A Möbius strip is a two-dimensional surface that has only one side and one edge. It’s non-orientable, meaning that within the strip, there’s no way to distinguish “clockwise” from “counter-clockwise”.
Wikipedia page on Möbius strips.
A flexagon is a strip, folded so that it can be “flexed” to bring different polygonal panels to the visible surface. These are usually topologically equivalent to a Möbius strip.
The Sierpiński triangle is a fractal that can be produced by starting with a filled triangle, removing the sub-triangle formed by connecting the midpoints of the sides, and repeating this on each of the remaining triangular pieces.
It can also be produced by the chaos game. Draw a triangle. Plot a starting point. Pick a vertex at random, and plot the next point half way between the current point and that vertex. Repeat until the picture emerges.
Wikipedia page on the Sierpiński triangle. It includes an animation of the chaos game.
Wikipedia page on the Chaos game.
Video by Numberphile on the Chaos game.
Geogebra implementation of the Chaos game.
Chaos game by Wang-Hoyer. Configured for the Sierpiński triangle. Page on the Experiments with Google collection.
Scratch implementation of the Chaos game.
Mathenaeum page with the Chaos game.
This is possibly the most confusing problem in elementary probability!
Wikipedia page on the Monty Hall problem.
Mythbusters episode S11.E7 “Wheel of Mythfortune” with the Monty Hall problem . The IMDB page has the wrong season. You may be able to watch it on Discovery or Amazon. Mythbusters aftershow with comments. Contrary to what Adam says here, there’s actually a simple calculation that proves that under the appropriate assumptions, the probability of winning increases if you switch doors. The detail that makes the most difference is where the host knows where the prize is.
Gerrymandering is the manipulation of elections by drawing voting districts to produce results that are out of proportion with the overall makeup of the population. It’s an area of active mathematical research: How can gerrymandering be formally defined and identified? What modifications to the election process can make them resistant to gerrymandering?
Wikipedia article on gerrymandering. Article on the US.
Princeton gerrymandering project.
The Gerrymadering Project, a podcast series by FiveThirtyEight.
A kenken is a fill-in-the-grid puzzle. Each of the numbers 1 to n must appear once in each row and column of the n by n grid, forming a Latin square. Some grid cells are joined into groups called cages, with a number and an operation. The indicated number has to be the result of applying the operation to the numbers in the cage, which places strong constraints on which numbers can go where. A well-formed kenken has a unique solution.
Nim is a mathematical strategy game. The board consists of several piles of counters. Two players take turns. To move, a player chooses one pile, and removes at least one counter from that pile. You’re allowed to remove a whole pile if you like.
In the normal version, the player who takes the last counter wins. A winning strategy can be developed in terms of “nimbers” and modified arithmetic.
In the misère version, the player who is forced to take the last counter loses.
Game theory: A playful introduction, by DeVos and Kent.
The fold-and-cut theorem states that it is possible to cut any shape with straight sides out of a piece of paper by folding it somehow and making one straight cut across the folded paper. Given a particular shape, how do you come up with the folds and cut to make it?
Wikipedia article on the fold-and-cut theorem.
Video by Numberphile on the Fold-and-cut theorem.
Folding and one straight cut suffice, by Demaine, Demaine, and Lubiw.
Efron’s dice are the most famous example of this probability paradox. There are four dice, A, B, C, and D. Imagine that you roll two of the dice, and whichever one shows a higher number “wins.” If you roll A and B, A comes out greater than B with probability 2/3, so A wins against B with probability 2/3. Likewise, B wins against C and C wins against D all with probability 2/3. Remarkably, D also wins against A with probability 2/3!
That is, A appears to be “better” than B, which is “better” than C, which is “better” than D, so intuitively, A ought to be “better” than D, but that turns out not to be true in this game!
The name “intransitive” comes from the fact that if you define a relation > on these dice by saying that X > Y if the probability that a roll of X is higher than a roll of Y is greater than 1/2, then the relation is not transitive.
Wikipedia article on intransitive dice.
Infinity has all sorts of weird and wonderful properties!
E-books by Schwartz. Printed copies are also available: Life on the Infinite Farm, by Schwartz. Gallery of the Infinite, by Schwartz.
