A magic square is an arrangement of numbers from 1 to n2 in an n x n matrix. In a magic square each number occurs exactly once such that the sum of the entries of any row, column, or main diagonal is the same. You can make several magic squares and investigate the different properties of the square. Can you make an algorithm for constructing a Magic Square? Can you show that the sum of the entries of any row, column, or main diagonal must be n(n2+1)/2? Are there any other hidden properties of a…
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Almost all of the games we play are based on math in some way or another. Card games, board games, and computer games are designed using statistics, probabilities, and algorithms. Begin by reading about games and game theory. Then you can choose your favorite game and investigate the mathematical principles behind how it works. Can combinatorial game theory help you to win two-player games of perfect knowledge such as go, chess, or checkers? (Weisstein, 2006; Watkins, 2004) In a multi-player…
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Imagine a symmetrical grid of nine points superimposed over the ball. Kicking the ball squarely on the center point imparts no spin, but kicking on any of the other points will impart spin on the ball. How will the resulting spin affect the trajectory of the ball for each of the 8 outer grid points? Kicking the ball with a sliding motion of the foot is another way to impart spin. Once you've made your predictions, you can set up to test them with a soccer ball, video camera and a tape…
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You may know Lewis Carroll as the author of Alice in Wonderland, but did you know that in real life he was a mathematician who studied symbolic logic and logical reasoning? How can math help you solve Lewis Carroll's Logic Game? (Bogomolny, 2006) How are algorithms for solving the game Sudoku similar to solving a logic problem? (Hayes, 2006) For the super-advanced mathematical genius, try to evaluate currently available, logic-based computational tools, or design a better one!…
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What do knots, maps, mazes, driving directions, and doughnuts have in common? The answer is topology, a branch of mathematics that studies the spatial properties and connections of an object. Topology has sometimes been called rubber-sheet geometry because it does not distinguish between a circle and a square (a circle made out of a rubber band can be stretched into a square) but does distinguish between a circle and a figure eight (you cannot stretch a figure eight into a circle without…
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You can measure the diameter of the Sun (and Moon) with a pinhole and a ruler! All you need to know is some simple geometry and the average distance between the Earth and Sun (or Moon). An easy way to make a pinhole is to cut a square hole (2-3 cm across) in the center of a piece of cardboard. Carefully tape a piece of aluminum foil flat over the hole. Use a sharp pin or needle to poke a tiny hole in the center of the foil. Use the pinhole to project an image of the Sun onto a wall or piece…
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Start with 7 drinking straws and 14 paper clips. Use the paper clips to fasten the straws together. Here's how: 1) Clip two paper clips together, narrow end to narrow end. 2) Push the wide ends of each clip into the end of a straw. That's it! Connect four straws to make a square, and three straws to make a triangle. Now test which shape is stronger. Hold the shapes vertically, with an edge or a vertex resting on the tabletop. Have a helper push on the opposite side or vertex. Which shape…
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Want to send coded messages to your friends? Can you write a simple letter-substitution encryption program in JavaScript? How easy is it to break the simple code? Can you write a second program that "cracks" the letter-substitution code? Investigate other encryption schemes. What types of encryption are least vulnerable to attack?
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Students who are mathematically inclined can use the student version of a program like MatLab or Mathematica to convert a digital image into numbers, then perform operations such as sharpening or special effects. This is a great way to learn about image processing algorithms.
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Your digital photo comprises a certain number of dots in the x and y directions. What happens to the print image quality as you "stretch" those dots out to larger and larger pictures? (Note: This experiment studies the dots per inch in the image itself, not the number of dots per inch that is output by your printer.)
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