Interactive Mathematics and Puzzles

The purpose of this site is set out in the Manifesto. You can navigate through the site with the help of red/yellow buttons at the top of every page, the left navigational bar, or a few buttons to the critical pages placed at the bottom of the site.

The site is a collection of K16 topics drawn from arithmetic,algebra,geometry,probability, logic, social sciences, math fallacies, visual illusions and a good deal of miscellany. More than 950 interactive Java illustrations help enliven some topics, while making others more comprehensible. One can get a good introduction at the contents page.

I am convinced that mathematics should be studied because of its intrinsic beauty, not because of its utility.

https://www.cut-the-knot.org/Games/Cram/ThreeChain.shtml
https://www.cut-the-knot.org/Games/Vectors.shtml
https://www.cut-the-knot.org/Games/Vowel-Movement.shtml
https://www.cut-the-knot.org/Games/Hounds.shtml
https://www.cut-the-knot.org/Games/SeeSaw.shtml
https://www.cut-the-knot.org/Games/Caterpillar.shtml
https://www.cut-the-knot.org/Games/MatchBugz.shtml
https://www.cut-the-knot.org/Games/Torque.shtml
https://www.cut-the-knot.org/Games/WolfRabbit.shtml
https://www.cut-the-knot.org/Games/Thugs-Bowmen.shtml
https://www.cut-the-knot.org/Games/SuperNim/SNim.shtml
https://www.cut-the-knot.org/Games/Numbers.shtml
https://www.cut-the-knot.org/Games/cball.shtml
https://www.cut-the-knot.org/Games/Mahjong.shtml
https://www.cut-the-knot.org/Games/Bulls-Cows.shtml
https://www.cut-the-knot.org/Games/MirrorMaze/MirrorMaze.shtml
https://www.cut-the-knot.org/Games/BlockNRoll.shtml
https://www.cut-the-knot.org/Games/MineSweeper.shtml
https://www.cut-the-knot.org/Games/Reversi.shtml
https://www.cut-the-knot.org/Games/PushIt.shtml
https://www.cut-the-knot.org/latimes.shtml
https://www.cut-the-knot.org/pythagoras/Pyth107.shtml
https://www.cut-the-knot.org/pythagoras/BottemaInThreeRotations.shtml
https://www.cut-the-knot.org/pythagoras/LipogrammaticProof.shtml
https://www.cut-the-knot.org/pythagoras/EuclidRef.shtml
https://www.cut-the-knot.org/pythagoras/lines.shtml
https://www.cut-the-knot.org/pythagoras/PPower.shtml
https://www.cut-the-knot.org/pythagoras/Constructions/index.shtml
https://www.cut-the-knot.org/pythagoras/Constructions/CyclicQuadrilateral.shtml
https://www.cut-the-knot.org/pythagoras/Constructions/TrisectSegment.shtml
https://www.cut-the-knot.org/pythagoras/Constructions/ParallelLineThroughPoint.shtml
https://www.cut-the-knot.org/pythagoras/ButterflyByMidline.shtml
https://www.cut-the-knot.org/pythagoras/ButterflyInHyperbola.shtml
https://www.cut-the-knot.org/pythagoras/SLIDER.shtml
https://www.cut-the-knot.org/pythagoras/fifteen.shtml
https://www.cut-the-knot.org/pythagoras/ThalesTheorems.shtml
https://www.cut-the-knot.org/pythagoras/ModernDaySangaku.shtml
https://www.cut-the-knot.org/pythagoras/FanSangaku.shtml
https://www.cut-the-knot.org/pythagoras/proof89.shtml
https://www.cut-the-knot.org/pythagoras/MechanicalProofs.shtml
https://www.cut-the-knot.org/pythagoras/two_coins.shtml
https://www.cut-the-knot.org/pythagoras/KidsSangaku.shtml
https://www.cut-the-knot.org/pythagoras/InscribedQuadri.shtml
https://www.cut-the-knot.org/pythagoras/CircleSquareCircle.shtml
https://www.cut-the-knot.org/pythagoras/Sina.shtml
https://www.cut-the-knot.org/pythagoras/pythPerfect.shtml
https://www.cut-the-knot.org/pythagoras/SangakuArt.shtml
https://www.cut-the-knot.org/pythagoras/PythagorasBySimilarity.shtml
https://www.cut-the-knot.org/pythagoras/AnalyticBottema.shtml
https://www.cut-the-knot.org/pythagoras/Carnot.shtml
https://www.cut-the-knot.org/pythagoras/IMO2007-2.shtml
https://www.cut-the-knot.org/pythagoras/cos15.shtml
https://www.cut-the-knot.org/pythagoras/TwoChessboards.shtml
https://www.cut-the-knot.org/pythagoras/EuclidRef_I.shtml
https://www.cut-the-knot.org/pythagoras/PencilOfCubics.shtml
https://www.cut-the-knot.org/pythagoras/PTcom3.shtml
https://www.cut-the-knot.org/pythagoras/Chasles/SimpleChasles.shtml
https://www.cut-the-knot.org/pythagoras/Chasles/Chasles.shtml
https://www.cut-the-knot.org/pythagoras/Chasles/Pascal.shtml
https://www.cut-the-knot.org/pythagoras/EuclidRef_V.shtml
https://www.cut-the-knot.org/pythagoras/AfterEudoxus.shtml
https://www.cut-the-knot.org/pythagoras/GeoLimit.shtml
https://www.cut-the-knot.org/pythagoras/Proof117.shtml
https://www.cut-the-knot.org/pythagoras/UrquhartElementary.shtml
https://www.cut-the-knot.org/pythagoras/CosineLawBySimilarity.shtml
https://www.cut-the-knot.org/pythagoras/GemMorsel.shtml
