two points in $S_n$ ...
- 135k
8
votes
Accepted
Determine all convex polyhedra with $6$ faces
At Canonical Polyhedra. you can get the seven hexahedra and their duals. These are your 11, 2, 1, 3, XX, 7, 4. You are missing the (3,3,4,4,4,4) case. Vertices {{-0.930617,0,-1.00},{0.930617,0,-1....
- 21.4k
8
votes
How many different graphs of order $n$ are there?
In general these counts do not have nice closed formulas, but some satisfy nice recurrence relations.
Graphs
Graphs on $n$ nodes is OEIS A000088: $$1, 1, 2, 4, 11, 34, 156, 1044, 12346, 274668, \...
- 102k
8
votes
Accepted
How many walks are there from $(0,0)$ to $(N, r)$ on $\mathbb Z^2$ along diagonals?
We consider OP's problem in a slightly more convenient (symmetrical) setting:
Let $0\leq n\leq m$. We are looking for the number $L_{m,n;r,s}$ of lattice paths starting in $(0,0)$ and ...
- 109k
8
votes
Given an $n\times n\times n$ cube, what is the largest number of $1\times 1\times 1$ blocks that a plane can cut through?
This answer solves the $3\times 3\times 3$ case and makes a conjecture about further cases.
To give an answer, first imagine how we can create the given $n\times n\times n$ cube in the first place: ...
- 61.1k
8
votes
Accepted
Combinatorial Geometric proof of $\binom{\binom{n}{2}}{3} > \binom{\binom{n}{3}}{2}$
The set we are counting
This is a geometric proof, so I'm avoiding e.g. set counting or any algebra. I started this proof before a comment appeared as well which was related to it, so it's quite ...
8
votes
Does a random set of points in the plane contain a large empty convex polygon?
No, there are only small empty convex polygons
@BillyJoe has discovered that Balogh, Gonzalez-Aguilar and Salazar (1) solved this question in 2012. They showed that a random set of points contains, on ...
7
votes
Geometric way to view the truncated braid groups?
An idea to search for a structural answer would be as follows.
The five groups are complex reflection groups, i.e. generated by reflection symmetries $s$ in a finite dimensional vector space $V$ over $...
- 34.2k
7
votes
Polyhedron, understanding face vs facet.
A facet is just a special type of face.
According to Wikipedia:
The facets of an $n-$polytope are the faces of the polytope with dimension $n-1$.
However a face can have many more dimensions. So, ...
- 21.3k
7
votes
Accepted
Can every polyhedron be inscribed in a sphere?
This is not possible in general, or even in particular when $n=3$. See this paper of Ziegler for a reference.
I would guess that this is possible if I allow the sphere to have arbitrarily high ...
- 461
7
votes
Accepted
How many $n$-pointed stars are there?
What we seek are: nontrivial cyclic permutations, with shifts and reversals counting as identical.
Without the shift requirement, there are $(N-1)!$ different cyclic permutations. Shifting allows us ...
- 11.2k
Only top scored, non community-wiki answers of a minimum length are eligible