Math Puzzles and Paradoxes

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25 People in a Square Grid Puzzle
Twenty-five people stand in a square 5x5 grid, one person per cell. At the blow of a whistle, each person is to take a single step, horizontally or vertically and move into a neighboring cell to end up as a new configuration of 25 people standing in a 5x5 grid, one person per cell. Can it be done? This little video gives away the answer, so perhaps try playing with this first before watching! When does Perimeter equal Area?
The 3x6 rectangle has the property that its area has the same numeric value as its perimeter. (They are both 18). A circle of radius 2 has this property, as does the 6-8-10 right triangle. In this video we explore - briefly - the question of when area equals perimeter. A fuller discussion takes place in the written essay "Integer Triangles" available on the website. What is Fractional Dimension and What is a Fractal?
Take a picture to a photocopier and set it to scale by a factor of three. Then all lengths in the picture triple in size (factor of three to the one-th power) and all areas increase by a factor of nine (three to the two-th power). But there are objects that scale by fractional powers! Pi for a Square
Everyone talks about the value of pi for a circle, but what is the value of pi for a square, or a triangle, or a septagon? Do the formulas for circumference and area, which involve pi, still work? (They do!)