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Title: Math/Recreations - HAKMEM A collection of problems from MIT. Work reported herein was conducted at the Artificial Intelligence Laboratory, a Massachusetts Institute of Technology research program.

How_To_Manually_Find_A_Square_Root The way to do find a square root without a calculator.

Iamond A page on polyiamond puzzles. Includes many pages on tessellation.

Jlpe\'s_Number_Recreations_Page Features original number recreations by the author, such as generalized perfect numbers, digital diversions, diophantine equations, didactic numbers, and number theory.

Look_and_Say_Sequence_Generator Creates a special kind of summation formula created by John Conway.

MAA_Library_List_in_Recreational_Mathematics Book list split into categories. Includes title, author, publisher and date information about each book.

MAT_007_I_News A newsletter edited by undergraduates of the Department of Mathematics at the University of Toronto. Includes some online copies.

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Math_Forum__2001_Mathematics_Game A contest that asks to write all integers from 1 to 100 using only the digits 2,0,0,1 and arithmetic operations.

Math_Forum__2002_Mathematics_Game A contest where the contestants have to write all integers from 1 to 100 using only the digits 2,0,0,2 and arithmetic operations.

Math_Forum__Algebra_Problem_of_the_Week View this week's algebra problem or those of previous weeks.

Math_in_the_Movies A guide to major motion pictures with scenes of real mathematics.

Math_Magic_By_Computer Interactive java puzzles and activities in different mathematical topics.

Mathematical_Diversions Mostly original diversions in mathematics and word play.

Mathematical_Fiction A list of mathematical fictional movies, books, stories, plays and shows. Split into categories including children's books. Can be sorted by mathematical content and literary quality.

Mathematical_Induction A page of uncommon problems, most closely connected with number theory.

Mathematical_Lego_Sculptures Designed and built by Andrew Lipson. Images and LDraw files.

Mathematical_Problems_-_Problem_Solving Mathematics Hots (Problems) by Bruno Kevius

Mathematical_Quotation_Server Furman University Mathematical Quotation Server - random or searchable math quotes.

Mathematical_Quotation_Server A collection of mathematical quotations culled from many sources. You may conduct a keyword search through the quotation database.

Mathematical_Spectrum Mathematical Spectrum is a magazine for students and teachers of mathematics in universities, colleges and schools worldwide. It may be read by anybody interested in mathematics as a recreation.

A__Mathematician\'s_Aesthetics In his classic A Mathematician's Apology, G. H. Hardy likened mathematics to poetry and painting. This site elaborates on Hardy's remark with quotations from Stevens, Klee, Fry, and Focillon. Links to

Mathematics_in_Art_and_Architecture An interdisciplinary course on mathematics in art and architecture.

Mathematics_Museum_(Japan) At Mathematics Museum (Japan) you would be surprised how interesting mathematics is. You will find exhibition rooms produced by Japanese researchers and educators.

Mathematik Individual pages on different topics in Mathematics. Examples : group theory, dynamical systems theory, geometry or number theory.

Mathematische_Basteleien Topics include Flexagon, Soma Cube, Pentominos, Cube-it, Rubik's Cube, Froebel's Star, Tangram, House of Santa Claus, Chronogram, Numeric Palindromes, Latticework of Letters. English/German.

Mathematrix Devoted to exploring the more entertaining (and generally lesser known) areas of mathematics. Can be enjoyed by anyone, from individuals with little or no math background to professors of the subject.

Mathmos Includes puzzles, jokes, quotations, poetry, and FAQs.

Maze_Classification_and_Algorithms A short description of mazes and how to create them. Definition of different mazetypes and their algorithms.

Mudd_Math_Fun_Facts An archive of interesting math facts for use in the classroom or just for fun. Browse by subject, difficulty, keywords, or try the "random" feature. Based at Harvey Mudd College.

Narcissistic_Numbers Those that are representable, in some way, by mathematically manipulating the digits of the numbers themselves.

The_Nine_Digits_Page Puzzles and problems connected with numbers using the digits 1-9.

Number_Recreations_by_Shyam_Sunder_Gupta Features interesting facts about different numbers. Includes favorite related links.

On_the_Puzzles_with_Polyhedra_and_Numbers This is an article on a set of didactical games edited by the Portuguese Mathematical Society (SPM).

