Recent changes to Interactive Mathematics Miscellany and PuzzlesInteractive Mathematics Miscellany and Puzzles is a frequently updated site. Here are the most recent addtions. For other changes please visit the site.Bubbling of SortsWhat is the probability that on a pass of a bubble sort the number in position 20 will move into position 30?
https://www.cut-the-knot.org/Probability/BubblingOfSorts.shtml
Equilateral Triangle from Three CentroidsABC is inscribed in the unit circle. Ga, Gb, Gc are the centroids of OBC, OCA, OAB. Prove that ABC is equilateral iff AGa + BGb + CGc = 4
https://www.cut-the-knot.org/m/Geometry/InequalityWithThreeCentroids.shtml
Cutting a Cube into Smaller OnesIs it possible to cut a cube into
https://www.cut-the-knot.org/m/Geometry/CuttingCubeIntoSmallerOnes.shtml
The Circle of the OrthocentersGiven a circle and a chord BC. As A varies over the circle the orthocenter of triangle ABC describes a circle which is a reflection of the given circle in BC
https://www.cut-the-knot.org/m/Geometry/CircleOfOrthocenters.shtml
A 2-Variable Optimization From a China Competitionfind maximum of x+y provided x^3+y^3 +(x+y)/4=15/2
https://www.cut-the-knot.org/Optimization/ChinaOptimizationInTwoVariables.shtml
Probability of Majorization IIGiven three a's and three b's such that sum(a) ≥ sum(b). What is the probability that a majorizes b, if necessary after a suitable rearrangement
https://www.cut-the-knot.org/Probability/ProbabilityOfMajorization2.shtml
Inheriting InequalityGiven three a's and three b's such that sum(a) ≥ sum(b). If p is greater than 1, i s it true that sum (a^p) ≥ sum (b^p)?
https://www.cut-the-knot.org/Probability/InheritingInequality.shtml
Training Bicyclists on a Mountain RoadBicyclists in a line on a mountain road a scared by the racing James Bond. What is the expectation of the number of bicyclists that tripped
https://www.cut-the-knot.org/Probability/AfterJamesBond.shtml
A Simple Integral of a Peculiar FunctionFinding a value of a simple definite integral without Calculus, an extra simple example
https://www.cut-the-knot.org/proofs/SimpleIntegralWithASpecialFunction.shtml
Pythagorean Theorem by ContradictionPythagorean Theorem by Contradiction
https://www.cut-the-knot.org/pythagoras/proof122.shtml