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(I assume that the OP wants to minimize is the number of turns rather than the total amount of absolute turning angles.) If we restrict to moving only parallel to the axes, here is an elementary proof …

Define the Pythagorean Graph as having nodes $a,b\in \mathbb{N}_{\ge 3}$ and an edge $a\rightarrow b$ if and only if $a^2+b^2$ is a square. After much searching I found the example in the picture, pro …

Consider an $n\times n$ real matrix $A=(a_{ij})$ with non negative entries. Assume that - the sum of the three largest entries in each row is a constant $R$ (the same for all rows), - the sum of the …

This answers question 2 as well. But I think both questions are way more suitable for math.stackexchange. I add another example because, unlike the first, it has the property that multiplied by $I^{n …

Not an answer - but I decided to delete a prior comment and repost as an answer, because I think it puts the 2-4-6-8 conjecture in a different light than considered so far, hopefully leading to some o …

The first two equalities imply $x>m$ and $y>n$ so one can substitute $x=m+X$, $y=n+Y$ and $k=X+Y$, with still $X,Y \in \mathbb N$: ${X \choose 2}=nX+nY-mX\tag{1}$ ${Y \choose 2}=mX+mY-nY\tag{2}$ Fr …

Let $S$ be a set with $n$ elements and $\Sigma_k= \{ R\subseteq S \mid |R|=k \}$. For $k\le n/2$ how many bijections $f$ are there between $\Sigma_k$ and $\Sigma_{n-k}$, such that $x\subseteq f(x)$? …

I think $\big\{-1, 0, \frac{1+\sqrt{5}}{2}\big\}$ is a counterexample. THIS IS WRONG, see comments, but I'll leave it up as a warning.

There is a new upper bound of 254/67 (= 3.79104477...). Define 6 sets of cardinality 4: X1={-B+A, 0, A, B} Y1={0, A, B-A, B} X2={-B, -B+3A, B-2A, B+A} Y2={-2B+A, -A, B+A, 3B-A} X3={-6B+2A, -2B-2A, …

Here is a summary for the $\mathbb{R}^2$ situation. Upper limit: no improvement over the 3.8 known on $\mathbb{Z}^2$. To create specific examples I modified the $\mathbb{Z}^2$ ones by uniformely spr …

The upper bound is <3.95. I hope the code below will show correctly... It proves that assuming a sum >=3.95 in the central AxB rectangle of the grid ({-B,-B+A,-2A,-A,0,A,2A,B-A,B}+{0,A}) x ({-2B,-B- …

Almost 25 years ago a professor at Indiana U showed me the following problem: given a map $\mathbb{Z}^2\rightarrow\mathbb{R}$ such that the sum inside every square (parallel to the axes) is $\leq1$ i …