Last week we looked at problems about counting diagonals in a polygon, and the very similar problem of counting handshakes when everyone in a group shakes with everyone else. In the course of searching for those problems, I also found some very different problems that are also about handshakes. We’ll look at those here, just …
(A new question of the week) Last week we looked at a recent question about an attempt to write a proof using the contrapositive, which was foiled by difficulty in negating a statement. Two weeks later, we had another question about the same sort of issue, but with a different problem in the negation. In …
(A new question of the week) A recent question raised some interesting issues related to the contrapositive of a logical statement, and how to negate a statement, similar to some past discussions. What universe you are in makes a big difference! Proof by contrapositive The question came from Kalyan, in June: My question is this: …
(A new question of the week) A recent question provided an opportunity to examine some ideas about ratios, and also ways to tame a potentially huge product. Proving one proportion from another Here is the question, received from Sithum in late April: Hi Dr. Math! There are some problems that involve a given ‘equation with …
(A new question of the week) Although we focus in this blog on questions at early college level and below, we do get questions at higher levels. This one deals with finding an invariant for a finite state machine, with possible movements of a robot as the example. First problem (relatively easy) The question came …
Last week we looked at several proofs of the Law of Sines. Here we will see a couple proofs of the Law of Cosines; they are more or less equivalent, but take different perspectives – even one from before trigonometry and algebra were invented! Proof using coordinates First, here is a question we looked at …
Two of the most useful facts in trigonometry are the Law of Sines and the Law of Cosines (sometimes called the Sine Rule or Sine Formula, and the Cosine Rule or Cosine Formula). Over the years we were often asked where they come from (or are just asked about them, and reflexively offer proofs). There …
A classic problem we’ve seen hundreds of times involves students opening and closing lockers. I have often told people that, believe it or not, they could find the answer by searching the Ask Dr. Math site for the word “locker”. But I prefer to give them a reference to one of the answers in which …
(A new question of the week) Last week we looked at a recent question about basic trigonometric equations. That discussion continued into the subject of identities, which we’ll look at here. We’ll be sitting in on an extended chat about many important aspects of this kind of work. It’s still very long, even after extensive …
We have been looking at some classic “false proofs” or “fallacies”, where a seemingly valid proof shows something clearly false to be true. The goal is to learn from these, how to distinguish a valid proof from an error. In a post from last year, What Role Should a Figure Play in a Proof?, I …