Grid Colouring
How many ways are there to colour a 3x3 grid using 2 colours such that no rectangle in the grid has the same colour in all 4 corners?
How many ways are there to colour a 3x3 grid using 2 colours such that no rectangle in the grid has the same colour in all 4 corners?
If we randomly select 3 vertices of a regular hexagon to form a triangle, what are the respective probabilities that the center of the hexagon is i) inside, ii) outside, iii) on the edge of the triangle?
Let’s consider the 3D version of regular planar angles, called solid angles. If an angle can be thought of as a portion of a full circle, then a solid angle is a portion of a full sphere.
If a quadrilateral has 1 vertex on each side of a square of side 1, its smallest possible perimeter is 2sqrt2.
In the square below, points that are closer to the center than one of the vertices is coloured yellow. The area of the yellow region is 1/2 of the area of the square.
A circle of radius 5 has chord AB with length 6. A second chord is drawn here and is bisected by AB.
Let’s remove the central portion of a regular tetrahedron such that what remains are 4 tetrahedrons each with half of the side length of the original.
Here’s a way to put 6 dots into a 3x3 grid such that no 3 dots form a straight line.
For any point inside a square, the sum of the distances from this point to the four sides is half of the perimeter.
Let’s draw a smaller square inside a unit square by dividing out a fixed fraction of each side and connecting the dividing point to an opposite angle.