(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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We have seen a number of questions recently about combinatorics: the study of methods for counting possibilities. These topics are studied at all levels of mathematical education, from elementary (where they might just be called counting) to high school (where they are often learned along with probability) to college (where they are part of “discrete …
(An archive problem of the week) While gathering sequence/pattern questions for my last post, I ran across a very different problem. Here we are told what the pattern is (a good example of one that you would probably never discover on your own), and asked some questions about later terms. It can be understood either …
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.
Having just talked about definition issues in geometry, I thought a recent, short question related to a definition would be of interest. We know what the Greatest Common Divisor (GCD, also called the Greatest Common Factor, GCF, or the Highest Common Factor, HCF) of two numbers is; or do we?
(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 …
(An archive problem of the week) A couple weeks ago, in discussing the value of estimates, I included one example of a (very simple) Fermi problem: one in which it is necessary to invent the data as well as the method of solution. Today, I will examine one answer in which we dug deeper into …
(An archive question of the week) Last time I looked at reasons for learning to estimate. In searching for answers on that topic, I ran across a question that touches not just on reasons for estimation, but on other ways to check an answer, and on some of the specific ideas we will be looking …
Many questions we have received have been about various aspects of estimation. Often this topic has been downplayed, because we tend to think of math as being all about precision; but it is essential in many applications, sometimes because there is nothing else to do, and other times because exactness wastes effort. I am starting …
I want to close out this series on multiplication with a very different kind of question. We have seen that multiplication of natural numbers can be modeled as a repeated sum of the multiplicand, taken the number of times indicated by the multiplier; and that the terms “multiplier” and “multiplicand” reflect only this model, not …