One_Metaphor_Fits_All Explains Conway's audioactive decay that is generated by a particular kind of sequence. Includes illustrations and related resources.

Properties_of_Dice Polyhedral dice and their properties.

Recmath Includes pages on magic squares and polyomino patterns and contains related java applets.

Recreational_and_Educational_Computing A newsletter with programs, including optional supplemental PC disk. All back issues are available. Topics include puzzles and teasers, BASIC programming, letters, graphics, fractals, challenges, r

Recreational_Mathematics Links collected at CAMEL, the Canadian Mathematical Society website.

Recreational_Mathematics_(David_Eppstein) An extensive list of web resources for recreational math.

Recreational_Mathematics_Forum A forum for posting messages about math recreations. Hosted at Delphi.

Recreational_Mathematics_Topics By Steven Dutch. Symmetry, Crystals, Polyhedra and Tilings; Pythagorean triplets and other things about sums of powers; Geometry Classics.

Roman_Numerals Contains a introduction to Roman numerals including a translation of the digits used and a converter which can convert decimal to Roman numerals and vice versa.

Rubik\'s_Cube_Lecture_Notes Notes on the mathematics of the Rubik's cube.

SimonSingh_net Home of Simon Singh: author, journalist and TV producer, specialising in science and mathematics. Cryptography is one of his specialties, and his site has a lot of educational and fun content about c

Sir_Roger_Penrose A article about him and his interests and contributions to recreational mathematics.

Skytopia_-_Super_Magnet A colourful world built entirely using mathematical atoms and molecules. Pictures and animations demonstrate structures colliding and interacting. Animated GIF demonstrations.

The_Sound_of_Mathematics Algorithmic music determined by mathematics and by the musical preferences of a human. General MIDI files.

This is sites2007.com cache of m/ as retrieved on 2008.12.02 sites2007.com's cache is the snapshot that we took of the page as we crawled the web. The page may have changed since that time.

HAKMEM -- CONTENTS -- DRAFT, NOT YET PROOFEDMassachusetts Institute of TechnologyA. I. LaboratoryArtificial Intelligence Memo No. 239February 29, 1972

HAKMEM

contents indexM. Beeler [beeler@bbn.com]R. W. Gosper [rwg@newton.macsyma.com]R. Schroeppel [rcs@cs.arizona.edu][Retyped and formatted in'html'('Web browser format) byHenry Baker,April, 1995. The goal of this 'html' document is to make HAKMEMavailable to the widest possible audience -- including those withoutbitmapped graphics browsers. Therefore, equations have been formattedto be readable even on ASCII browsers such as'lynx'. Clickhereto get original AI Memo 239 in 400 dots/inch, 1 bit/pixel, Group 4 facsimileTIFFformat (a single 5 megabytegzipcompressed tar file, AIM-239.tiff.tar.gz).]Work reported herein was conducted at the Artificial IntelligenceLaboratory, a Massachusetts Institute of Technology research programsupported in part by theAdvanced Research Projects Agencyof the Department of Defense and monitored by the Office of NavalResearch under Contract Number N00014-70-A-0362-0002.Reproduction of this document, in whole or in part, is permitted for anypurpose of the United States Government.Compiled with the hope that a record of the random things people do around herecan save some duplication of effort -- except for fun.Here is some little known data which may be of interest to computer hackers.The items and examples are so sketchy that to decipher them may require moresincerity and curiosity than a non-hacker can muster. Doubtless, little ofthis is new, but nowadays it's hard to tell. So we must be content to give youan insight, or save you some cycles, and to welcome further contributions ofitems, new or used.The classification of items into sections is even more illogical thannecessary. This is because later elaborations tend to shift perspective onmany items, and this elaboration will (hopefully) continue after publication,since this text is retained in "machinable" form. We forgive in advance anyonedeterred by this wretched typography.People referred to are from the A. I. Lab:Marvin Minsky [minsky@ai.mit.edu]Bill Gosper [rwg@newton.macsyma.com]Michael Beeler [beeler@bbn.com]Joe CohenJerry FreibergJohn RoeRich Schroeppel [rcs@cs.arizona.edu]David SilverMichael SpecinerRichard Stallman [rms@ai.mit.edu]Gerald Sussman [gjs@ai.mit.edu]David WaltzOnce at the A. I. Lab but now elsewhere:Jan Kok William HennemanRici Liknaitzky George MitchellPeter Samson Stuart NelsonRoger Banks Rollo SilverMike Paterson [Mike.Paterson@dcs.warwick.ac.uk]at Digital Equipment Corporation:Jud Lenard Dave PlumerBen Gurley (deceased) Steve Rootelsewhere at M.I.T.:Gene Salamin PDP-1 hackersEric Jensen Frances YaoEdward Fredkinonce at M.I.T., but now elsewhere:Jackson Wright Steve BrownMalcolm Rayfieldin France:Marco Schutzenberger Henry Cohenat Computer Corporation of America:Bill Mannat BBN:Robert ClementsCAVEATS:Some of this material is very inside -- many readers will have to excusecryptic references.The label "PROBLEM" does not always mean exercise; if no solution is given, itmeans we couldn't solve it. If you solve a problem in here, let us know.Unless otherwise stated, all computer programs are inPDP-6/10 assembly language.

