Combined Chords
Two circles intersect at P and Q. Let’s call line AB through P a combined chord if AP and PB are chords of the respective circles. Let’s define the split product of AB as the length of AP times the length of PB.
Linear Flooring
The floor function [a] denotes the greatest integer less than or equal to a. For x and y strictly between 0 and 1, the probability of [2x]+[2y] being 1 is 1/2.
Intersecting Circles
If we draw pairs of circles of the same sizes centered at two distinct points and mark where each pair of circles intersects, we’d trace out a straight line.
Digit Removal Division
Which digit can we remove from an n-digit number such that all that are divisible by the resulting number would end in 0s?
Stepwise Codes
Let’s arrange all 3 digit binary codes in a list starting with 000 such that each code differs from its previous code by only 1 digit.
Connected Lights Continued
Here are 4 connected lights (where flipping one switch on a light will change the state of itself and its neighbours) in a circle, initially all off.
Connected Lights Revisited
Flipping the switch of one light in this chain of connected lights will change the state (on O or off X) of the corresponding light (highlighted) and its immediate neighbours.
Enclosing the Center
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?
Lucas Sequences
A Lucas Sequence is made from two starting terms and a recursive formula for the next term using the two previous ones.
Solid Angles
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.