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Browse Classroom Capsules and Notes

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Displaying 971 - 980 of 1214

The set of all "even up" card games is modeled by a group of binary strings, which shows a game outcome is determined by luck, not skill. The probability for each outcome is obtained...

The author proves visually that the square of the \(n\)th triangular number equals the sum of the cubes of the first \(n\) positive integers.
The authors determine a function that relates 6 pairs of hyperbolic and trigonometric functions, and explain the resemblance and some coincidences from the corresponding graphs.

The author revisits formulas of measuring solid angles that he could find only in centuries-old literature, and provides modern versions of the proofs.

The author verifies that a scalar field associated to a vector field, satisfying spherical symmetry and Laplace's equation, implies Newton's inverse square law.

The author presents a characterization for a ring to be Boolean ring.

Using Haruki's lemma, the author provides an easy proof of the Double Butterfly Theorem in plane geometry regarding a circle and its chords.

The author presents a uniform way of studying the weighted \(r\)th power means and their relations, as well as a simple proof for the limits of the means discussed in previous articles....

The author provides, with derivations, a list of common groups (subgroups) associated to the \(2\times 2\) matrices over \(Z_{p}\) either additively or multiplicatively.

The author finds visually sums of triangular numbers.