We can view braids mathematically as n strings hanging from a horizontal bar. Each piece of string runs downward and can cross neighboring strings. In the 1920s Emil Artin observed that braids of n strings form an algebraic group. To “multiply” two braids, we append the bottom of one braid with the top of another braid. The identity element in this group…

# Tag: Puzzle

## A game for budding knot theorists

Thanks to Sam Shah for introducing me to this fascinating online game: Entanglement. The rules are simple. You are given hexagonal tiles, one at a time, each adorned with six short segments of rope. Use them to construct the longest possible knot (measured in segments) before running into a wall. Entanglement is fun and addicting!…

## A card trick that will probably amaze your friends (solution)

Warning! Spoiler alert! This post contains the secret behind the card trick that I described in my last post. Read that post before reading this one. First the bad news: this card trick is not fool-proof; it is a probabilistic card trick. The good news is that in my experience, it has a high probability…

## A card trick that will probably amaze your friends

Here’s a neat card trick that I learned a few years ago. I can’t remember where I read about it. If anyone knows the source of trick, please post it in the comments. [Update: I now know more about the origin of this trick. I’ll write more in my follow-up post.] Thoroughly shuffle an ordinary…

## The famous trick donkeys: a Sam Loyd puzzle

Several years ago a colleague and I made paper copies of this famous old puzzle to distribute to prospective mathematics students. It is a fantastic puzzle, so I thought I’d post it here again. From what I can tell, the puzzle was invented in 1871 by Sam Loyd (1842–1911), probably history’s most famous “puzzler.” It…

## Freeman Dyson and a mathematical puzzle in the NY Times

On March 25 the New York Times Magazine had an article about Freeman Dyson, “The Civil Heretic.” It contains an interesting mathematical problem. At Jason, taking problems to Dyson is something of a parlor trick. A group of scientists will be sitting around the cafeteria, and one will idly wonder if there is an integer…

## KenKen

I just discovered a cool new Sudoku-like game called KenKen. A KenKen board is an nxn grid, and the object is to place the numbers 1 to n in each square subject to the following rules. Do not repeat a number in any row or column. The numbers in each heavily outlined set of squares,…

## Four color theorem applets

The four color theorem is a beloved result with a long and fascinating history. The theorem says that four colors suffice to color any map so that no two bordering regions are the same color. The conjecture was made in 1852 by Francis Guthrie. After many, many failed proofs, the conjecture was finally put to…

## Kuratowski’s closure-complement theorem (solution)

Stop! This post contains spoilers. This page has the solution to the problem posed in yesterday’s post. We challenged you to find a set from which we can make as many new sets as possible using only the closure and complement operations. In 1922 Kuratowski proved the following theorem. Theorem. At most 14 sets can…

## Kuratowski’s closure-complement theorem

One of my favorite theorems in elementary topology is Kuratowski’s closure-complement theorem. First some notation. For any set let denote the complement of and denote the closure of . (Recall that and is the union of and all the limit points of ). Here’s the problem. Find a set so that we can construct as many…