Let's flip a coin, over and over. Beforehand, the players each pick a sequence of flips. The sequence that occurs first wins. With HH vs. TH, HH will win if the first two flips are HH and will lose if any of those flips are tails. HH vs. TH has a 1/4 vs. 3/4 possibility of winning. Phrased a different way: Suppose I offer a bet on a series of coin flips. One of these bets would be bad for you. Which one? The odds of the event occurring are given at the end. (The second bet is the bad bet to take.) HTT appears before TTT. If it does, I give you $1. If not, you give me $4. (7/8) HHT appears before TTT. If it does, I give you $1. If not, you give me $3. (7/10) THH appears before HHT. If it does, I give you $1. If not, you give me $2. (3/4) HTH appears before THH. If it does, I give you $1. If not, you give me $1. (1/2) This is the strange world of Penney's game. Here is a table of odds and facts for various 3-flip games. Calculating these odds can be both tedious and mathematically demanding—a natural job for Mathematica or Mathematica Home Edition.

Buried deep in the list of new technology in the Mathematica development pipeline was the item "integration of oscillatory functions (univariate, multivariate)---new algorithm". I expect most people will overlook it, as I did, in favor of the new functions, new directions, big infrastructure, and the eye candy. Even worse, most people who will use it won't even know---it will be selected automatically when needed, like many of Mathematica's algorithms. So I think it's my duty to share my discovery that this algorithm is actually really cool. Why is it so cool? The first clue I had was when I read in the notes that this was the first time anyone had fully automated the algorithm into a very wide class of problems. Second, that it was a hybrid numeric-symbolic method (putting it beyond the reach of most numerical systems). And finally, that it was developed by the talented Wolfram Research developer Andrew Moylan.

Back in 325 BC, Aristotle talked about which polyhedra can fill space, and noted that regular tetrahedra could fill space. Around 1470 AD, Regiomontanus showed that Aristotle was wrong. He also found the spot where a statue on a pedestal appears the largest, as shown in the Demonstration “The Statue of Regiomontanus”. In 1896, Minkowski tried to solve the problem of how well tetrahedra could pack. He failed. But he did introduce many valuable tools to math, such as “The Minkowski Sum of Two Triangles”. In 1900, Hilbert tried the problem of tetrahedra packing and included it as a part of problem 18 in his list of unsolved problems. Hilbert is also famous for the Hilbert curve and “The Hilbert Hotel”.

Is it possible to have a pair of nonstandard dice with the same odds as regular dice? Sure. You just need to know how to calculate the odds, and how to determine what different numbers could be on the faces to give the same odds. Let's start with some tables. The addition table is one of the first tables learned in school. Here is one way to present an addition table in Mathematica.

Bruce Torrence, PhD, chair of the Department of Mathematics at Randolph-Macon College, says he's engaging his students in mathematics more than ever before thanks to a single Mathematica command. That command is Manipulate. Professor Torrence calls the ability to create instant dynamic interfaces a "real game changer" for helping students understand mathematics. He says, "Once you play with a Manipulate and interact with the sliders and buttons, you really develop your intuition as to how the underlying mechanisms are interacting and working." In this video, Professor Torrence shares an example of how he used Mathematica to turn a previously tedious lesson into a highly compelling, interactive classroom activity.

Somewhere, you've likely been forced to learn how fractions work, and how to calculate 2/7 + 2/5. To some extent, fractions have been falling out of favor in the world, losing out to decimals. The New York Stock Exchange gave up fractions on April 9, 2001. Much of the time, a decimal is okay. Sometimes, though, especially in mathematics, exact values are desired. Instead of a value being 3.00000000...00727..., it is exactly 3. Or exactly 10/35 + 14/35 = 24/35. For fractions themselves, the Farey sequence is quite interesting—the reduced fractions between 0 and 1 where the denominator is less than or equal to a particular value, like 7. For example, the F7 Farey sequence is the the first row in the following block. The next row has the denominator. The third row is twice the reciprocal of the denominator squared. The fourth row is the denominator from the third row.

April is Mathematics Awareness Month, and this year's theme is "Mathematics and Sports." It's sponsored by the Joint Policy Board for Mathematics to promote the importance of math, and schools and organizations nationwide are participating by hosting presentations, competitions, and poster contests for students from elementary school through graduate school. Wolfram Research is proud to support Mathematics Awareness Month again this year. To remind students everywhere that math can be fun, we have provided complimentary Mathematica for Students licenses to several competitions this month to be distributed as prizes, including these:

Thousands of universities around the world take advantage of Mathematica's revolutionary developments for engineering, science, economics, mathematics, and more, for a vast number of courses across campus. One of those schools is Truman State University. Dana Vazzana, an associate professor of mathematics at Truman, integrates Mathematica into every course she teaches. She says using Mathematica with her students creates a dynamic classroom where students gain deeper understanding of concepts and richer insights into real-world applications of mathematics. "Anything that gets them that involved and that excited and makes them want to go and work some more has just got to be a good thing," explains Professor Vazzana.