Who doesn’t love a good card trick? Many card tricks rely on math and can be a way to demonstrate mathematical concepts, or they can just be a fun way to amaze your friends. Here’s one that was invented by Bob Hummer.

This trick illustrates the concept of permutations. A permutation is basically the order of arrangement in a set. For the set {1,2,3}, there are 6 possible permutations: {1,2,3}, {1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}, and {3,2,1}. In the first step of the trick, the magician asks the volunteer to make an unknown transposition, which is when you swap one element with another. For example, transposing 1 with 3 in {1,2,3} gives {3,2,1}.

**Procedure:**

- Place an ace, 2, and 3 on the table as shown above. (You can use any three cards, but it is easiest to remember 1,2,3.)
- Turn your back and ask your friend to choose one of the cards without telling you, remember what it is, and flip it over. Then ask your friend switch the positions of the 2 remaining cards and flip those over.
- Now, pick up the cards in this order: left card on the bottom, middle card in the middle, right card on the top.
- “Shuffle” the cards, by moving top cards to the bottom 1 or 2 cards at a time. Stop when you have moved 4, 7, or 10 cards from the top to the bottom.
- Place the cards facedown on the table in this order: top card in the middle, next card on the right, and last card on the left.
- Imagine the cards in the reverse order as the original positions (3, 2, 1).
- Have the friend guess which card is theirs and flip it over, but not tell you whether they are right or not.
- If the card is in the expected position (i.e., they flip over the first card and it is the 3), you know they picked the correct card and you can say “Congratulations!”
- If the card your friend flips is not in the expected position, it must have been one of the switched cards. This card, and the card in the place you would have expected the flipped card to be in, are not the correct cards. For example, if they flip over the first card and it is the ace, this card and the 3
^{rd}card (the expected position of the ace) are not the correct cards. (Remember, the cards should be in reverse order at this point.) So, flip the 2^{nd}Say, “I think you were looking for this one.”

This trick works because you are picking up the cards in a way that reverses their order, which is equivalent to the transposing 1 and 3. By revealing one the cards allows, you can determine which two cards were switched by your friend.

]]>- thick cardboard, cork board, or other surface that you can put a push pin into (I’ve tried foam board insulation, and that works well.)
- paper
- push pins
- loops of thread or string
- pens or pencils

Before doing this activity at my after-school math club, I asked the kids to describe how to draw a circle without using the word “round”. They struggled with this, so we talked about the fact that every point on a circle is equidistant from the center.

To draw a circle, place your paper on top of the cardboard and put a push pin in the center. Place a loop of string around the push pin and the pencil. Pull the loop tight and draw a circle around the push pin as shown in the video below. Make sure you string isn’t too long or you will run off the paper, and be careful that the loop doesn’t slip off of the push pin.

Vary the length of the string to see how that changes the circle.

Now, place two push pins on the board, and loop the string around both pins. The resulting shape is an ellipse. The two push pins are the foci of the ellipse. Try varying the distance between the push pins and the length of the string.

The circle is the case where both foci are in the same spot. Moving the foci apart, give the ellipse. Now, imagine moving one of the foci out to infinity. Of course, this isn’t possible in practice, but you can move one end of the loop far away. Make a long loop of string, and attach one end to a chair or table, or have a friend hold it, making it almost tight. Now use your pen to pull it tight to the side of the paper and draw around the other push pin. This creates a parabola – the conic section with one foci at infinity.

]]>Here is a video from The Mathematical Association of America. It is “The Great pi/e Debate”, presented at Williams College during their First Year Family Weekend, in which two professors debate the relative merits of the numbers pi and e. Which is better? Watch the videos and decide for yourself. Part 1 is mostly introductions, so if you prefer, just start in with part 2.

]]>Also, here are a few cute pi cartoons from Foxtrot:

]]>If you watched the documentary about fractals that I shared a while ago, you may recognize that Cantor Set, illustrated below. It was one of the earliest fractals, known at the time as “monsters”. The Cantor Set is formed by taking a line, removing the middle third, then removing the middle third of the remaining lines, and so on. It would seem that eventually there would be nothing left, but there is always a set of remaining points, just ever smaller and smaller.

]]>Today (February 15, 2014) is the 450th anniversary of the birth of Galileo Galilei. Galileo made many contributions to science including describing the motion of pendulums and falling objects, his improvements to the telescope, his astronomical discoveries (including 4 of Jupiter’s moons), and his support of heliocentrism (the astronomical model wherein the planets move around the sun, described by Copernicus).

