# Classroom Capsules and Notes

Capsules By Courses. We are organizing the capsules into courses, when possible using the same topics as are used in Course Communities. So far we have organized capsules for the following courses:

You may select topics within each course.

Notes: Sequences and Series is part of One-Variable Calculus. We felt that since this topic had so many capsules associated with it, we wanted to introduce sub-topics. Also, the Number Theory collection of capsules does not correspond to a course in Course Communities, but has topics selected by the Editorial Board for Classroom Capsules and Notes.

## Featured Items

##### Complementary Coffee Cups

This capsule  started with two coffee cups that are complementary, i.e., their profiles fit together. The author then explores the condition in which the two cups, obtained as solids of revolution, have the same volume.

##### Magic "Squares" Indeed!

A real matrix is called square-palindromic if, for every base $b$, the sum of the squares of its rows, columns, and four sets of diagonals (as described in the article) are unchanged when the numbers are read "backwards" in base $b$.  The authors prove that all $3 \times 3$ magic squares are square-palindromic.  They also give sufficient conditions for  $n \times n$ magic squares to be square-palindromic, which include all circulant matrices and all symmetrical magic squares.

##### A Quick Change of Base Algorithm for Fractions

This note is on the digital (floating-point) representation in various arithmetic bases of the reciprocal of an integer $m$. An algorithm is given to change the representation of $1/m$ in base $b$ to its representation in base $b+mt$ for any integer $t$.

##### The Fresnel Integrals Revisited

Without using double integrals, a new method is presented to evaluate the two Fresnel integrals about sine and cosine that has a wider applications than previous methods.

##### Proof Without Words: Ordering Arithmetic, Geometric, and Harmonic Means

A visual proof of the ordering of the means in the title is presented.

##### Arc Length and Pythagorean Triples

A family of curves whose lengths are closely related to Pythagorean Triples and therefore rational