A beautiful analytic proof of a number theoretic result — For today’s interesting problem, we are looking to find all positive integer pairs n and m which satisfy the equation above, when n and m are distinct. If we play around with small n and m we can quickly see that 2⁴ = 16 = 4² , so the pair…

A formula that seems to be taught less and less in school today — The formula above gives you the surface area of the shape that is formed from rotating a curve y = f(x) around the x-axis (excluding the surfaces at each end of that shape, if any). It used to be very commonly taught in high school curricula back in the day…

How the hunt for an Irish cook led to today’s rich understanding of network propagation — In the summer of 1906, the Warrens — a large, well-to-do New York family — were having dinner in their vacation house in Oyster Bay, Long Island. The most anticipated part was dessert. …

Rigorous mathematical thinking is missing in most decision making environments — Recently I was having a conversation with a friend who was frustrated with his current situation at work. He was responsible for driving a major software rollout in his sales force, but he didn’t believe in what he was doing. Neither, it turns out, did any of his colleagues. …

What is the average area of all triangles transcribed inside a semicircle with one vertex at the center? — I spent an hour last weekend looking at a problem related to the average area of all triangles transcribed inside a semicircle of unit radius. …

How a new field of discrete mathematics helped London become less smelly — One July morning in the mid-19th century, the people of London awoke to a disgusting, gut-twisting stench. They couldn’t leave their homes without being sick. The more well-off laced their handkerchiefs with perfume and walked around permanently covering their faces. Many of the poor left town to find work in…

From humble origins, graph theory is shaping our world — In the early 1700s, the good people of Königsberg in Prussia had a problem on their minds. …

Pushing past the formula to build a visual understanding of the concept of integration by parts — Recently I asked myself: if I was a mathematics high school teacher, how would I push my brightest students and prepare them to be great mathematicians? I thought back to my time studying math in high school, and I realised that much of the math curriculum is fed to students…

Test your understanding of density and distribution — I recently came across this slightly mad looking problem which I thought was a nice test of knowledge of elementary statistics and I thought I would share it with you. The problem comes in three parts. Let’s start with part one. Part One The prevailing winds blow in a constant southerly direction…