Bongard problems are a type of puzzle invented by Russian computer scientist M. M. Bongard, intended as challenges for pattern recognition algorithms. Regardless of the possible relevance to computer science, they are much fun simply as puzzles created by people for other people to solve. They were popularized in the Gödel, Escher, Bach book and Hofstadter himself designed many Bongard problems. Harry Foundalis has several pages dedicated to these problems, with an extensive index of example Bongard problems.

The idea of a Bongard problem is the following: given a set A of six figures (examples) and another set B of six figures (counter-examples), discover what is the rule that the figures in A obey and figures in B violate.

Here are some very nice examples of varying difficulty and a much more detailed description:

http://www.foundalis.com/res/diss_research.html

Inspired by these examples I tried my hand at making some Bongard puzzles myself. The modest result is below. If you can solve them, leave the answer in the comments.

**Puzzle 1.**

**Puzzle 2.**

**Puzzle 3.**

**Puzzle 4.**

Inequalities are the bread and butter of mathematical proofs, especially those with an “approximate” flavor. I found many useful sources that describe the most important inequalities, there exist even some books dedicated exclusively to inequalities. However, I couldn’t find a concise “cheat-sheet” style summary of the most important ones, so I wrote one myself, collecting information from the sources I found. Check out my useful inequalities cheat sheet.