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I'm an electrical engineer and having a tough problem with... math :) geometry and probability...
Here's the problem :
We have an infinite chessboard. Each square of the chessboard is of known height/width (x).
I have four points, distances between these points being FIXED. They can be arranged as a rectangle, square, diamond, on a circle...
Question : I want to throw the rectangle (or square, or diamond...) on the chessboard and get a 100% probability for two points of the rectangle to land on a white square and the two other points on a black square.
- What shape/dimensions regarding to x, do I have to adopt for this condition to be fullfilled ?
Thanks for your help :) !
Take any set of four vertices, $\{A,B,C,D\}$. We will only be focusing on three of them, however, as only three need to land on the same color in order to contradict your wishes.
$AB$ defines a line. Draw a line perpendicular to $AB$ that passes through $C$. You now have a cross that passes through $A$, $B$ and $C$. Now the cross can be placed on the grid like so:
$\hskip 250 px$
$A, B, C$ will all be on black squares. If only one or zero of the vertices is on the corner of a square, the cross can be translated or rotated within a probabilistically non-negligible area. This is also the case after the following transformations:
If all three vertices are on corners, the cross can be rotated slightly about its center so all are now on white. If only $A,B$ are on corners, the cross can be translated along $AB$ to get them off the corners. If only $C$ and one of $A,B$ are on corners, then the cross can be translated slightly horizontally or vertically and rotated slightly about its center.
Therefore, for every quadrilateral there is a nonzero probability that three of its vertices will land on the same color.