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  • Title: Nick's Mathematical Miscellany
    Descriptive info: .. Nick's Mathematical Miscellany.. Puzzles.. Nick's Mathematical Puzzles.. - A collection of puzzles, with hints, full solutions, and links to related mathematical topics.. Wu's Riddles.. - Riddles, puzzles, and technical interview questions, with a.. forum.. University 'problem of the week', with extensive archives:.. Macalester College Problem of the Week.. - from.. Professor Stan Wagon.. Purdue University Problem of the Week.. - from the mathematics department.. Bradley University Problem of the Week.. - from Professor Alberto L.. Delgado.. Math Contest.. Columbus State University.. Mathpuzzle.. com.. - A cornucopia of math puzzles and mathematical recreations.. Art of Problem Solving.. - A community for avid students of mathematics, with forums, articles, and other mathematical activities.. Wisconsin Mathematics, Engineering and Science Talent Search.. - Archive of problem sets and solutions.. Project Euler.. - A collection of mathematical/computer programming problems, ranging from easy to challenging, with a forum thread for each problem.. Other mathematical resources and links.. Interactive Mathematics Miscellany and Puzzles.. - A treasure trove of proofs, games and puzzles, and other eye openers.. Mathematical Association of America: MAA Online.. -  ...   of quotations culled from many sources.. Mudd Math Fun Facts.. - An archive of mathematical snippets, arranged by subject, difficulty, and keyword.. The Mathematical Atlas.. - A gateway to modern mathematics.. QuickMath.. - Automated solutions in algebra, equations, inequalities, calculus, matrices, graphs, and numbers.. Dario Alpern's Calculators.. - A variety of number theoretic calculators, equation solvers, and factorization engines.. Frequently Asked Questions in Mathematics.. - Covers a wide range of topics, ranging from trivia and the trivial to advanced subjects such as Wiles' recent proof of Fermat's Last Theorem.. Mathematical discussions.. - Essays attempting to show how various mathematical ideas arise naturally.. Written in the spirit of.. George Polya.. MathPages.. - Reflections on miscellaneous mathematical topics.. The Most Common Errors in Undergraduate Mathematics.. - A comprehensive survey, discussing likely causes of errors, and their remedies.. Math Help.. - Articles, proofs, puzzles, jokes, and math tutoring services.. Oxford University Undergraduate Students Lecture Notes and Problem Sheets.. - A wealth of useful material.. Reciprocal links -.. Educational.. and.. Puzzle Sites.. Nick Hobson.. nickh@qbyte.. org.. Last updated: April 3, 2008..

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  • Title: Nick's Mathematical Puzzles
    Descriptive info: Puzzles:.. 1-10.. 11-20.. 21-30.. 31-40.. 41-50.. 51-60.. 61-70.. 71-80.. 81-90.. 91-100.. 101-110.. 111-120.. 121-130.. 131-140.. 141-150.. 151-160.. Index.. |.. Home.. About.. Accessibility.. Site search:.. RSS.. What's this?.. Welcome to my selection of mathematical puzzles.. What's new?.. See.. puzzle 160.. The math puzzles presented here are selected for the deceptive simplicity of their statement, or the elegance of their solution.. They range over geometry, probability, number theory, algebra, calculus, trigonometry, and logic.. All require a certain ingenuity, but usually only pre-college math.. Some puzzles are original.. Explaining how an answer is arrived at is more important than the answer itself.. To this end, hints, answers, and fully worked solutions are provided, together with links to related mathematical topics.. Further references are provided with many of the solutions.. The puzzles are intended  ...   each problem is given at the bottom of the solution page.. I usually provide only a proximate credit; some math puzzles have been around for so long it can be difficult to trace their provenance.. However, if I know the original source, such as when a puzzle is a recognized mathematical theorem, I credit that source.. I welcome feedback of any kind.. Should you find a puzzle ambiguous, a hint misleading, an answer incorrect, or a solution unclear, please let me know.. I also welcome.. new.. math puzzles.. Some brief biographical information.. I gained a maths degree from a British university in the 1980s.. I've worked in computer software since then, but I retain a keen interest in mathematics and in education.. Nick.. Back to top.. Last updated: February 26, 2009..