CONTENTS, HAKMEM 140

Geometry, Algebra, CalculusItem 1 Fractional factorialsItem 2 N-gon diagonalsItem 3 Convergence of Newton's square rootsItem 4 Quartic discriminantItem 5 General discriminantItem 6 Symmetric functionsItem 7 Symmetric functionsItem 8 Cubic solutionItem 9 Quartic solutionItem 10 Trigonometric cubic solutionItem 11 Conformal mappings of N-spaceRecurrence RelationsItem 12 Fast Fibonacci TransformItem 13 Linear Recurrence RelationsItem 14 Rapid RecurrencesItem 15 Chebychev polynomialsItem 16 Tangents and arctangentsBoolean AlgebraItem 17 Minimize AND'sItem 18 Count monotonic Boolean functionsItem 19 2-NOT'sItem 20 Venn DiagramsItem 21 Cover character rasterItem 22 Binary masksItem 23 Plus and XORItem 24 Venn DiagramsRandom NumbersItem 25 Random number generatorsItem 26 Gaussian distributionItem 27 Random unit vectors in N-spaceNumber Theory, Primes, ProbabilityItem 28 Mersenne 125 is compositeItem 29 Probability of largest prime factorItem 30 Twin PrimesItem 31 A Ramanujan problemItem 32 Distribution of fractionsItem 33 Russian doll primesItem 34 Factorial and integer square rootItem 35 Arithmetic progressionsItem 36 Squares with no zero digitsItem 37 Numbers with each digitItem 38 Forward differencesItem 39 Sequences of digitsItem 40 Variance of a sumItem 41 Number of primes less than 2^18Item 42 Probability of PingPong winnerItem 43 Multinomial coefficientsItem 44 Recurrences for multinomial coefficientsItem 45 Locus of stepsItem 46 Distribution of bridge handsItem 47 Fibonacci series mod PItem 48 Visibility on 2D latticeItem 49 Magic hexagonItem 50 No magic cubes of order 4Item 51 No magic tesseract of order 5Item 52 Relatively prime probabilityItem 53 Lack of common divisorsItem 54 Probability of even denominatorsItem 55 Gaussian integersItem 56 "Length" of a numberItem 57 Zero digits in powers of twoItem 58 Sum of cubesItem 59 Interesting squareItem 60 Perfect numbersItem 61 Amicable pairsItem 62 Amicable pairsItem 63 Joys of 239Automata TheoryItem 64 2-counter machinesItem 65 Complexity of pi(X)Item 66 Space-filling curvesGamesItem 67 "Poker coins"Item 68 BlackoutItem 69 DotsItem 70 Chess problemItem 71 Instant InsanityItem 72 Window-diceItem 73 SimItem 74 TactixItem 75 Peg solitaireItem 76 Hi-QProposed Computer ProgramsProblem 77 Counting polyominosProblem 78 Solve minichessProblem 79 Solve the tiger puzzleProblem 80 Find smallest squared squareProblem 81 Counting magic squaresProblem 82 Counting semigroupsProblem 83 Counting inscribed circlesProblem 84 Solve pentominosProblem 85 Geometric dissectionsProblem 86 Domino coveringsProblem 87 Analyze giveaway chessProblem 88 Analyze escalation chessProblem 89 Analyze "4 pawns"Problem 90 Solve Scarne's game TeekoProblem 91 Solve "five-in-a-row"Problem 92 Solve 4x4x4 Tic-Tac-ToeProblem 93 Solve checkersProblem 94 Solve HexProblem 95 Solve chessProblem 96 Solve GoContinued FractionsItem 97 Sqrt(2), Sqrt(3)Item 98 Minimum of factorial functionItem 99 Partial quotients in arithmetic progressionItem 100 ProductItem 101A Continued fraction representationItem 101B Continued fraction arithmeticItem 101C Rationals in intervalsGroup TheoryItem 102 Group-like definitionsItem 