Browse Classroom Capsules and Notes

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Proof without Words: Sum of Squares

The sum of squares formula is proved visually.

The Reduced Row Echelon Form of a Matrix Is Unique: A Simple Proof

Using mathematical induction, the author provides a simple proof that the reduced row echelon form of a matrix is unique.

Proof without Words: Sum of Integers

A visual derivation of the formula for the sum of the first \(n\) integers is presented.

Guess a Number ─ with Lying

The author discusses the game "guess a number in 20 questions" if some incorrect answers are allowed.

Periodic Equilibria under Periodic Harvesting

The authors investigate the maximum number of periodic solutions for a population growth differential equation with two examples.

The Circumdisk and Its Relation to a Theorem of Kirszbraun and Valentine

The author addresses the task: "Given a finite set of points in Euclidean m-space, characterize the radius R of the smallest disk which contains those points."

An Infinite Series for \(\pi\) with Determinants

The author gives an expression for \(\pi\) involving an infinite sequence of determinants, each representing the area of a triangle.

Characteristic Polynomials of Magic Squares

An \(n \times n \) matrix whose rows, columns, and diagonal all sum to the same number \(m\) is called magic, and the number \(m\) is called the magic sum. If \(A\) is a magic square matrix...

Integral Solutions to the Equation \(x^2 + y^2 + z^2 = u^2\): A Geometrical Approach

Six identities, each of which gives infinitely many (but not all) integral solutions to the equation in the title, are shown to be special cases of a more general identity.

Archimedes Revisited

A generalization of a geometric theorem by Archimedes is proven.