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  • Title: Nick's Mathematical Miscellany: Reciprocal Links - Educational
    Descriptive info: Reciprocal Links - Puzzle Sites.. Accessibility statement.. Reciprocal Links - Educational.. Many thanks to the following people and educational organizations/sites for linking to this site:.. University-related links.. Juha Puranen.. , from his.. mathematical puzzles.. page.. Oxford University Invariant Society.. , from.. Other Mathematical Links.. Heriot-Watt University, Department of Mathematics.. External Links.. Nina Newlin, from.. Games, Puzzles, Problems, Brain Teasers More Fun.. Whatcom Community College Online Math Center.. Diane L.. Judd, Ph.. D.. Integrating Technology into the Curriculum (Mathematics).. Department of Mathematics, Uppsala University.. Mathematical Links.. Les Reid.. Links to Other Puzzle and Problem Pages.. Artur Gorka.. , of.. Clemson University.. Mathematical Web Resources.. University of Wisconsin - Marathon County.. A Catalog of Mathematics Resources on the WWW and the Internet.. Kansas State University, Department of Mathematics.. Mathematical Sites.. The Math Forum @ Drexel.. an entry.. in its.. Internet Mathematics Library.. Torsten Sillke at the University of Bielefeld.. Mathematics Education Resources.. (Math links by.. Bruno Kevius.. ).. David Wilkinson.. Recreational Mathematics Links.. Richard Kaye.. Stepping up to University Maths: mathematics links.. University of Plymouth: Centre for Innovation in Mathematics Teaching.. A Guide to Sources of Information about Recreational Mathematics.. Cornell Theory Center Math and  ...   Web Sites.. Mathematics Problem Solving Task Centres.. Other Problem Solving Sites.. The Canadian Mathematical Society.. Knot a Braid of Links.. New Zealand Mathematics Enrichment Trust.. Interesting Sites.. International Mathematics Olympiad (IMO).. Archives and Link Collections.. Other links.. mathschallenge.. net.. Puzzles and Games.. mathwizz.. Fun and Games.. Treebeard.. More Stumpers on the Web.. The Frightening World of Mathematics.. Kentucky Council of Teachers of Mathematics.. Mathematics Resources on the Web.. MathNerds.. Useful Web Sites.. Spartacus Educational.. Education on the Internet (Number 59).. e-Learning Centre.. School e-Learning Showcase.. MathsNet.. MathsNet Links.. Homeschooling On A Shoestring.. Ray Schroeder.. Challenging your Gifted Students through the Internet.. Sites for Teachers.. internet4classrooms.. Daily Activities (Mathematics).. Angela's Mathematical Links.. Study Stack.. Educational Resources.. Burlington County Library System (NJ).. Mathematics Sites.. Intute: Science, Engineering and Technology.. Science, Engineering and Technology.. ELApro.. net Web Portal.. Mathematics: Puzzles, Problems Games.. Homeschool Math.. Online math resources.. Computing Technology for Math Excellence.. Math resources.. The Art of Problem Solving.. Math Contest Problems.. Mathematics in Education and Industry.. MEI Online Resources: Useful sites.. The Further Mathematics Network.. Links.. Math Puzzles.. Eric Schechter.. About "This Week in Mathematics".. Lifesmith Classic Fractals.. Hotlinks.. Waldo's Interactive Maths Pages.. Waldo's Links..

