# Strategies

## Overlapping Square Tiles

(A new question of the week) An interesting geometry question came to us in July, about the area of overlap between two squares. The discussion was not long, but leads to some interesting ideas. Here is the initial question, from Vignesh: This is the problem I need help with: Two square tiles of area 9 …

## Challenging Inverse Trig and Polynomial Equations

(A new question of the week) We have had a lot of interesting questions recently. This one involved inverse trigonometric equations that led to cubic and quartic equations. We’ll observe here one of the benefits of embedding the original discussion in a blog format where I can add information that will help you, the reader, …

## Proof by Contrapositive with Quantifiers

(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 …

## Proving Proportions, Problematic Products

(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 Rate Problem: Two Speeds, Two Ways

(A new question of the week) A question we got at the end of March asked about a standard kind of algebra word problem that can be solved in a couple very different ways. It illustrates several choices that can be made (both about the meaning of the problem and how to solve it), as …

## Impossible? Try anyway!

(An archive question of the week) Here’s a little problem with some big lessons for problem solving. Not enough equations? This question is from Kenanga in 2017, not long before the Ask Dr. Math service stopped taking questions: Insufficient Information In Question Given A + B + C = 1 B + C + D …

## Logic Puzzles: Who Owns the Fish?

Having looked at methods for solving four logic puzzles, I’ll close the series with a classic, this one involving five houses, five colors, five countries, five drinks, five cigarettes, and five pets – that’s six attributes to juggle! (Last time, we had only 3 or 5.) Once again, the challenge is to solve it without …

## Logic Puzzles: Five Murders, Five Friends

Here are two more logic puzzles similar to those we looked at last time. Each is partly solved, enough to teach how to approach such problems (this time, using tables), but leaving enough of a challenge for you to finish up. What’s new here is that we will be using tables to keep track of …

## Logic Puzzles: Five Jars, Five Teams

We have received a number of questions about various kinds of logic puzzles. This week I want to collect several for which we gave hints or partial solutions that demonstrate in detail how to think, without taking away all the fun. These all involve a set of people or objects that have a set of …

## “Order of Operations” Puzzles

We have often received questions about things called “Order of Operations problems”, or some similar name. Generally, what that means is simply that they are puzzles to give lots of practice evaluating expressions using the order of operations. I have collected a few quite different puzzles in this broad category. A free-form expression We’ll start …