Questions tagged [combinatorial-geometry]

Combinatorial geometry is concerned with combinatorial properties and constructive methods of discrete geometric objects. Questions on this topic are on packing, covering, coloring, folding, symmetry, tiling, partitioning, decomposition, and illumination problems.

74 questions
378 views

Tiling problem: 100 by 100 grid and 1 by 8 pieces

Why can't I tile a $100 \times 100$ table with $1$ by $8$ pieces? If we look at the number of tiles, $100^2$ is divisible by $8$. So this does not contradict existence of such tiling. The standard ...
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Cases where ANY 2 of 3 +/- choices select one of four possible elements

Edited 1/21/2018 to add the following: Here is a DropBox link https://www.dropbox.com/s/7rtt0iqmgimsgzu/Zumkeller_edge-magic.pdf?dl=0 to a PDF showing how my team used biomolecular first ...
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Squaring the plane, with consecutive integer squares. And a related arrangement

Q1. I was fiddling around with squaring-the-square type algebraic maths, and came up with a family of arrangements of $n^2$ squares, with sides $1, 2, 3\ldots n^2$ ($n$ odd). Which seems like it would ...
201 views

Is there an equidissection of a unit square involving irrational coordinates?

An equidissection of a square is a dissection into non-overlapping triangles of equal area. Monsky's theorem from 1970 states that if a square is equidissected into $n$ triangles, then $n$ is even. ...
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Number of ways to distribute objects, some identical and others not, into identical groups

The question I initially thought of that prompted this was "How many distinct integer-sided cuboids are there with a volume of $60^3$?". A small example to clarify: There are $3$ integer-sided ...
532 views

Shooting Game for Fun

Trigger Warning: Murder is mentioned. Let there be $n>1$ people (players) on a plane, each having a loaded gun and each being a perfect shot (assuming that each bullet is laced with one gram of ...
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Positivity of the alternating sum associated to at most five subspaces

Let $V_1 , V_2 , \dots , V_n$ be vector subspaces of $\mathbb{C}^m$ and let $$\alpha = \sum_{r=1}^n (-1)^{r+1} \sum_{ \ i_1 < i_2 < \cdots < i_r } \dim(V_{i_1} \cap \cdots \cap V_{i_r})$$ ...
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Coloring $\mathbb R^n$ with $n$ colors always gives us a color with all distances.

I wanted to share a really cool but simple problem. Consider a coloring of the points of $\mathbb R^n$ with $n$ colors. Prove that there is a color $c$ such that for any $r>0$ there are two points ...
For what positive real numbers $x,y$ can an $x\times y$ rectangle be partitioned into a finite number of squares? When $\dfrac{x}{y}$ is a rational number, it is not hard to see that we can partition ...