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  • Title: Nick's Mathematical Miscellany: Reciprocal Links - Puzzle Sites
    Descriptive info: Many thanks to the following people and puzzle sites for linking to this site:.. Cliff Pickover.. one of his many puzzle pages.. Carlos Paula Simões.. Inteligência e Q.. I.. Steve Schaefer.. , puzzlemaster of.. MathRec.. Older Topics and Links.. The Open Directory Project.. , from its.. Science: Math: Recreations: Games and Puzzles.. category.. Paul Hsieh's Puzzles.. John Hanna.. Useful Links.. Skytopia.. Super impossible math questions.. (offsite links).. The Prime Puzzles and Problems Connection.. Michael Shackleford.. Math  ...   Cohn.. Henry Cohn's random mathematical thoughts.. Uncle Bob.. Uncle Bob's Puzzle Corner.. Thinks.. Mathematical Puzzles.. PuzzleUp.. The Mind Breakers - Puzzles, Riddles, Brainteasers and Card Tricks.. Links to other fun sites.. Math, Puzzles, Magic, Mazes, Illusions.. Recreational Math.. Puzzles of Leonid Mochalov.. Link Partners.. Puzzles in Education - Educational Links.. Archimedes' Laboratory.. Interesting Links.. Puzzle Express.. Puzzle Links.. Livewire Puzzles.. Puzzle JumpStation: Miscellaneous Puzzles.. A good collection of.. Brain Teasers.. Tony Heyes.. Last updated: June 29, 2012..

    Original link path: /reciprocalpuzzle.html
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  • Title: Nick's Mathematical Puzzles: 1 to 10
    Descriptive info: 1-10.. About.. Nick's Mathematical Puzzles: 1 to 10.. 1.. Folded sheet of paper.. A rectangular sheet of paper is folded so that two diagonally opposite corners come together.. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side?.. Hint.. -.. Answer.. Solution.. 2.. Triangular area.. In.. ABC, produce a line from B to AC, meeting at D, and from C to AB, meeting at E.. Let BD and CE meet at X.. Let.. BXE have area a,.. BXC have area b, and.. CXD have area c.. Find the area of quadrilateral AEXD in terms of a, b, and c.. 3.. Two logicians.. Two perfect logicians, S and P, are told that integers x and y have been chosen such that 1 x y and x+y 100.. S is given the value x+y and P is given the value xy.. They then have the following conversation.. P: I cannot determine the two numbers.. S: I knew that.. P: Now I can determine them.. S: So can I.. Given that the above statements are true, what are the two numbers? (Computer assistance allowed.. ).. 4.. Equatorial belt.. A snug-fitting belt is placed around the Earth's equator.. Suppose you added an extra 1 meter of length to the belt, held it at a point, and lifted until all the slack was gone.. How high above the Earth's surface would you then be? That is, find h in the diagram below.. Assume that the Earth is a perfect sphere of radius 6400 km, and that the belt material does not stretch.. An approximate solution is acceptable.. 5.. Confused bank teller.. A confused bank teller transposed the dollars and cents when he cashed a check  ...   crawl under the box.. 7.. Five men, a monkey, and some coconuts.. Five men crash-land their airplane on a deserted island in the South Pacific.. On their first day they gather as many coconuts as they can find into one big pile.. They decide that, since it is getting dark, they will wait until the next day to divide the coconuts.. That night each man took a turn watching for rescue searchers while the others slept.. The first watcher got bored so he decided to divide the coconuts into five equal piles.. When he did this, he found he had one remaining coconut.. He gave this coconut to a monkey, took one of the piles, and hid it for himself.. Then he jumbled up the four other piles into one big pile again.. To cut a long story short, each of the five men ended up doing exactly the same thing.. They each divided the coconuts into five equal piles and had one extra coconut left over, which they gave to the monkey.. They each took one of the five piles and hid those coconuts.. They each came back and jumbled up the remaining four piles into one big pile.. What is the smallest number of coconuts there could have been in the original pile?.. 8.. 271.. Write 271 as the sum of positive real numbers so as to maximize their product.. 9.. Reciprocals and cubes.. The sum of the reciprocals of two real numbers is 1, and the sum of their cubes is 4.. What are the numbers?.. 10.. Farmer's enclosure.. A farmer has four straight pieces of fencing: 1, 2, 3, and 4 yards in length.. What is the maximum area he can enclose by connecting the pieces? Assume the land is flat.. Last updated: December 19, 2002..