Until Galileo, scientists typically conducted qualitative studies, meaning they relied on descriptions of characteristics rather than on measured quantities or values. One of Galileo’s most important contributions to science was that he used mathematics to describe his scientific observations. He was one of the first thinkers to state that the laws of nature are mathematical. He also understood the relationship between mathematics and physics. He understood that the parabola is a conic section , a mathematical function where the y-value is a function of the square of the x-value (for example, y=x^{2}), and the trajectory of a falling object. If you throw ball, it follows a parabolic path.

Galileo mostly applied the standard mathematics of the day to his scientific pursuits, but he did produce the Galileo’s paradox, which shows that there are as many perfect squares as there are whole numbers, even though most numbers are not perfect squares.

Galileo also proved that objects fall at the same rate, regardless of weight. Before that, people believed that heavier objects fell faster than lighter objects. Of course, we know that sometimes lighter objects fall more slowly, but this is due to air resistance. On the Apollo 15 mission, Commander David Scott showed that in the absence of air, a feather falls at the same rate as a hammer.

Legend has it, Galileo performed the falling object experiment by dropping objects off the Tower of Pisa. In the video below, Galen Weitkamp, professor of mathematics at Western Illinois University, explains some of the math describing falling objects.

]]>If this seems like too much work, there are tons of cute math-themed pictures on the internet you can turn into Valentines. Last year, we found cute a Sierpinski Valentine.

]]>Over the Christmas break, while on vacation with extended family, my son learned how to solve the Rubik’s Cube (a.k.a. magic cube) from his older cousin. He became somewhat obsessed and worked on it until he could solve it himself. One of the things he told me when he got his new cube was, “Mom, the colors can be arranged in over 43 quintillion ways!” This is why it is so hard to solve; there are too many different permutations. You’d never land on the solution by randomly turning the sides. The number of permutations is calculated with the equation:

The first term, 8!, comes from the fact that there are 8! (read 8 factorial) ways to arrange the 8 corner “cubies” (8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1). The next term is 3^{7}, because there are 3 colors on a corner cubie and 7 of the corners can be oriented independently. (“Oriented” differs from “arranged” in that arranged refers to moving a cubies to different locations on the cube, while oriented refers to the cubie being at the same corner but rotated to show a different color on top.) The orientation of the eighth cubie depends on the preceding seven. The next term, 12!, is the number of ways to arrange the edges. This term must be divided by 2 because it is an even permutation. Eleven edge cubies (with 2 colors each) can be flipped independently, with 12th depending on the preceding ones, giving 2^{11}.

If you are interested in a more in-depth explanation, there is a powerpoint presentation here, and paper here.

Most people solve the Rubik’s cube by memorizing sequences of moves called “algorithms”. In mathematics, the term algorithm means a step-by-step procedure for calculations. Many algorithms, especially those used later in the solution, are designed to move a cubie to a specific spot without moving parts of the cube that have already been moved into place. If you are interested in learning basic algorithms for solving the cube, there is a solving guide available at Rubiks.com. People into “speedcubing” learn algorithms that make it possible to solve the cube in fewer moves, but requires learning more algorithms used with specific patterns on the cube.

Puzzles like the Rubik’s Cube can also be investigated through mathematical group theory. Group theory is the study of the algebraic structures called groups. A group is a set of elements together with an operation that combines any two of its elements to form a third element also in the set while satisfying the group axioms (these are closure, associativity, identity and invertibility). For example, the set of integers together with the addition operation; the addition of any two integers forms another integer. The Rubik’s group is the group corresponding to possible moves.

There is a lot of cool math cooking tools and dining ware available. Here’s a few items:

i 8 Sum Pi Dish, $38 at Uncommon Goods

Math Glasses – set of 4, $38 at Uncommon Goods

Obsessive Chef Cutting Board, $25 at Uncommon Goods

Pizza Pi Cutter, $25 at Amazon

Pi Bottle Opener, $30 at Uncommon Goods

For your geeky guy…

Math Formulas Tie, $45 at Uncommon Goods

If you are looking for a tie, but not sure about this one, there are a bunch of math-themed ties at Zazzle for around $32-$36.

Math Cufflinks, on sale for $14 (regular $40) at Uncommon Goods

Geek Wrist Watch, $65-$78 at Uncommon Goods

There is a ton of cool math jewelry at Etsy…

…like this Laws of Motion Cuff Bracelet, $40,

…or these Math Earrings, $25,

…or this Fibonacci Nautilus Necklace, $30.

Pop Quiz Clock, $27 at ThinkGeek

I came across this Irrational Numbers Wall Clock in the SkyMall catalog while traveling for Thanksgiving, $35.

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Of course, it is always possible to find lots of wonderfully geeky math t-shirts. I like the selection at the Neato Shop. For example…

]]>Here are 4 videos from Vi Hart to show you how to make green bean matherole, Borromean onion rings, optimal potatoes, and turduckenen-duckenen.

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