NQOTW

Minimizing a Function of Two Variables: Multiple Methods

(A new question of the week) A recent question from a student working beyond what he has learned led to an interesting discussion of alternative methods for solving a minimization problem, both with and without calculus. The problem The question came from Kurisada a couple months ago: f(x, y) = x2 – 4xy + 5y2 – 4y …

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Awkward Sequence Problems

(A new question of the week) Having just discussed some pattern or sequence problems that were poorly posed, let’s look at some recent questions about sequences, some of which are quite complicated, and others seem to be just wrong. Two extreme patterns Here is the initial question from Zehra: Please make me understand how I …

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Permutations and Combinations: Undercounts and Overcounts

(A new question of the week) We have been looking at some combinatorics questions, both easy and challenging. Some questions have come to us in recent weeks that can illustrate how to think your way through relatively difficult problems, including catching errors and interpreting a textbook’s solutions. We’ll see yet again that there are usually …

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Distances to an Arc: Exact and Approximate Formulas

(A new question of the week) It can be an interesting challenge to be presented with a formula and asked how it was derived. This becomes a bigger challenge when the formula is only approximate, so we have to figure out how to arrive at this particular approximation. But it is impressive when several different …

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Greatest Common Divisor: Extending the Definition

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? Negative GCD? Here is the …

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