https://www.cut-the-knot.org/pythagoras/ChildsPlay.shtml
https://www.cut-the-knot.org/pythagoras/AxisOfSimilitude.shtml
https://www.cut-the-knot.org/pythagoras/herons2.shtml
https://www.cut-the-knot.org/pythagoras/SangakuCritique.shtml
https://www.cut-the-knot.org/pythagoras/HomogeneousCoordinates/homogeneous.shtml
https://www.cut-the-knot.org/pythagoras/Proof80p.shtml
https://www.cut-the-knot.org/pythagoras/PythLoomis27G.shtml
https://www.cut-the-knot.org/pythagoras/Plato.shtml
https://www.cut-the-knot.org/pythagoras/EightCircles.shtml
https://www.cut-the-knot.org/pythagoras/paradox.shtml
https://www.cut-the-knot.org/pythagoras/GMSangaku.shtml
https://www.cut-the-knot.org/pythagoras/UrquhartsCabart.shtml
https://www.cut-the-knot.org/pythagoras/FibonacciCos.shtml
https://www.cut-the-knot.org/pythagoras/corollary.shtml
https://www.cut-the-knot.org/pythagoras/FalseProofs.shtml
https://www.cut-the-knot.org/pythagoras/BrokenChordPythagoras.shtml
https://www.cut-the-knot.org/pythagoras/FaultyPythPWW.shtml
https://www.cut-the-knot.org/pythagoras/HarmonicRatioProperties.shtml
https://www.cut-the-knot.org/pythagoras/index.shtml
https://www.cut-the-knot.org/pythagoras/UrquhartLeningrad.shtml
https://www.cut-the-knot.org/pythagoras/bath.shtml
https://www.cut-the-knot.org/pythagoras/mech.shtml
https://www.cut-the-knot.org/pythagoras/Copernicus.shtml
https://www.cut-the-knot.org/pythagoras/StereographicProjection.shtml
https://www.cut-the-knot.org/pythagoras/ellipse.shtml
https://www.cut-the-knot.org/pythagoras/Orthodiagonal2.shtml
https://www.cut-the-knot.org/pythagoras/ButterflyViaCeva.shtml
https://www.cut-the-knot.org/pythagoras/cosine.shtml
https://www.cut-the-knot.org/pythagoras/QOptimizationSangaku.shtml
https://www.cut-the-knot.org/pythagoras/pentagon.shtml
https://www.cut-the-knot.org/pythagoras/Extra/geometry_problem.shtml
https://www.cut-the-knot.org/pythagoras/happy8.shtml
https://www.cut-the-knot.org/pythagoras/Samosvatov.shtml
https://www.cut-the-knot.org/pythagoras/AncientMorsel.shtml
https://www.cut-the-knot.org/pythagoras/ComputingInTetrahedron.shtml
https://www.cut-the-knot.org/pythagoras/Bottema.shtml
https://www.cut-the-knot.org/pythagoras/Stevens.shtml
https://www.cut-the-knot.org/pythagoras/PythAG.shtml
https://www.cut-the-knot.org/pythagoras/KungCosLaw.shtml
https://www.cut-the-knot.org/pythagoras/train_wheel.shtml
https://www.cut-the-knot.org/pythagoras/IMO2004-5.shtml
https://www.cut-the-knot.org/pythagoras/PTForReciprocals.shtml
https://www.cut-the-knot.org/pythagoras/AreaOfTrapezoid.shtml
https://www.cut-the-knot.org/pythagoras/ButterflyInConics.shtml
https://www.cut-the-knot.org/pythagoras/HarmonicRatio.shtml
https://www.cut-the-knot.org/pythagoras/Ch6Pr3Sangaku.shtml
https://www.cut-the-knot.org/pythagoras/UnusualLocus.shtml
https://www.cut-the-knot.org/pythagoras/SphereAndPlanes.shtml
https://www.cut-the-knot.org/pythagoras/TrigProof.shtml
https://www.cut-the-knot.org/pythagoras/history15.shtml
https://www.cut-the-knot.org/pythagoras/725Sangaku.shtml
https://www.cut-the-knot.org/pythagoras/FaultyLuneQuadrature.shtml
https://www.cut-the-knot.org/pythagoras/GeneralizedCarnot.shtml
https://www.cut-the-knot.org/pythagoras/sliders.shtml
https://www.cut-the-knot.org/pythagoras/PythStarOfDavid.shtml
https://www.cut-the-knot.org/pythagoras/arctans.shtml
https://www.cut-the-knot.org/pythagoras/tricky.shtml
https://www.cut-the-knot.org/pythagoras/GarfieldsGhost.shtml
https://www.cut-the-knot.org/pythagoras/SangakuInSquare2.shtml
https://www.cut-the-knot.org/pythagoras/Proof69Complete.shtml
https://www.cut-the-knot.org/pythagoras/Proof37.shtml
https://www.cut-the-knot.org/pythagoras/Pyth9PWW.shtml
https://www.cut-the-knot.org/pythagoras/PythRectSangaku.shtml
https://www.cut-the-knot.org/pythagoras/YinYangBisection.shtml
https://www.cut-the-knot.org/pythagoras/MiquelPlens.shtml
https://www.cut-the-knot.org/pythagoras/PentagonSangaku.shtml
https://www.cut-the-knot.org/pythagoras/Proof109.shtml
https://www.cut-the-knot.org/pythagoras/Proof118.shtml
https://www.cut-the-knot.org/pythagoras/PentagonProportion.shtml
https://www.cut-the-knot.org/pythagoras/bicolor.shtml
https://www.cut-the-knot.org/pythagoras/Pyth30.shtml
https://www.cut-the-knot.org/pythagoras/Pyth301.shtml
https://www.cut-the-knot.org/pythagoras/GeometricMean.shtml
https://www.cut-the-knot.org/pythagoras/casey.shtml
https://www.cut-the-knot.org/pythagoras/CosineLawMolokach.shtml
https://www.cut-the-knot.org/pythagoras/proof41.shtml
https://www.cut-the-knot.org/pythagoras/ApolloniusCircleInEquilateralTriangle.shtml
