It's well known that the famous Delian problem admits no solution in the classical construction framework with ruler and compass
http://www.cut-the-knot.org/Curriculum/Geometry/Delian.shtml
The applet presents a famous dissection "paradox": take a 8×8 square and cut it as shown in the left portion of the applet.
http://www.cut-the-knot.org/Curriculum/Fallacies/FibonacciCheat.shtml
The pieces of the 8×8 square dissection that combine into a 5×13 rectangle can also be arranged into a shape comprising 63 small squares.
http://www.cut-the-knot.org/Curriculum/Fallacies/SamLoydSon.shtml
According to J. dePillis, George Pólya used to define Geometry as the science of correct reasoning on incorrect figures.
http://www.cut-the-knot.org/Curriculum/Fallacies/GeoFallacy.shtml
There are some half-dozen puzzles, as old as the hills, that are perpetually cropping up, and there is hardly a month in the year that does not bring inquiries as to their solution.
http://www.cut-the-knot.org/do_you_know/3Utilities.shtml
A number of students sit in a circle while their teacher gives them candy. Each student initially has an even number of pieces of candy.
http://www.cut-the-knot.org/Curriculum/Algebra/IntergerIterationsOnACircle.shtml
The area of a rectangle equals the product of its sides. The area of a parallelogram equals the product of one of its sides times the distance between that side and its parallel (and equal) mate.
http://www.cut-the-knot.org/Curriculum/Geometry/AreaOfParallelogram.shtml
A Mathematical Droodle
http://www.cut-the-knot.org/Curriculum/Geometry/AsymmetricPropeller.shtml
A fellow travels from city A to city B. For the first hour, he drove at the constant speed of 20 miles per hour.
http://www.cut-the-knot.org/arithmetic/HarmonicMean.shtml
On two separate occasions (Magic Squares and Nim) I had an opportunity to discuss the Binary System and Boolean Algebra. These two algebraic structures that both use only two digits (0 and 1) to represent their elements, are still much different.
http://www.cut-the-knot.org/Curriculum/Algebra/BinaryColorDevice.shtml