_1: ...that outstanding mathematicianGrigori Perelmanwas offered aFields Medalin 2006, in part for his proof of thePoincar conjecture, which he declined? _2: ...that a regularheptagonis theregular polygonwith the fewest number of sides which is notconstructiblewith acompass and straightedge? _3: ...that theGudermannian functionrelates the regulartrigonometric functionsand thehyperbolic trigonometric functionswithout the use ofcomplex numbers? _4: ...that theCatalan numberssolve a number of problems incombinatoricssuch as the number of ways to completely parenthesize an algebraic expression withn+1 factors? _5: ...that aballcan be cut up and reassembled into two balls the same size as the original (Banach-Tarski paradox)? _6: ...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitraryquintic equation? _7: ...thatEulerfound 59 moreamicable numberswhile for 2000 years, only 3 pairs had been found before him? _8: ...that you cannotknot stringsin 4-dimensions? You can, however, knot 2-dimensionalsurfaceslikespheres. _9: ...that there are6 unsolved mathematics problemswhose solutions will earn you one million US dollars each? _10: ...that there are different sizes ofinfinite setsinset theory? More precisely, not all infinitecardinal numbersare equal? _11: ...that everynatural numbercan be written as the sum offour squares? _12: ...that thelargest known prime numberis over 17milliondigits long? _13: ...that the set ofrational numbersis equal in size to the subset ofintegers; that is, they can be put inone-to-one correspondence? _14: ...that there are precisely sixconvex regular polytopesin fourdimensions? These are analogs of the fivePlatonic solidsknown to theancient Greeks. _15: ...that it is unknown whetherandearealgebraically independent? _16: ...that a nonconvexpolygonwith threeconvexvertices is called apseudotriangle? _17: ...that it is possible for a three dimensional figure to have a finitevolumebut infinitesurface area? An example of this isGabriel's Horn. _18: ... that as thedimensionof ahyperspheretends to infinity, its "volume" (content) tends to 0? _19: ...that the primality of a number can be determined using only a single division usingWilson's Theorem? _20: ...that the line separating thenumeratoranddenominatorof afractionis called asolidusif written as a diagonal line or avinculumif written as a horizontal line? _21: ...that a monkey hitting keys atrandomon a typewriter keyboard for an infinite amount of time willalmost surely typethe complete works of William Shakespeare? _22: ... that there are 115,200 solutions to themnage problemofpermutingsix female-male couples at a twelve-person table so that men and women alternate and are seated away from their partners? _23: ... thatmathematicianPaul Erdscalled theHadwiger conjecture, a still-open generalization of thefour-color problem, "one of the deepest unsolved problems ingraph theory"? _24: ...that the sixpermutationsof thevector(1,2,3) form aregular hexagonin3d space, the 24 permutations of (1,2,3,4) form atruncated octahedroninfour dimensions, and both are examples ofpermutohedra? _25: ...thatOstomachionis a mathematicaltreatiseattributed toArchimedeson a 14-piecetiling puzzlesimilar totangram? _26: ...that some functions can be written as aninfinite sumoftrigonometric polynomialsand that this sum is called theFourier seriesof that function? _27: ...that theidentity elementsforarithmetic operationsmake use of the only twowhole numbersthat are neithercompositesnorprime numbers,0and1? _28: ...that as of April 2010 only 35 even numbers have been found that are not the sum of two primes which are each in aTwin Primespair?ref _29: ...thePiphilologyrecord (memorizing digits ofPi) is in excess of 67000 as of Apr 2010? _30: ...with aPerrin numberdenoted P(i), i=1,2,3..., when i is prime then P(i) is composite, being divisible by i? _31: ...thatAuction theorywas successfully used in 1994 to sellFCCairwave spectrum, in a financial application ofgame theory? _32: ...properties ofPascal's trianglehave application in many fields ofmathematicsincludingcombinatorics,algebra,calculusandgeometry? _33: ...work inartificial intelligencemakes use ofSwarm intelligence, which has foundations in the behavorial examples found in nature of ants, birds, bees, and fish among others? _34: ...that statistical properties dictated byBenford's Laware used in auditing of financial accounts as one means of detecting fraud? _35: ...thatModular arithmetichas application in at least ten different fields of study, including the arts, computer science, and chemistry in addition to mathematics? _36: ... that according toKawasaki's theorem, anorigamicrease patternwith onevertexmay befolded flatif and only if the sum of every other angle between consecutive creases