# Alternatives

## Fine Points on Polygons and Polyhedra

Last time, looking at degenerate polygons, I mentioned some other issues pertaining to the definition of a polygon. Let’s take the opportunity to look at them. This post supplements what was said previously in What is a Polyhedron … Really?

## Combinatorics: Multiple Methods, Subtle Wording

(A new question of the week) With few new questions of general interest available this week, I thought I’d go back a few months to a couple little questions on a topic we haven’t dealt with lately, combinatorics. We’ll have one question each on permutations and combinations, showing some subtlety in both the methods we …

## A Hole in a Cube

(A new question of the week) Here is an interesting little question about how drilling a hole affects volume and surface area. We’ll have one answer, and several explanations.

## Two Sinusoidal Models

(A new question of the week) Two recent questions involved using trigonometric functions to model real-life (or nearly so) situations, one about breathing, the other about a Ferris wheel. Both can be done by writing a sinusoidal function; the second can be done in other interesting ways as well.

## Function Transformations Revisited (I)

(A new question of the week) Transformations of functions, which we covered in January 2019 with a series of posts, is a frequent topic, which can be explained in a number of different ways. A recent discussion brought out some approaches that nicely supplement what we have said before. Here, the focus will be on …

## Solving a Triangle: What Went Wrong?

Trigonometry can be a powerful tool for solving sides and angles in triangles. But you have to be careful with it! We’ll look at a classic type of error in solving an SSA triangle, get three explanations, and then see how knowing the context of a question can change our answer – to the point …

## A Triangle Problem: With Trig, and Without

(A new question of the week) When you are given a problem about a triangle, there can be many ways to approach it: pure geometry, trigonometry, and analytic geometry come to mind. When the context doesn’t dictate a method (as turns out to be true here), you just have to try what feels right to …

## Summing Squares: Finding or Proving a Formula

Last week we looked at problems about counting the squares of all sizes in a checkerboard. Some solutions required finding the sum of consecutive squares, $$1^2+2^2+3^2+\dots+n^2$$, for which we used a formula whose derivation I deferred to this week. Here we’ll see a couple proofs that require knowing the formula ahead of time, and a …

## Application of Vectors: Airplane in the Wind

A recent question about the resultant velocity of an airplane illustrates different ways to make a diagram showing the bearings of air velocity and wind velocity, and to work out angles without getting too dizzy.

## How Many Paths from A to B?

A popular kind of question in combinatorics is to count the number of paths between two points in a grid (following simple constraints). This can be done by very different methods at different levels. We’ll look at several problems of this type, starting with the simplest.