    Original link path: /puzzles/puzzle01.html
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  • Title: Nick's Mathematical Puzzles: 11 to 20
    Descriptive info: 11-20.. Nick's Mathematical Puzzles: 11 to 20.. 11.. Dice game.. Two students play a game based on the total roll of two standard dice.. Student A says that a 12 will be rolled first.. Student B says that two consecutive 7s will be rolled first.. The students keep rolling until one of them wins.. What is the probability that A will win?.. 12.. Making $1.. You are given n 0 of each of the standard denomination US coins: 1 , 5 , 10 , 25 , 50 , $1.. What is the smallest n such that it is impossible to select n coins that make exactly a dollar?.. 13.. Coin triplets.. Two players play the following game with a fair coin.. Player 1 chooses (and announces) a triplet (HHH, HHT, HTH, HTT, THH, THT, TTH, or TTT) that might result from three successive tosses of the coin.. Player 2 then chooses a different triplet.. The players toss the coin until one of the two named triplets appears.. The triplets may appear in any three consecutive tosses:.. (1st, 2nd, 3rd),.. (2nd, 3rd, 4th),.. and so on.. The winner is the player whose triplet appears first.. What is the optimal strategy for each player? With best play, who is most likely to win?.. Suppose the triplets were chosen in secret? What then would be the optimal strategy?.. What would be the optimal strategy against a randomly selected triplet?.. 14.. Two ladders in an alley.. Two ladders are placed cross-wise in an alley to form a lopsided X-shape.. The walls of the alley are not quite vertical, but are parallel to each other.. The ground is flat and horizontal.. The bottom of each ladder is placed against the opposite wall.. The top of the longer ladder touches the alley wall 5 feet vertically higher than the top of the shorter ladder touches the opposite wall, which in turn is 4 feet vertically higher than the intersection of the two ladders.. How high vertically above the ground is that intersection?.. 15.. Infinite product.. Find the value of the infinite product.. 16.. Zero-sum game.. Two players take turns choosing one number at a time (without replacement) from the set.. { 4, 3, 2, 1, 0, 1, 2, 3, 4}.. The first player to obtain three numbers (out of three, four, or five) which sum to 0 wins.. Does either player have a forced win?.. 17.. Three children.. On the first day of a new job, a colleague invites you  ...   a colleague with two boys and a girl would be more likely to have invited you to dinner than one with two girls and a boy.. If so, this would affect the probabilities of the two possibilities.. But if your imagination is that good, you can probably imagine the opposite as well.. You should assume that any such extra information not mentioned in the story is not available.. 18.. One extra coin.. Player A has one more coin than player B.. Both players throw all of their coins simultaneously and observe the number that come up heads.. Assuming all the coins are fair, what is the probability that A obtains more heads than B?.. 19.. Five card trick.. Two information theoreticians, A and B, perform a trick with a shuffled deck of cards, jokers removed.. A asks a member of the audience to select five cards at random from the deck.. The audience member passes the five cards to A, who examines them, and hands one back.. A then arranges the remaining four cards in some way and places them face down, in a neat pile.. B, who has not witnessed these proceedings, then enters the room, looks at the four cards, and determines the missing fifth card, held by the audience member.. How is this trick done?.. Note: The only communication between A and B is via the arrangement of the four cards.. There is no encoded speech or hand signals or ESP, no bent or marked cards, no clue in the orientation of the pile of four cards.. Hint 1.. Hint 2.. 20.. Five card trick, part 2.. The two information theoreticians from puzzle 19 now attempt an even more ambitious trick.. It is in fact the same trick, but performed this time with a pack of 124 cards! How does.. this.. trick work?.. (Note: 124 cards is the maximum number of cards for which this trick can be performed.. The cards may be thought of as four suits with 31 cards each, or perhaps as days from a calendar, using the months January, March, May, and July.. Or they may be thought of as a double deck, with 20 extra cards from a third deck thrown in, bearing in mind that the magicians must be able to tell, perhaps from the design on the back of the cards, from which pack a given card is taken.. Or they may simply be numbered from 1 to 124.. Last updated: June 9, 2004..

