The typographic scale has been used for centuries as a means of creating balanced and harmonious font sizing. It has been likened to a harmonious musical scale. However, the standard typographic scale is *not* a musical scale. It contains several flaws, where it deviates from the perfect mathematical ideal.

The Liar’s paradox is a unsettling logical paradox that has caused no end of trouble. However, there is a deeply satisfying solution.

The egg problem is a classic programming challenge. However, a mathematical solution exists. It is possible to solve this problem analytically, even in the general case with an arbitrary number of eggs.

The natural way to serialize a binary tree requires

*2n* bits to store a binary tree with

*n* nodes. However, it’s possible to reduce this to

*2n - (3/2)log_2(n) - (1/2)log_2(pi)* bits, with the help of the Catalan numbers. This is the

*minimal* space needed to serialize an arbitrary binary tree with

*n* nodes.

Symmetric senary is a tiny numeral system which is intended for people to use in their everyday life. It is especially useful for dividing things up in practical ways, and easily estimating things like tips and mileage without ever needing a calculator.

Bézier curves are often used to generate smooth curves because Bézier curves are computationally inexpensive and produce high-quality results. The circle can be closely approximated using a cubic Bézier curve that is constructed to minimize the radial deviation from a circle.

The crystal calendar is an almost perfectly-regular calendar that tracks the Earth’s climate and constellations more closely than the Gregorian calendar. The same calendar can be used year after year. Every month starts on the same day of the week, and the year is nearly evenly divisible into halves, thirds, quarters, sixths, and twelfths. The weekend comes a day early, without reducing your wages, and you get an extended weekend four times every year.