https://www.cut-the-knot.org/pythagoras/ComplexHarmonicRatio.shtml
https://www.cut-the-knot.org/pythagoras/WilliamWallaceButterfly.shtml
https://www.cut-the-knot.org/pythagoras/ButterflyInQuad2.shtml
https://www.cut-the-knot.org/pythagoras/EuclidRef_III.shtml
https://www.cut-the-knot.org/pythagoras/PT_matrix.shtml
https://www.cut-the-knot.org/pythagoras/ButterflyDoc/AlgebraicButterfly.shtml
https://www.cut-the-knot.org/pythagoras/HeronsDerivative.shtml
https://www.cut-the-knot.org/pythagoras/PythI47PWW.shtml
https://www.cut-the-knot.org/pythagoras/BQTLemma.shtml
https://www.cut-the-knot.org/pythagoras/Ch6Pr3Unger.shtml
https://www.cut-the-knot.org/pythagoras/RectifiedEyeball.shtml
https://www.cut-the-knot.org/pythagoras/Proof114.shtml
https://www.cut-the-knot.org/pythagoras/CirclesInCircularSegment.shtml
https://www.cut-the-knot.org/pythagoras/DonMcConnell.shtml
https://www.cut-the-knot.org/pythagoras/Proof103.shtml
https://www.cut-the-knot.org/pythagoras/slider_proof.shtml
https://www.cut-the-knot.org/pythagoras/GeneralButterflyInPictures.shtml
https://www.cut-the-knot.org/pythagoras/Apollonius.shtml
https://www.cut-the-knot.org/pythagoras/RegularHeptagon.shtml
https://www.cut-the-knot.org/pythagoras/8CirclesSangaku.shtml
https://www.cut-the-knot.org/pythagoras/1331Sangaku.shtml
https://www.cut-the-knot.org/pythagoras/RichmondPentagon.shtml
https://www.cut-the-knot.org/pythagoras/Loyds.shtml
https://www.cut-the-knot.org/pythagoras/VanKhea5.shtml
https://www.cut-the-knot.org/pythagoras/GionShrineProblem.shtml
https://www.cut-the-knot.org/pythagoras/PappusPyth.shtml
https://www.cut-the-knot.org/pythagoras/ArealButterflies.shtml
https://www.cut-the-knot.org/pythagoras/colorchange2.shtml
https://www.cut-the-knot.org/pythagoras/StraightEdgePolar.shtml
https://www.cut-the-knot.org/pythagoras/a_series.shtml
https://www.cut-the-knot.org/pythagoras/cosine2.shtml
https://www.cut-the-knot.org/pythagoras/StereoProAndInversion.shtml
https://www.cut-the-knot.org/pythagoras/NuBe-Emet.shtml
https://www.cut-the-knot.org/pythagoras/Hoehn.shtml
https://www.cut-the-knot.org/pythagoras/proof91.shtml
https://www.cut-the-knot.org/pythagoras/Pappus.shtml
https://www.cut-the-knot.org/pythagoras/MakeMove.shtml
https://www.cut-the-knot.org/pythagoras/UrquhartsTheorem.shtml
https://www.cut-the-knot.org/pythagoras/MascheroniPentagon.shtml
https://www.cut-the-knot.org/pythagoras/CosLawMolokach.shtml
https://www.cut-the-knot.org/pythagoras/Proof113.shtml
https://www.cut-the-knot.org/pythagoras/Pyth3Trapezoids.shtml
https://www.cut-the-knot.org/pythagoras/BimediansAndDiagonals.shtml
https://www.cut-the-knot.org/pythagoras/ProjectiveButterfly.shtml
https://www.cut-the-knot.org/pythagoras/PythDroodle.shtml
https://www.cut-the-knot.org/pythagoras/AbulWafaPythagoras.shtml
https://www.cut-the-knot.org/pythagoras/PropertiesI47.shtml
https://www.cut-the-knot.org/pythagoras/RopeStretchersSangaku.shtml
https://www.cut-the-knot.org/pythagoras/EquiIn3D.shtml
https://www.cut-the-knot.org/pythagoras/PointOnCircumcircle.shtml
https://www.cut-the-knot.org/pythagoras/Butterfly.shtml
https://www.cut-the-knot.org/pythagoras/colorchange.shtml
https://www.cut-the-knot.org/pythagoras/Hanner.shtml
https://www.cut-the-knot.org/pythagoras/Ch6Pr3Cabart.shtml
https://www.cut-the-knot.org/pythagoras/Transforms/index.shtml
https://www.cut-the-knot.org/pythagoras/Transforms/ProblemsByReflection.shtml
https://www.cut-the-knot.org/pythagoras/Transforms/Isometries.shtml
https://www.cut-the-knot.org/pythagoras/Transforms/ComplexIsometries.shtml
https://www.cut-the-knot.org/pythagoras/Transforms/ProblemsByHomothety.shtml
https://www.cut-the-knot.org/pythagoras/PythInPara.shtml
https://www.cut-the-knot.org/pythagoras/proof79.shtml
https://www.cut-the-knot.org/pythagoras/TrigProof2.shtml
https://www.cut-the-knot.org/pythagoras/hypo.shtml
https://www.cut-the-knot.org/pythagoras/CosineLawHoehn.shtml
https://www.cut-the-knot.org/pythagoras/FloorBuiTwins.shtml
https://www.cut-the-knot.org/pythagoras/HuygensPyth.shtml
https://www.cut-the-knot.org/pythagoras/CosLawFor60and120.shtml
https://www.cut-the-knot.org/pythagoras/CalculusProofCorrectedVersion.shtml
https://www.cut-the-knot.org/pythagoras/RepetitionSangaku.shtml
https://www.cut-the-knot.org/pythagoras/RationalPointsOnCircle.shtml
https://www.cut-the-knot.org/pythagoras/midpoints.shtml
https://www.cut-the-knot.org/pythagoras/Proof102.shtml
https://www.cut-the-knot.org/pythagoras/PentagonConstruction.shtml
https://www.cut-the-knot.org/pythagoras/ButterflyInQuad.shtml
https://www.cut-the-knot.org/pythagoras/WallaceButterfly.shtml
