(An archive question of the week) Last time we looked at some elementary problems in combinatorics, where we counted the number of ways to choose or arrange elements of a set. Let’s look at a somewhat more complicated problem, which will demonstrate issues that come up in interpreting such a problem and in choosing a …
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Back in May, I wrote about pattern and sequence puzzles, and didn’t have the space to cover all that I would have liked. It’s time to revisit the topic, looking at a couple different types of sequences, and then the “input/output” or “function” puzzles that add an extra twist to the idea.
(A new problem of the week) Having discussed counting earlier this week, let’s take a look at a different kind of counting. The subject of combinatorics (the study of counting) arises in many guises: probability, sets, geometry. Here, we look at a relatively basic type of problem that involves the same sort of organized thinking …
Over the years, we have had many questions, often from young students, asking how to count the parts (faces, edges, vertices) of a polyhedron (cube, prism, pyramid, etc.). The task requires understanding of terms, visualization of three-dimensional objects, and organizing the parts for accurate counting — all important skills. How can we help with this?
Using the epsilon-delta definition of a limit in calculus can be challenging. (That’s why, after using it for a few examples, we derive some easier techniques, and never use the definition directly unless we have to!) We’ll start with an overview of what the definition means, and then look at several examples of how it …
(A new question of the week) We have had a number of challenging questions about inequalities recently. I want to show one of those here, because it involved a useful discussion about how to prove them. A proof problem Here is the question: Q) Show that a^4 + b^4 >= (a^3)b + (b^3)a for all …
(A new question of the week) Since we’ve been doing a little trigonometry this week, let’s look at a recent set of questions from a student in the Philippines, all about compositions of trig and inverse trig functions. This is a topic with some interesting little surprises, and several of us answered, to give a …
(An archive question of the week) Last time we looked at various 2- and 3-set Venn diagram problems (and alternative methods). One discussion I found while collecting them deserved to be set aside for special examination, if only because it would scare the beginner. A mixture of tools will make the work easier. Here it …
Puzzles involving two or three properties of people or objects, commonly solved by Venn diagrams, are popular with teachers, and perhaps not so popular with students! Several have asked about them recently, leading me to catalog our past answers about them, to find the most useful examples to point to. Some of these are very …
(A new question of the week) Sometimes we give only minimal help on a question, to let a student puzzle over the problem and learn to figure it out herself. That may be enough, or we may go back and forth for some time, guiding her thinking until she finds the answer. In the latter …