103 Hamiltonian pathsItem 104 Permuting bits on PDP-6/10Set TheoryItem 105 Closure and complementQuaternionsItem 107 QuaternionsPolyominos, etc.Item 108 Counting polyominosItem 109 Tessellating the planeItem 110 Covering rectanglesItem 111 Domino coveringItem 112 PolyiamondsTopologyItem 113Item 114Item 115SeriesItem 116Item 117 GammaItem 118 N^NItem 119 Pi seriesItem 120 Accelerating seriesItem 121 Bernoulli polynomialsItem 122 Parity numberItem 123 Fourier clocksItem 124Item 125 Radius of convergenceFlows and Iterated FunctionsItem 126 Flow for Newton's square rootItem 127 Polynomial functions which commuteItem 128 Binary/negative binary radix flowItem 129 Flow power seriesItem 130 Flow power seriesItem 131 Arithmetic geometric meanItem 132 Loop detectorItem 133 3 N + 1 problemItem 134 Numbers to English words flowItem 135 The "C" CurvePiItem 136 Gaussian integersItem 137 Arctangent formulas for piItem 138 Arctangent formulas for piItem 139 Ramanujan formulas for piItem 140 Continued fraction for piItem 141 Pi digitsItem 142 Pi from fast trig functionsItem 143 AGM for elliptic integrals, log, and piItem 144 AGM for piProgramming HacksItem 145 Display/soundItem 146 Munching squaresItem 147 Munching squaresItem 148 Munching squaresItem 149 Circle algorithmItem 150 Circle algorithmItem 151 Circle algorithmItem 152 Circle algorithmItem 153 3D solids on 2D displayItem 154 Twos-complement arithmeticItem 155 Subtract right from leftItem 156 Make AOBJN pointerItem 157 Make AOBJN pointerItem 158 Recursive SIN/COSItem 159 Recursive SIN/COSItem 160 Log base 2Item 161 Swap two locationsItem 162 Swap two bitsItem 163 Exchange two Lisp variablesItem 164 Max of byte pointersItem 165 Byte pointersItem 166 Rotate 3 accumulatorsItem 167 Parity, count, reverse bitsItem 168 PDP-1 soundsItem 169 Count onesItem 170 Decimal print routinesItem 171 Division by zeroItem 172 Two word Lisp CONSItem 173 Fix floatItem 174 Fix point of float functionItem 175 Next higherProgramming Algorithms, HeuristicsItem 176 BANANA PhenomenonItem 177 Drawing Curves IncrementallyItem 178 Evaluating FunctionsItem 179 Searching Character StringsItem 180 Finding Corners in ImagesHardwareItem 181 Floating PointItem 182 Voltage regulatorsItem 183 Current mirrorsItem 184 One-shotItem 185 OscillatorsItem 186 FM Radio LinkItem 187 Phone Line XMTR, RCVRItem 188 DC Motor Velocity ServoItem 189 Optical CouplerItem 190 Photocathode Current OscillatorItem 191 Deflection AmplifierFigures1a, 1bBinary integers radix i-1, i+1.2 Radix i-1 fraction parts (Knuth).3a, 3bSquared square, rectangle.4 Square, hexagon dissection.5 Hexiamond solutions.6a, 6b "clock hands" series.7 Binary numbers radix -2.8 "C" curves.9 Incremental curve drawing.

A	collection	of	problems
from	MIT.		Work	reported
herein	was	conducted	at	the
Artificial	Intelligence	Laboratory,	a	Massachusetts
Institute	of	Technology	research	program.

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A collection of problems from MIT. Work reported herein was conducted at the Artificial Intelligence Laboratory, a Massachusetts Institute of Technology research program.

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