    Original link path: /puzzles/puzzle02.html
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  • Title: Nick's Mathematical Puzzles: 21 to 30
    Descriptive info: 21-30.. Nick's Mathematical Puzzles: 21 to 30.. 21.. Birthday line.. At a movie theater, the manager announces that a free ticket will be given to the first person in line whose birthday is the same as someone in line who has already bought a ticket.. You have the option of getting in line at any time.. Assuming that you don't know anyone else's birthday, and that birthdays are uniformly distributed throughout a 365 day year, what position in line gives you the best chance of being the first duplicate birthday?.. 22.. Isosceles angle.. Let ABC be an isosceles triangle (AB = AC) with.. BAC = 20.. Point D is on side AC such that.. DBC = 60.. Point E is on side AB such that.. ECB = 50.. Find, with proof, the measure of.. EDB.. 23.. Length of hypotenuse.. Triangle ABC is right-angled at B.. D is a point on AB such that.. BCD =.. DCA.. E is a point on BC such that.. BAE =.. EAC.. If.. AE = 9 inches.. CD = 8.. inches,.. find AC.. 24.. Die: median throws.. What is the minimum number of times a fair die must be thrown for there to be at least an even chance that all scores appear at  ...   What is the 1000th digit to the right of the decimal point in the decimal representation of.. (1 +.. 3000.. ?.. 28.. Making 24.. Using only the numbers 1, 3, 4, and 6, together with the operations +, , , and , and unlimited use of brackets, make the number 24.. Each number must be used precisely once.. Each operation may be used zero or more times.. Decimal points are not allowed, nor is implicit use of base 10 by concatenating digits, as in.. 3 (14 6).. As an example, one way to make 25 is: 4 (6 + 1) 3.. 29.. x.. If x is a positive rational number, show that x.. is irrational unless x is an integer.. 30.. Two pool balls.. A cloth bag contains a pool ball, which is known to be a solid ball.. A second pool ball is chosen at random in such a way that it is equally likely to be a solid or a stripe ball.. The ball is added to the bag, the bag is shaken, and a ball is drawn at random.. This ball proves to be a solid.. What is the probability that the ball remaining in the bag is also a solid?.. Last updated: December 13, 2002..

