$40$ can be written as $7^2 - 3^2.$ Which other numbers can be written as the difference of squares of odd numbers?
Can you prove that triangles are right-angled when $a^2+b^2=c^2$?
Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the relationship between Euler's formula and angle deficiency of polyhedra.
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?
