Most easily digestible posts
You know about electricity, and you know about magnetism, and you’ve probably heard the word “electromagnetism” somewhere before to refer to electricity and magnetism combined in some sense. But why do these two seemingly different phenomena get lumped together into one word, whereas there’s no such thing (yet?) as “electrogravity”? If you know some more … Read More “Magnetism from Relativity” »
A sultan has granted a commoner a chance to marry one of his $N$ daughters. The commoner will be presented with the daughters one at a time and, when each daughter is presented, the commoner will be told the daughter’s dowry (which is fixed in advance). Upon being presented with a daughter, the commoner must … Read More “The Sultan’s Daughters Problem” »
Several competitive decks in Vintage Magic: The Gathering are powered by the card Bazaar of Baghdad. Deck construction rules mandate at least 60 total cards in a deck with at most four copies of any given card. The game starts by each player drawing seven cards then performing, if he wishes, a series of “mulligans” … Read More “Mulligans in the Bazaar” »
In celebration of the unofficial end of summer (Labor Day in the U.S.), consider a snowpile on a trash bin: Observe how the shape of the lid (straight edges, rounded corners) seems to propagate upward, and how the sides of each pile are nearly planar with nearly the same slope on each side. Is this … Read More “Snowpile Math” »
Ridders’ method is a single-variable root-finding method that is more efficient than the basic regula falsi method. The formula is easy enough to find online (e.g. the first link), but its derivation is not. This post fills in the blanks. Suppose the function $f$ has a zero between $x_L$ and $x_R$, i.e. $f(x_L)f(x_R) < 0$. … Read More “Notes on the Derivation of Ridders’ Method” »
In multivariable calculus courses, one usually first encounters the divergence as: $$\displaystyle \nabla\cdot\vec{u} = \frac{\partial{u_x}}{\partial x} + \frac{\partial{u_y}}{\partial y} + \frac{\partial{u_z}}{\partial z} $$ where $\vec{u} = [u_x,\,u_y,\,u_z]$ in cartesian coordinates. Then one learns the divergence theorem: $$\displaystyle \int_\Omega \nabla\cdot\vec{u}\, dx = \int_{\partial\Omega} \vec{u}\cdot\hat{n}\,dS $$ and then, since cylindrical coordinates have usually already been introduced, one … Read More “Divergence in Cylindrical Coordinates – The Right Way” »
The Arithmetic and Geometric Means You’re probably familiar with the arithmetic mean, which is most people mean (heh) when they say “average”: $$\displaystyle A(x_i) = \frac{1}{n}\sum_i x_i $$ where $x_i$ is a set of $n$ real numbers. The arithmetic mean answers the question “If all these numbers were equal, what would they have to be … Read More “Means and Concavity” »
A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a circle, like this one: The center of this circle (the “circumscribed circle” or “circumcircle”) is called the “circumcenter”, marked by a dot in the above figure. Quadrilaterals have many special points that get called centers, such as the circumcenter, but the most … Read More “When is the centroid of a cyclic quadrilateral also its circumcenter?” »
Suppose you have a set $S$ of $N$ numbers: $$\displaystyle S = \{x_1, x_2, x_3, \cdots, x_N\}$$ How many different products can be formed by multiplying at most $F$ members of $S$, allowing repetitions? For example, take $S = \{a,b,c,d\}$. Then the distinct products are: Number of factors Products 1 $$\displaystyle a, b, c, d$$ … Read More “Counting Products When Factors Count” »
1. Introduction One might remember from high-school precalculus class something called the Law of Sines, and might even remember what it is because it has a memorable pattern: $ \displaystyle \frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c} \ \ \ \ \ (1)$ where $a$, $b$, and $c$ (lowercase) are the side lengths and $A$, $B$, and … Read More “Law of Sines for Tetrahedra” »