    Original link path: /puzzles/puzzle03.html
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  • Title: Nick's Mathematical Puzzles: 31 to 40
    Descriptive info: 31-40.. Nick's Mathematical Puzzles: 31 to 40.. 31.. Area of a rhombus.. A rhombus, ABCD, has sides of length 10.. A circle with center A passes through C (the opposite vertex.. ) Likewise, a circle with center B passes through D.. If the two circles are tangent to each other, what is the area of the rhombus?.. 32.. Differentiation conundrum.. The derivative of x.. , with respect to x, is 2x.. However, suppose we write x.. as the sum of x x's, and then take the derivative:.. Let f(x) = x + x +.. + x (x times).. Then f'(x).. = d/dx[x + x +.. + x] (x times).. = d/dx[x] + d/dx[x] +.. + d/dx[x] (x times).. = 1 + 1 +.. + 1 (x times).. = x.. This argument appears to show that the derivative of x.. , with respect to x, is actually x.. Where is the fallacy?.. 33.. Harmonic sum.. Let H.. 0.. = 0 and H.. n.. = 1/1 + 1/2 +.. + 1/n.. Show that, for n 0, H.. = 1 + (H.. + H.. +.. n 1.. )/n.. (That is, show that H.. is one greater than the arithmetic mean of the n preceding values, H..  ...   internal cubes.. Find all cuboids such that the number of external cubes equals the number of internal cubes.. (That is, give the dimensions of all such cuboids.. 36.. Composite numbers.. Take any positive composite integer, m.. We have m = ab = cd, where ab and cd are distinct factorizations, and a, b, c, d.. Show that a.. + b.. + c.. + d.. is composite, for all integers n.. 37.. Five marbles.. Five marbles of various sizes are placed in a conical funnel.. Each marble is in contact with the adjacent marble(s).. Also, each marble is in contact all around the funnel wall.. The smallest marble has a radius of 8mm.. The largest marble has a radius of 18mm.. What is the radius of the middle marble?.. 38.. Twelve marbles.. A boy has four red marbles and eight blue marbles.. He arranges his twelve marbles randomly, in a ring.. What is the probability that no two red marbles are adjacent?.. 39.. Prime or composite?.. Is the number 2438100000001 prime or composite? No calculators or computers allowed!.. 40.. No consecutive heads.. A fair coin is tossed n times.. What is the probability that no two consecutive heads appear?.. Last updated: April 30, 2005..

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  • Title: Nick's Mathematical Puzzles: 41 to 50
    Descriptive info: 41-50.. Nick's Mathematical Puzzles: 41 to 50.. 41.. Crazy dice.. Roll a standard pair of six-sided dice, and note the sum.. There is one way of obtaining a 2, two ways of obtaining a 3, and so on, up to one way of obtaining a 12.. Find all other pairs of six-sided dice such that:.. The set of dots on each die is not the standard {1,2,3,4,5,6}.. Each face has at least one dot.. The number of ways of obtaining each sum is the same as for the standard dice.. 42.. Multiplicative sequence.. Let {a.. } be a strictly increasing sequence of positive integers such that:.. a.. = 2.. mn.. = a.. m.. for m, n relatively prime (multiplicative property).. = n, for every positive integer, n.. 43.. Sum of two powers.. Show that n.. + 4.. is composite for all integers n 1.. 44.. Sum of two powers 2.. If x and y are positive real numbers, show that x.. y.. + y.. 45.. Area of regular 2.. -gon.. Show that the area of a regular polygon with 2.. sides and unit  ...   second element.. 47.. 1000 divisors.. Find the smallest natural number greater than 1 billion (10.. ) that has exactly 1000 positive divisors.. (The term.. divisor.. includes 1 and the number itself.. So, for example, 9 has three positive divisors.. 48.. Exponential equation.. Suppose x.. = y.. , where x and y are positive real numbers, with x y.. Show that.. x = 2, y = 4.. is the only integer solution.. Are there further rational solutions? (That is, with x and y rational.. ) For what values of x do real solutions exist?.. 49.. An odd polynomial.. Let p(x) be a polynomial with integer coefficients.. Show that, if the constant term is odd, and the sum of all the coefficients is odd, then p has no integer roots.. (That is, if.. p(x) = a.. + a.. x +.. ,.. is odd, and.. is odd, then there is no integer k such that.. p(k) = 0.. 50.. Highest score.. Suppose n fair 6-sided dice are rolled simultaneously.. What is the expected value of the score on the highest valued die?.. Last updated: March 15, 2003..

