Last time, we looked at the Pigeonhole Principle, applying it to geometrical problems, largely about distances, gradually working from almost literal “balls and boxes” (“pigeons and pigeonholes”) to more abstract applications that are harder to see. Here, we will go beyond that, proving facts about sets.
A recent question involved the Pigeonhole Principle; we’ll start with an older question to introduce the idea, then the new question and a few old ones.
(A new question of the week) Suppose we have a question that can be answered with Yes, No, or Maybe, and that whenever two people with different opinions meet, their discussion convinces each of them that neither can be right, so they both change to the other opinion. Given initial numbers of people with each …
A recent question from a student demonstrates that not everything on the Internet should be taken at face value – and that it’s easy to think you are right when you are not.
(A new question of the week) Last week we looked at how the adjugate matrix can be used to find an inverse. (This was formerly called the [classical] adjoint, a term that is avoided because it conflicts with another use of the word, but is still used in many sources.) I posted that as background …
(A new question of the week) I find it interesting to observe the process of problem-solving, particularly for proofs: how we discover a solution initially, and then how we turn that into a final answer. Sometimes we can see the main idea in a flash, but the process of writing it as a formal proof …
A new question of the week A simple question about the Basic Proportionality Theorem (which goes by several other names, including Thales’ Theorem) and its relationship to similar triangles leads to some helpful ideas about how to choose a suitable manipulation at each step in a proof, a skill central to good problem solving. In …
(A new question of the week) Proving a trigonometric identity can be a challenge; sometimes even when we read someone else’s proof, we can fail to see how they came up with a seemingly magical step. We’ll look at two such identities here, and consider how to bridge a gap when we are stuck.
(A new question of the week) Two weeks ago, in Proving Certain Polynomials Form a Group, we joined a beginner in learning about groups. Here we will pick up where that left off, learning how to prove that the group we saw there, a subset of polynomials, is isomorphic to a group of matrices. As …
Last week’s exploration of a problem involving the Fibonacci sequence, has led me to delve into that and related concepts. In order to say much about the Fibonacci numbers, we have to first explore the concept of proof by mathematical induction. We’ll introduce it here, and then dig deeper next time.