https://www.cut-the-knot.org/pythagoras/Bottcher.shtml
https://www.cut-the-knot.org/pythagoras/Proof56.shtml
https://www.cut-the-knot.org/pythagoras/CircleCircleEllipse.shtml
https://www.cut-the-knot.org/pythagoras/lucky7.shtml
https://www.cut-the-knot.org/pythagoras/Proof99Solution.shtml
https://www.cut-the-knot.org/pythagoras/Tangent225.shtml
https://www.cut-the-knot.org/pythagoras/NatureOfPi.shtml
https://www.cut-the-knot.org/pythagoras/TeenSangaku.shtml
https://www.cut-the-knot.org/pythagoras/SangakuInSquare.shtml
https://www.cut-the-knot.org/pythagoras/Sangaku.shtml
https://www.cut-the-knot.org/pythagoras/PointOnCircumcircle2.shtml
https://www.cut-the-knot.org/pythagoras/DistanceEquation.shtml
https://www.cut-the-knot.org/pythagoras/ChainInSemicircle.shtml
https://www.cut-the-knot.org/pythagoras/blithe12.shtml
https://www.cut-the-knot.org/pythagoras/AMatterOfCollinearity.shtml
https://www.cut-the-knot.org/pythagoras/RoundCarpets.shtml
https://www.cut-the-knot.org/pythagoras/PentagonByKnop.shtml
https://www.cut-the-knot.org/pythagoras/Munching/inscribed.shtml
https://www.cut-the-knot.org/pythagoras/Munching/circle.shtml
https://www.cut-the-knot.org/pythagoras/Munching/MissingCircle.shtml
https://www.cut-the-knot.org/pythagoras/Munching/DiameterChord.shtml
https://www.cut-the-knot.org/pythagoras/Proof98.shtml
https://www.cut-the-knot.org/pythagoras/PythagoreanPerimeterTheorem.shtml
https://www.cut-the-knot.org/pythagoras/SangakuViaPeru.shtml
https://www.cut-the-knot.org/pythagoras/LikelySangaku.shtml
https://www.cut-the-knot.org/pythagoras/ButterflyViaScaleFactor.shtml
https://www.cut-the-knot.org/pythagoras/ChainsInDoubleArbelos.shtml
https://www.cut-the-knot.org/pythagoras/LightingTheBall.shtml
https://www.cut-the-knot.org/pythagoras/withoff.shtml
https://www.cut-the-knot.org/pythagoras/pythTriple.shtml
https://www.cut-the-knot.org/pythagoras/ButterflyInInscriptibleQuadrilateral.shtml
https://www.cut-the-knot.org/pythagoras/DistanceFormula.shtml
https://www.cut-the-knot.org/pythagoras/SSA.shtml
https://www.cut-the-knot.org/pythagoras/PythagoreanMach.shtml
https://www.cut-the-knot.org/pythagoras/SteinerSangaku.shtml
https://www.cut-the-knot.org/pythagoras/AreaOfParallelogrambyShearing.shtml
https://www.cut-the-knot.org/pythagoras/TrigPythIMO.shtml
https://www.cut-the-knot.org/pythagoras/QuadrilateralFromASegment.shtml
https://www.cut-the-knot.org/pythagoras/EuclidRef_VI.shtml
https://www.cut-the-knot.org/pythagoras/BrodiePythODE.shtml
https://www.cut-the-knot.org/pythagoras/torque.shtml
https://www.cut-the-knot.org/pythagoras/3DExercise.shtml
https://www.cut-the-knot.org/pythagoras/PythFromBottema.shtml
https://www.cut-the-knot.org/pythagoras/JordanKober.shtml
https://www.cut-the-knot.org/pythagoras/DurerPentagon.shtml
https://www.cut-the-knot.org/pythagoras/5-star.shtml
https://www.cut-the-knot.org/pythagoras/PaperFolding/345Triangle.shtml
https://www.cut-the-knot.org/pythagoras/PaperFolding/arctan123.shtml
https://www.cut-the-knot.org/pythagoras/PaperFolding/index.shtml
https://www.cut-the-knot.org/pythagoras/PaperFolding/345Triangle2.shtml
https://www.cut-the-knot.org/pythagoras/PaperFolding/SegmentDivision.shtml
https://www.cut-the-knot.org/pythagoras/PaperFolding/KungAngleTrisection.shtml
https://www.cut-the-knot.org/pythagoras/PaperFolding/EquilateralTriangleInSquare.shtml
https://www.cut-the-knot.org/pythagoras/PaperFolding/Octagon.shtml
https://www.cut-the-knot.org/pythagoras/PaperFolding/345Triangle3.shtml
https://www.cut-the-knot.org/pythagoras/PaperFolding/PigeonholeExtensions.shtml
https://www.cut-the-knot.org/pythagoras/PaperFolding/PaperFoldingSangaku.shtml
https://www.cut-the-knot.org/pythagoras/PaperFolding/AngleTrisection.shtml
https://www.cut-the-knot.org/pythagoras/PaperFolding/Pentagon.shtml
https://www.cut-the-knot.org/pythagoras/AngleInSquareCojointWithPentagon.shtml
https://www.cut-the-knot.org/pythagoras/TwoButterfliesByKung.shtml
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https://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/StarElementary.shtml
https://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/FlankTriangles1.shtml
https://www.cut-the-knot.org/Curriculum/Geometry/SierpinskiCurve.shtml
https://www.cut-the-knot.org/Curriculum/Geometry/EuclidI43.shtml
https://www.cut-the-knot.org/Curriculum/Geometry/CyclicIncenters.shtml
https://www.cut-the-knot.org/Curriculum/Geometry/RightInEquilateral.shtml
https://www.cut-the-knot.org/Curriculum/Geometry/Contrast2.shtml
https://www.cut-the-knot.org/Curriculum/Magic/MindReaderNine.shtml