    Original link path: /puzzles/puzzle05.html
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  • Title: Nick's Mathematical Puzzles: 51 to 60
    Descriptive info: 51-60.. Nick's Mathematical Puzzles: 51 to 60.. 51.. Greatest common divisor.. Let a, m, and n be positive integers, with a 1, and m odd.. What is the greatest common divisor of a.. 1 and a.. + 1?.. 52.. Floor function sum.. Let x be a real number and n be a positive integer.. Show that [x] + [x + 1/n] +.. + [x + (n 1)/n] = [nx], where [x] is the greatest integer less than or equal to x.. 53.. The absentminded professor.. An absentminded professor buys two boxes of matches and puts them in his pocket.. Every time he needs a match, he selects at random (with equal probability) from one or other of the boxes.. One day the professor opens a matchbox and finds that it is empty.. (He must have absentmindedly put the empty box back in his pocket when he took the last match from it.. ) If each box originally contained.. matches, what is the probability that the other box currently contains.. k.. matches?.. (Where 0.. k.. 54.. Diophantine squares.. Find all solutions to c.. + 1 = (a.. 1)(b.. 1), in integers a, b, and c.. 55.. Area of a trapezoid.. A trapezoid is divided into four triangles by its diagonals.. Let the triangles adjacent to the parallel sides  ...   distinct parts.. column shows how many distinct numbers occur in each partition.. The sum for each column, over all the partitions of 5, is shown at the foot of the table.. Partition.. Number of 1s.. Number of distinct parts.. 4 + 1.. 3 + 2.. 3 + 1 + 1.. 2 + 2 + 1.. 2 + 1 + 1 + 1.. 1 + 1 + 1 + 1 + 1.. Total:.. Let a(n) be the number of 1s in all the partitions of n.. Let b(n) be the sum, over all partitions of n, of the number of distinct parts.. The above table demonstrates that.. a(5) = b(5).. Show that, for all n, a(n) = b(n).. 57.. Binomial coefficient divisibility.. Show that, for n 0, the binomial coefficient.. is divisible by.. n + 1.. and by.. 4n 2.. 58.. Fifth power plus five.. Consecutive fifth powers (or, indeed, any powers) of positive integers are always relatively prime.. That is, for all.. n 0.. , n.. (n + 1).. are relatively prime.. Are.. + 5.. always relatively prime? If not, for what values of n do they have a common factor, and what is that factor?.. 59.. Triangle inequality.. A triangle has sides of length a, b, and c.. 60.. Sum of reciprocals.. Last updated: May 6, 2003..

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  • Title: Nick's Mathematical Puzzles: 61 to 70
    Descriptive info: 61-70.. Nick's Mathematical Puzzles: 61 to 70.. 61.. Two cubes?.. Let n be an integer.. Can both n + 3 and n.. + 3 be perfect cubes?.. 62.. Four squares on a quadrilateral.. Squares are constructed externally on the sides of an arbitrary quadrilateral.. Show that the line segments joining the centers of opposite squares lie on perpendicular lines and are of equal length.. 63.. Cyclic hexagon.. A hexagon with consecutive sides of lengths 2, 2, 7, 7, 11, and 11 is inscribed in a circle.. Find the radius of the circle.. 64.. Balls in an urn.. An urn contains a number of colored balls, with equal numbers of each color.. Adding 20 balls of a new color to the urn would not change the probability of drawing (without replacement) two balls of the same color.. How many balls are in the urn? (Before the extra balls are added.. 65.. Consecutive integer products.. Show  ...   random number generator generates integers in the range 1.. n, where n is a parameter passed into the generator.. The output from the generator is repeatedly passed back in as the input.. If the initial input parameter is one googol (10.. 100.. ), find, to the nearest integer, the expected value of the number of iterations by which the generator first outputs the number 1.. That is, what is the expected value of x, after running the following pseudo-code?.. = 10.. = 0.. do while (n 1).. n = random(n) // Generates random integer in the range 1.. n.. x = x + 1.. end-do.. 68.. Difference of powers.. Find all ordered pairs (a,b) of positive integers such that.. |.. b.. = 1.. 69.. Combinatorial sum.. Find a closed form expression for.. 70.. One degree.. Show that cos 1 , sin 1 , and tan 1 are irrational numbers.. Last updated: September 17, 2003..

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