Summary We provide some fractal-based irreducibility criteria for integer polynomials. The results are obtained using the self-similar pattern of Pascal’s triangle modulo two and by applying a restatement of the Schönemann–Eisenstein’s irreducibility criterion.

Summary We use Ptolemy’s theorem to prove Apollonius’s theorem for the length of a triangle median.

Summary We give yet another visual proof for the sum of cubes formula using two stacks of the sum of the first n cubes.

Summary We present a refined, simplified, and unified proof of Aristarchus’ inequality, without resorting to trigonometry, calculus or extended geometric arguments.

Summary We give short proofs that ℝ is uncountable directly from the definition of ℝ as the set of Dedekind cuts of ℚ .

Summary We show how a cake can optimally be split symmetrically among seven people using only three cuts.

Summary Wepresent a proof of Apollonius’s theorem via Heron’s formula.

Summary If we consider the Madhava–Gregory–Leibniz series π 4 = 1 − 1 3 + 1 5 − ⋯ , which is commonly referred to as the Gregory series, and then compare it to Euler’s formula π 2 8 = 1 + 1 3 2 + 1 5 2 + ⋯ , the following question arises: Can this latter formula be derived by squaring both sides of the former? There have been several proofs of Euler’s formula, or its equivalent formulation ζ ( 2 ) = π 2 / 6 , based on the idea of squaring 1 − 1 3 + 1 5 − ⋯ = π 4 , including a proof presented in a letter from Euler to Goldbach dating from 1742. We consider the history of proofs of this form, and we offer another simple proof of ζ ( 2 ) = π 2 / 6 that also relies on squaring Gregory’s series.

Summary We provide a concrete example of two constructible polygons in the plane whose common area bisecting line is not constructible.

Summary The convolution of the Fibonacci numbers and the Lucas numbers is equal to a simple multiple of a Fibonacci number. We show how this can be generalized to other Fibonacci-type and Lucas-type numbers. Along the way, we discuss generating functions and Newton’s Identities.