https://www.cut-the-knot.org/Curriculum/Magic/MagicInSquare.shtml
https://www.cut-the-knot.org/Curriculum/Magic/GergonneMagic.shtml
https://www.cut-the-knot.org/Curriculum/Magic/MindReader.shtml
https://www.cut-the-knot.org/Curriculum/Magic/MathTelepathy.shtml
https://www.cut-the-knot.org/Curriculum/Magic/Bachet.shtml
https://www.cut-the-knot.org/Curriculum/Magic/MandelCycle.shtml
https://www.cut-the-knot.org/Curriculum/Fallacies/FibonacciCheat.shtml
https://www.cut-the-knot.org/Curriculum/Fallacies/LangmanParadox.shtml
https://www.cut-the-knot.org/Curriculum/Fallacies/TwoFallacies.shtml
https://www.cut-the-knot.org/Curriculum/Fallacies/JohnSharpParadox.shtml
https://www.cut-the-knot.org/Curriculum/Fallacies/AllTrianglesIsosceles.shtml
https://www.cut-the-knot.org/Curriculum/Fallacies/SamLoydSon.shtml
https://www.cut-the-knot.org/Curriculum/Fallacies/CurryParadox.shtml
https://www.cut-the-knot.org/Curriculum/Fallacies/HooperParadox.shtml
https://www.cut-the-knot.org/Curriculum/Fallacies/CircleWithTwoCenters.shtml
https://www.cut-the-knot.org/Curriculum/Fallacies/GeoFallacy.shtml
https://www.cut-the-knot.org/Curriculum/Probability/BOW3.shtml
https://www.cut-the-knot.org/Curriculum/Probability/BOW1.shtml
https://www.cut-the-knot.org/Curriculum/Probability/Conditionalprobability.shtml
https://www.cut-the-knot.org/Curriculum/Probability/brouwer.shtml
https://www.cut-the-knot.org/Curriculum/Probability/TriProbCartesian.shtml
https://www.cut-the-knot.org/Curriculum/Probability/Triprobability.shtml
https://www.cut-the-knot.org/Curriculum/Probability/Indy.shtml
https://www.cut-the-knot.org/Curriculum/Probability/BinomialDistribution.shtml
https://www.cut-the-knot.org/Curriculum/Probability/PlainMontyHall.shtml
https://www.cut-the-knot.org/Curriculum/Probability/BOW2.shtml
https://www.cut-the-knot.org/Curriculum/Probability/Buffon.shtml
https://www.cut-the-knot.org/Curriculum/Probability/ChessTournament.shtml
https://www.cut-the-knot.org/Curriculum/Probability/BirdsOnWire.shtml
https://www.cut-the-knot.org/Curriculum/Probability/Parrondo.shtml
https://www.cut-the-knot.org/Curriculum/Probability/Ratchet.shtml
https://www.cut-the-knot.org/Curriculum/Probability/MontyHall.shtml
https://www.cut-the-knot.org/Curriculum/Probability/FamilyStats.shtml
https://www.cut-the-knot.org/Curriculum/Probability/BOW4.shtml
https://www.cut-the-knot.org/Curriculum/Algorithms/EnemyAmbassadors.shtml
https://www.cut-the-knot.org/Curriculum/Algorithms/FastSpigotForPi.shtml
https://www.cut-the-knot.org/Curriculum/Algorithms/LuckyNumbers.shtml
https://www.cut-the-knot.org/Curriculum/Algorithms/TableToZeroGame.shtml
https://www.cut-the-knot.org/Curriculum/Algorithms/Sorts.shtml
https://www.cut-the-knot.org/Curriculum/Algorithms/BaseConversion.shtml
https://www.cut-the-knot.org/Curriculum/Algorithms/SpigotForPi.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/PowerIndices.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/SealedBids.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/ApportionmentApplet.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/VotingInstructions.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/ShapleyShubikIndex.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/BanzhafIndex.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/MethodOfMarkersInstructions.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/BanzhafInstructions.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/Jefferson.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/SocialChoice.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/AHamilton.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/Webster.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/ShapleyShubikInstructions.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/HH.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/Markers.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/LoneDivider.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/MethodOfMarkers.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/SealedBidsInstructions.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/ApportionmentInstructions.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/Voting.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/Adams.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/SealedBids2009.shtml
https://www.cut-the-knot.org/Curriculum/SocialScience/PowerIndex.shtml
https://www.cut-the-knot.org/Curriculum/classes/index.shtml
https://www.cut-the-knot.org/Curriculum/classes/Last.shtml
https://www.cut-the-knot.org/algebra.shtml
