There are $P$ points in the 2-dimensional plane. Through each point, we draw two orthogonal lines: one horizontal (parallel to x axis), one vertical (parallel to y axis). Obviously, some of these lines conicide. How can we arrange the points such as the total number of lines is minimized? I started with solving the dual […]

How many possible shapes can one make by rearranging $n$ square shaped blocks, with and without allowing rotational symmetry? For example, for $n = 4$, there are seven possible shapes after discounting rotational symmetry, as in Tetris. How does this number change if one of the blocks is coloured – distinguishable from the rest? For […]

Given $n$ points on the plane, no three of which are collinear, what is the maximum number of acute triangles formed by them? [Source: Based on Hungarian competition problem]

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. In 1968, John Thomas proved the following weaker statement: there is no equidissection of a unit square into an odd number of triangles […]

An L block that is 3 unit blocks high and 2 unit blocks wide . It is true that if an n by m rectangle can be covered by such L blocks with out overlap that we would require an even amount of L blocks, why?

If $S\subseteq \mathbb{R} ^2$ is closed and convex, we say $S$ is strictly convex if for any $x,y\in Bd(S)$ we have that the segment $\overline{xy} \not\subseteq Bd(S)$. Show that if $S$ is compact, convex and constant width then $S$ is strictly convex. Any hint? Than you.

(Brouwer) Any continuous function from a convex compact subset K of a Euclidian space to itself has a fixed point. Given this lemma, is there a simple proof of: (Borsuk-Ulam) Any continuous function $f \, : \, S^n \to R^n$ (where $S^n$ is the $n$-sphere) has a point $x$ for which $f(x) = f(-x)$. ?

Given $m \times n$ rectangle with area $A$, and $m,n \in \mathbb{N}$. Let $S_k(m,n)$ be the number of way to dissect this rectangle into $k$ non-overlapping triangles whose area is $\frac{A}{k}$. It is known that when $k$ is odd $S_k(n,n)=0$ (For example see Paul Monsky, “On Dividing A Square Into Triangles“, American Mathematical Monthly, Vol […]

Given a positive integer $n\geq 2$, consider all the lattice points with each coordinate between $1$ and $n$, inclusive. At least how many points must we choose to guarantee that some four points form a rectangle with sides parallel to the axes? $2n-1$ points do not suffice, because we can select the points $$(1,1),(1,2),\dots,(1,n),(2,1),(3,1),\dots,(n,1),$$ then […]

Let $n$ be a positive integer. A subset $S$ of points in plane satisfies the following conditions: a) We can’t find $n$ lines in plane, such that every element of $S$ belongs to at least one of these lines. b) For every $X\in S$, we can find $n$ lines in plane, such that every element […]

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