is 180? _37: ... that, in theRule 90cellular automaton, any finite pattern eventually fills the whole array of cells with copies of itself? _38: ... that, while thecriss-cross algorithmvisits all eight corners of theKleeMinty cubewhen started at aworstcorner, it visits only three more cornerson averagewhen started at arandomcorner? _39: ...that insenary, allprime numbersother than 2 and 3 end in 1 or a 5? _40: ... if the integernisprime, then thenthPerrin numberis divisible byn? _41: ...that it is impossible totrisect a general angleusing only aruler and a compass? _42: ...that in a group of 23 people, there is a more than 50% chance that two peopleshare a birthday? _43: ...that statistical properties dictated byBenford's Laware used in auditing of financial accounts as one means of detecting fraud? _44: ...thehyperbolictrigonometricfunctions of thenatural logarithmcan be represented byrationalalgebraic fractions? _45: ... that economists blamemarketfailuresonnon-convexity? _46: ... that, according to thepizza theorem, acircularpizzathat is sliced off-center into eight equal-angled wedges can still bedivided equallybetween two people? _47: ... that theclique problemof programming a computer to findcompletesubgraphsin anundirected graphwas first studied as a way to find groups of people who all know each other insocial networks? _48: ... that theHerschel graphis the smallest possiblepolyhedral graphthat does not have aHamiltonian cycle? _49: ... that theLife without Deathcellular automaton, a mathematical model ofpattern formation, is a variant ofConway's Game of Lifein which cells, once brought to life, never die? _50: ... that one can list everypositiverational numberwithout repetition bybreadth-first traversalof theCalkinWilf tree? _51: ... that theHadwiger conjectureimplies that the external surface of anythree-dimensionalconvex bodycan beilluminatedby only eight light sources, but the bestprovenbound is that 16 lights are sufficient? _52: ... that anequitable coloringof agraph, in which the numbers of vertices of each color are as nearly equal as possible, may require far more colors than agraph coloringwithout this constraint? _53: ... that no matter howbiased a coinone uses,flipping a cointo determine whether eachedgeis present or absent in acountably infinitegraphwill always producethe same graph, theRado graph? _54: ...that it is possible tostack identical dominoesoff the edge of a table to create an arbitrarily large overhang? _55: ...that inFloyd's algorithmforcycle detection, thetortoise and haremove at very different speeds, but always finish at the same spot? _56: ...that ingraph theory, apseudoforestcan containtreesand pseudotrees, but cannot contain any butterflies, diamonds, handcuffs, or bicycles? _57: ...that it is not possible toconfiguretwo mutuallyinscribedquadrilateralsin theEuclidean plane, but theMbiusKantor graphdescribes a solution in thecomplex projective plane? _58: ...that the sixpermutationsof thevector(1,2,3) form ahexagonin3d space, the 24 permutations of (1,2,3,4) form atruncated octahedroninfour dimensions, and both are examples ofpermutohedra? _59: ...that theRule 184cellular automatoncan simultaneously model thebehavior of cars moving in traffic, theaccumulation of particles on a surface, and particle-antiparticleannihilationreactions? _60: ...that acyclic cellular automatonis a system of simple mathematical rules that can generate complex patterns mixing random chaos, blocks of color, and spirals? _61: ...that a nonconvexpolygonwith threeconvexvertices is called apseudotriangle? _62: ...that theaxiom of choiceislogically independentof the other axioms ofZermeloFraenkel set theory? _63: ...that thePythagorean Theoremgeneralizes to any three similar shapes on the three sides of a right-angled triangle? _64: ...that theorthocenter,circumcenter,centroidand the centre of thenine-point circleall line on one line, theEuler line? _65: ...that an arbitraryquadrilateralwilltessellate? _66: ...that it has not been proven whether or notevery even integer greater than two can be expressed as the sum of two primes? _67: ...that thesumof the firstnodd numbersdivided by the sum of the nextnodd numbers is always equal to one third? _68: ...thatito the power ofi, whereiis thesquare root of -1, is a real number? _69: ...that9814072356is the largest square number using each of the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0 exactly once? _70: ...there are (19) consecutive prime numbers ending in the digit 1, starting from 253931039382791? _71: ...that theElectronic Frontier Foundationfunds awards for the discovery ofprime numbersbeyond certain sizes? _72: ...aninfinite, nonrepeating decimalcan be represented using only the number 1 usingcontinued fractions?