https://www.cut-the-knot.org/changes2000.shtml
https://www.cut-the-knot.org/pitch.shtml
https://www.cut-the-knot.org/changes98.shtml
https://www.cut-the-knot.org/nim_st.shtml
https://www.cut-the-knot.org/basicR0Spot.shtml
https://www.cut-the-knot.org/SpecialBin/pythagoras/index.shtml
https://www.cut-the-knot.org/pauling4.shtml
https://www.cut-the-knot.org/changes2010.shtml
https://www.cut-the-knot.org/changes2005.shtml
https://www.cut-the-knot.org/Optimization/ShortestCrease.shtml
https://www.cut-the-knot.org/Optimization/ParallelogramAndRhombus.shtml
https://www.cut-the-knot.org/Optimization/CroatianTST.shtml
https://www.cut-the-knot.org/Optimization/Huygens.shtml
https://www.cut-the-knot.org/Optimization/MinimumUnderTwoConstraints.shtml
https://www.cut-the-knot.org/Optimization/DanielDanInThreeVariables.shtml
https://www.cut-the-knot.org/Optimization/MinimizationOfCyclicSum.shtml
https://www.cut-the-knot.org/Optimization/AreaOfIsoscelesTriangle.shtml
https://www.cut-the-knot.org/Optimization/CubicOptimizationWithLinearConstraint.shtml
https://www.cut-the-knot.org/Optimization/OptimalResidence.shtml
https://www.cut-the-knot.org/Optimization/WeightedMagic.shtml
https://www.cut-the-knot.org/Optimization/FourVariablesTwoConstraints.shtml
https://www.cut-the-knot.org/Optimization/CroatianTSTVariants.shtml
https://www.cut-the-knot.org/Optimization/CubicOptimizationWithLinearConstraint2.shtml
https://www.cut-the-knot.org/Optimization/OptimizationInParallelepiped.shtml
https://www.cut-the-knot.org/Optimization/TriangleOfMaximumArea2.shtml
https://www.cut-the-knot.org/Optimization/OldVietnameseProblem.shtml
https://www.cut-the-knot.org/Optimization/HellenicMO2006.shtml
https://www.cut-the-knot.org/Optimization/ConstrainedInequalityWithManyVariables.shtml
https://www.cut-the-knot.org/Optimization/CrookedTrapezoid.shtml
https://www.cut-the-knot.org/Optimization/MD.shtml
https://www.cut-the-knot.org/Optimization/DistanceBetweenProjections.shtml
https://www.cut-the-knot.org/Optimization/TriangleOfMaximumArea.shtml
https://www.cut-the-knot.org/Optimization/AreaOptimizationInTrapezoid.shtml
https://www.cut-the-knot.org/Optimization/QuadrilateralInSquare.shtml
https://www.cut-the-knot.org/Optimization/TwoSquaresInCircle.shtml
https://www.cut-the-knot.org/Optimization/GreatestDifference.shtml
https://www.cut-the-knot.org/pancakes.shtml
https://www.cut-the-knot.org/proofs/USAMO2001_6.shtml
https://www.cut-the-knot.org/proofs/InfinitudeOfProofs.shtml
https://www.cut-the-knot.org/proofs/half_sq.shtml
https://www.cut-the-knot.org/proofs/ChromaticNumber.shtml
https://www.cut-the-knot.org/proofs/Semicircles.shtml
https://www.cut-the-knot.org/proofs/FunctionInPlane.shtml
https://www.cut-the-knot.org/proofs/PtolemyDiagonals.shtml
https://www.cut-the-knot.org/proofs/chocolad.shtml
https://www.cut-the-knot.org/proofs/tessellation.shtml
https://www.cut-the-knot.org/proofs/ZeroDerivative.shtml
https://www.cut-the-knot.org/proofs/AveragesIII.shtml
https://www.cut-the-knot.org/proofs/average.shtml
https://www.cut-the-knot.org/proofs/OneLineProofOfInfinitudeOfPrimes.shtml
https://www.cut-the-knot.org/proofs/MorenoSqRt2.shtml
https://www.cut-the-knot.org/proofs/NapoleonLeo.shtml
https://www.cut-the-knot.org/proofs/ptolemy.shtml
https://www.cut-the-knot.org/proofs/eulers.shtml
https://www.cut-the-knot.org/proofs/SimpleIntegral.shtml
https://www.cut-the-knot.org/proofs/integral.shtml
https://www.cut-the-knot.org/proofs/checker.shtml
https://www.cut-the-knot.org/proofs/Averages.shtml
https://www.cut-the-knot.org/proofs/index.shtml
https://www.cut-the-knot.org/proofs/sine_cosine.shtml
https://www.cut-the-knot.org/proofs/Diagrams.shtml
https://www.cut-the-knot.org/proofs/Furstenberg.shtml
https://www.cut-the-knot.org/proofs/Ants.shtml
https://www.cut-the-knot.org/proofs/e-pi-spoof.shtml
https://www.cut-the-knot.org/proofs/AM-GMExtensionSecondLook.shtml
https://www.cut-the-knot.org/proofs/2Colors3Points.shtml
https://www.cut-the-knot.org/proofs/HalfQuarterCircle.shtml
https://www.cut-the-knot.org/proofs/PtolemyTheoremPWW.shtml
https://www.cut-the-knot.org/proofs/Nimbran.shtml
https://www.cut-the-knot.org/proofs/lattice.shtml
https://www.cut-the-knot.org/proofs/SumOfAllNatNum.shtml
https://www.cut-the-knot.org/proofs/PegSquare.shtml
https://www.cut-the-knot.org/proofs/YAAP.shtml
https://www.cut-the-knot.org/proofs/MorenoSqRt3.shtml
https://www.cut-the-knot.org/proofs/BadSolutionForInequalityFromMarocco.shtml
https://www.cut-the-knot.org/proofs/AllPowersOf2Equal1.shtml
https://www.cut-the-knot.org/proofs/BinetFormula.shtml
