Line Stitching
Consider all 1 unit long line segments that have an end point on the line y=0 and the other on x=0.
Inscribed Perimeters
If a quadrilateral has 1 vertex on each side of a square of side 1, its smallest possible perimeter is 2sqrt2.
Symmetrical Sets
A set of 3 real numbers satisfies the symmetry that any number added to the product of the other 2 will result in 2. There are 2 such sets: all 1s and all -2s.
Distances to Vertices
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.
Number Line Mirrors
Starting with some points on a number line, if we put a mirror somewhere on the number line, it will reflect all existing points in itself so that the resulting set of points are symmetrical around it.
Area-Bisecting Lines
Every line going through the center of a square bisects its area. Does every line going through the center of an equilateral triangle bisect its area too?
Circle-Bisecting Lines
All straight lines that bisect the area of a circle go through its center.
Chord-bisecting Chords
A circle of radius 5 has chord AB with length 6. A second chord is drawn here and is bisected by AB.
Friendly Factor Pairs
The number 6 has two pairs of factors: 2x3 and 1x6. These two pairs of factors are called friendly because the sum of one pair is equal to the difference of the other pair, ie. 2+3 = 6-1 = 5.
Sierpinski Pyramid
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.