https://www.cut-the-knot.org/proofs/threecircles.shtml
https://www.cut-the-knot.org/proofs/SumOfAllNatNumConventional.shtml
https://www.cut-the-knot.org/proofs/AllNumbersEqual.shtml
https://www.cut-the-knot.org/proofs/fords.shtml
https://www.cut-the-knot.org/proofs/AfterEuler.shtml
https://www.cut-the-knot.org/proofs/2Colors2Points.shtml
https://www.cut-the-knot.org/proofs/sq_root.shtml
https://www.cut-the-knot.org/proofs/FibonacciPrimes.shtml
https://www.cut-the-knot.org/proofs/CircleThroughIncenter.shtml
https://www.cut-the-knot.org/proofs/3ColorsBichromaticLines.shtml
https://www.cut-the-knot.org/proofs/AreasInCircle.shtml
https://www.cut-the-knot.org/proofs/HalfIsOne.shtml
https://www.cut-the-knot.org/proofs/PointInTriangle.shtml
https://www.cut-the-knot.org/proofs/inequality.shtml
https://www.cut-the-knot.org/proofs/primes.shtml
https://www.cut-the-knot.org/proofs/IntersectingChordsTheorem.shtml
https://www.cut-the-knot.org/proofs/mediant.shtml
https://www.cut-the-knot.org/proofs/FermatPrimes.shtml
https://www.cut-the-knot.org/proofs/LowBoundsOnNumberOfPrimes.shtml
https://www.cut-the-knot.org/proofs/IntegralIsArea.shtml
https://www.cut-the-knot.org/proofs/SimpleIntegral3.shtml
https://www.cut-the-knot.org/proofs/two_color.shtml
https://www.cut-the-knot.org/proofs/LinesDividePlane.shtml
https://www.cut-the-knot.org/proofs/jap.shtml
https://www.cut-the-knot.org/proofs/NineMorePoints.shtml
https://www.cut-the-knot.org/proofs/SylvesterLang.shtml
https://www.cut-the-knot.org/proofs/Lemma.shtml
https://www.cut-the-knot.org/proofs/Saidak.shtml
https://www.cut-the-knot.org/proofs/Proofreading.shtml
https://www.cut-the-knot.org/proofs/arbelos.shtml
https://www.cut-the-knot.org/proofs/2ColorsOnLine.shtml
https://www.cut-the-knot.org/proofs/SqRtParadox.shtml
https://www.cut-the-knot.org/proofs/GraphicalSqRoots.shtml
https://www.cut-the-knot.org/proofs/SqContinuedFraction.shtml
https://www.cut-the-knot.org/proofs/InfinitudeOfPrimes.shtml
https://www.cut-the-knot.org/proofs/WhatIsInfiniteDescent.shtml
https://www.cut-the-knot.org/proofs/PtolemyCrossRatio.shtml
https://www.cut-the-knot.org/proofs/ManyCircles.shtml
https://www.cut-the-knot.org/proofs/3circlesFormal.shtml
https://www.cut-the-knot.org/proofs/Pivot.shtml
https://www.cut-the-knot.org/proofs/NormalDistribution.shtml
https://www.cut-the-knot.org/proofs/Golomb.shtml
https://www.cut-the-knot.org/proofs/napoleon_complex.shtml
https://www.cut-the-knot.org/proofs/piles.shtml
https://www.cut-the-knot.org/proofs/DivisibilityByPowersOf2.shtml
https://www.cut-the-knot.org/proofs/WonderfulAlgebraOnUnitCircle.shtml
https://www.cut-the-knot.org/proofs/napoleon.shtml
https://www.cut-the-knot.org/proofs/TriangleIn3D.shtml
https://www.cut-the-knot.org/proofs/PtolemyByInversion.shtml
https://www.cut-the-knot.org/proofs/StarSets.shtml
https://www.cut-the-knot.org/proofs/napoleon_intro.shtml
https://www.cut-the-knot.org/proofs/ArbelosMorsel.shtml
https://www.cut-the-knot.org/proofs/InequalityInTriangle3.shtml
https://www.cut-the-knot.org/proofs/Sylvester.shtml
https://www.cut-the-knot.org/proofs/longest.shtml
https://www.cut-the-knot.org/proofs/AveragesII.shtml
https://www.cut-the-knot.org/proofs/UniversalInequalityForCevians.shtml
https://www.cut-the-knot.org/proofs/SylvesterSteinberg.shtml
https://www.cut-the-knot.org/proofs/WhicheverWay.shtml
https://www.cut-the-knot.org/proofs/GasStations.shtml
https://www.cut-the-knot.org/proofs/GeometricMeanInTrapezoid.shtml
https://www.cut-the-knot.org/proofs/TriangleInequality.shtml
https://www.cut-the-knot.org/proofs/Vandermonde.shtml
https://www.cut-the-knot.org/proofs/SqRtDifEq.shtml
https://www.cut-the-knot.org/proofs/Beatty2.shtml
https://www.cut-the-knot.org/proofs/Uncountability.shtml
https://www.cut-the-knot.org/proofs/2ColorsAllDistances.shtml
https://www.cut-the-knot.org/proofs/Mercer.shtml
https://www.cut-the-knot.org/proofs/ParaIsRect.shtml
https://www.cut-the-knot.org/proofs/Whittaker.shtml
https://www.cut-the-knot.org/proofs/NegativeCoconuts.shtml
https://www.cut-the-knot.org/proofs/ShiftingDigitsAndViewPoint.shtml
https://www.cut-the-knot.org/proofs/InfinitudeOfPrimesFromAQuickie.shtml
https://www.cut-the-knot.org/proofs/TwoVolumes.shtml
https://www.cut-the-knot.org/proofs/ExponentialInequalityForMeans.shtml
https://www.cut-the-knot.org/proofs/conics.shtml
https://www.cut-the-knot.org/proofs/InscriptibleQuadrilateral.shtml
https://www.cut-the-knot.org/proofs/bisect.shtml
https://www.cut-the-knot.org/proofs/DiagonalArgument.shtml
https://www.cut-the-knot.org/proofs/powers.shtml
https://www.cut-the-knot.org/proofs/FourWeighings.shtml
https://www.cut-the-knot.org/proofs/SylvesterKelly.shtml
https://www.cut-the-knot.org/proofs/Why17.shtml
https://www.cut-the-knot.org/proofs/BertrandPrimes.shtml
https://www.cut-the-knot.org/proofs/InjectivePrimes.shtml
https://www.cut-the-knot.org/proofs/circlesAndSpheres.shtml
https://www.cut-the-knot.org/proofs/AllIntegersAreEven.shtml
https://www.cut-the-knot.org/proofs/napoleon_complex2.shtml
https://www.cut-the-knot.org/proofs/GeneralFromMutuallyPrimes.shtml
https://www.cut-the-knot.org/proofs/Beatty.shtml
https://www.cut-the-knot.org/proofs/AnotherIntegral.shtml
https://www.cut-the-knot.org/proofs/rsolitaire.shtml
https://www.cut-the-knot.org/proofs/FalseSymmetry.shtml
https://www.cut-the-knot.org/proofs/ExtremesInQuadrilateral.shtml
https://www.cut-the-knot.org/proofs/SqRtFromLimitLemma.shtml
https://www.cut-the-knot.org/proofs/tictactoe.shtml
https://www.cut-the-knot.org/proofs/ThueInfinitudeOfPrimes.shtml
https://www.cut-the-knot.org/proofs/swivel.shtml
https://www.cut-the-knot.org/proofs/NapoleonNicollier.shtml
https://www.cut-the-knot.org/proofs/SylvesterGallai.shtml
https://www.cut-the-knot.org/proofs/FloydSqRt.shtml
https://www.cut-the-knot.org/proofs/HarukiTheorem.shtml
https://www.cut-the-knot.org/proofs/ClubsInVectorSpace.shtml
https://www.cut-the-knot.org/proofs/ShortestIsLongest.shtml
https://www.cut-the-knot.org/proofs/HarmonicPrimes.shtml
https://www.cut-the-knot.org/proofs/three_colors.shtml
https://www.cut-the-knot.org/proofs/Menelaus.shtml
https://www.cut-the-knot.org/proofs/even_spaced.shtml
https://www.cut-the-knot.org/proofs/ClassicalIntegral.shtml
https://www.cut-the-knot.org/proofs/AfterEulerProduct.shtml
https://www.cut-the-knot.org/proofs/InfinitudeOfPrimesViaPowersOfTwo.shtml
https://www.cut-the-knot.org/proofs/McDougallPtolemy.shtml
https://www.cut-the-knot.org/proofs/CarnotFromPtolemy.shtml
https://www.cut-the-knot.org/proofs/IntegerRabbits.shtml
https://www.cut-the-knot.org/proofs/carnot.shtml
https://www.cut-the-knot.org/proofs/nap_circles.shtml
https://www.cut-the-knot.org/proofs/MorenoSqRt1.shtml
https://www.cut-the-knot.org/proofs/3circles.shtml
https://www.cut-the-knot.org/proofs/orthocenter.shtml
https://www.cut-the-knot.org/proofs/pegsolitaire.shtml
https://www.cut-the-knot.org/proofs/PegsAndGroups.shtml
https://www.cut-the-knot.org/proofs/SqInSemi.shtml
https://www.cut-the-knot.org/MediaKit/MKRates.shtml
https://www.cut-the-knot.org/MediaKit/MKContact.shtml
https://www.cut-the-knot.org/MediaKit/MKUniq.shtml
https://www.cut-the-knot.org/MediaKit/MKVis.shtml
https://www.cut-the-knot.org/MediaKit/MKPrivacy.shtml
https://www.cut-the-knot.org/MediaKit/MKTerms.shtml
https://www.cut-the-knot.org/Probability/IndependentExperiments.shtml
https://www.cut-the-knot.org/Probability/NonTransitiveDice.shtml
https://www.cut-the-knot.org/Probability/ConditionalRecurrence.shtml
https://www.cut-the-knot.org/Probability/ContinuousSampleSpaces.shtml
https://www.cut-the-knot.org/Probability/RedBlueTaxicabs.shtml
https://www.cut-the-knot.org/Probability/LuckyWinners.shtml
https://www.cut-the-knot.org/Probability/ProbabilisticMethod.shtml
https://www.cut-the-knot.org/Probability/EmbarrassingQuestion.shtml
https://www.cut-the-knot.org/Probability/RainDropsX.shtml
https://www.cut-the-knot.org/Probability/IndexRange.shtml
https://www.cut-the-knot.org/Probability/Divisibility.shtml
https://www.cut-the-knot.org/Probability/PlainRedBlueBalls.shtml
https://www.cut-the-knot.org/Probability/Conditionalprobability.shtml
https://www.cut-the-knot.org/Probability/TenLetters.shtml
https://www.cut-the-knot.org/Probability/AmoebaSurvival.shtml
https://www.cut-the-knot.org/Probability/Expectation.shtml
https://www.cut-the-knot.org/Probability/SampleSpacesImportance.shtml
https://www.cut-the-knot.org/Probability/SampleSpaces.shtml
https://www.cut-the-knot.org/Probability/ObtuseTriangle.shtml
https://www.cut-the-knot.org/Probability/SeekAndFind.shtml
https://www.cut-the-knot.org/Probability/PokerSampleSpaces.shtml
https://www.cut-the-knot.org/Probability/GeometricProbabilities.shtml
https://www.cut-the-knot.org/Probability/dAlembert.shtml
https://www.cut-the-knot.org/Probability/Proofreading.shtml
https://www.cut-the-knot.org/Probability/TwoEnvelopes.shtml
https://www.cut-the-knot.org/Probability/BearBornOnTuesday.shtml
https://www.cut-the-knot.org/Probability/RuleOfSuccession.shtml
https://www.cut-the-knot.org/Probability/GladiatorGame.shtml
https://www.cut-the-knot.org/Probability/Paradox.shtml
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