In article <1991Jun1.044345.8194@hardy.u.washington.edu> creeds@hardy.acs.washington.edu (Ron Legere) writes: >How do you get the Frequently asked question list for this group? I have just doubled the size of the FAQL, and have decided that the volume of requests for the index justifies its regular posting. So here it is: ******** Index ******** Instructions for Accessing rec.puzzles Frequently Asked Questions List Below is a list of available puzzles, categorized by subject area. Each puzzle includes a solution, compiled from various sources, which is supposed to be definitive. To request a puzzle, send a letter to netlib@peregrine.com containing one or more lines of the form: send For example, to request decision/allais.p, send the line: send decision/allais.p or just: send allais The puzzle will be mailed via return email to the address in your request's "From:" line. If you are unsure of this address, and cannot edit this line, then include in your message BEFORE the first "send" line the line: return_address If you really must have the entire list, please do not request it via email. Instead, send a tape (9 track or cartridge) with a return envelope (including postage, please) to: Chris Cole Peregrine Systems 1959 Palomar Oaks Way Carlsbad, Ca 92009 Address corrections or comments to chris@peregrine.com. ==> analysis/bugs.p <== Amorous Bugs Four bugs are placed at the corners of a square. Each bug walks directly toward the next bug in the clockwise direction. The bugs walk with constant speed always directly toward their clockwise neighbor. Assuming the bugs make at least one full circuit around the center of the square before meeting, how much closer to the center will a bug be at the end of its first full circuit? ==> analysis/c.infinity.p <== A Problem with Taylor Series What function is zero at zero, strictly positive elsewhere, infinitely differentiable at zero and has all zero derivitives at zero? ==> analysis/cache.p <== Cache and Ferry (How far can a truck go in a desert?) A pick-up truck is in the desert beside N 50-gallon gas drums, all full. The truck's gas tank holds 10 gallons and is empty. The truck can carry one drum, whether full or empty, in its bed. It gets 10 miles to the gallon. How far away from the starting point can you drive the truck? ==> analysis/cats.and.rats.p <== If 6 cats can kill 6 rats in 6 minutes, how many cats does it take to kill one rat in one minute? ==> analysis/e.and.pi.p <== Which is greater, e^(pi) or (pi)^e ? ==> analysis/functional/distributed.p <== Find all f: R -> R, f not identically zero, such that (*) f( (x+y)/(x-y) ) = ( f(x)+f(y) )/( f(x)-f(y) ). ==> analysis/functional/linear.p <== Suppose f is non-decreasing with f(x+y) = f(x) + f(y) + C for all real x, y. Prove: there is a constant A such that f(x) = Ax - C for all x. (Note: continuity of f is not assumed in advance.) ==> analysis/integral.p <== If f is integrable on (0,inf), and differentiable at 0, and a > 0, show: inf ( f(x) - f(ax) ) Int ---------------- dx = f(0) ln(a) 0 x ==> analysis/period.p <== What is the least possible integral period of the sum of functions of periods 3 and 6? ==> analysis/rubberband.p <== A bug walks down a rubberband which is attached to a wall at one end and a car moving away from the wall at the other end. The car is moving at 1 m/sec while the bug is only moving at 1 cm/sec. Assuming the rubberband is uniformly and infinitely elastic, will the bug ever reach the car? ==> analysis/series.p <== Show that in the series: x, 2x, 3x, .... (n-1)x (x can be any real number) there is at least one number which is within 1/n of an integer. ==> analysis/snow.p <== Snow starts falling before noon on a cold December day. At noon a snowplow starts plowing a street. It travels 1 mile in the first hour, and 1/2 mile in the second hour. What time did the snow start falling?? You may assume that the plow's rate of travel is inversely proportioned to the height of the snow, and that the snow falls at a uniform rate. ==> analysis/tower.p <== A number is raised to its own power. The same number is then raised to the power of this result. The same number is then raised to the power of this second result. This process is continued forever. What is the maximum number which will yield a finite result from this process? ==> arithmetic/clock/day.of.week.p <== It's restful sitting in Tom's cosy den, talking quietly and sipping a glass of his Madeira. I was there one Sunday and we had the usual business of his clock. When the radio gave the time at the hour, the Ormolu antique was exactly 3 minutes slow. "It loses 7 minutes every hour", my old friend told me, as he had done so many times before. "No more and no less, but I've gotten used to it that way." ==> arithmetic/clock/thirds.p <== Do the 3 hands on a clock ever divide the face of the clock into 3 equal segments, i.e. 120 degrees between each hand? ==> arithmetic/consecutive.product.p <== Prove that the product of three or more consecutive natural numbers cannot be a perfect square. ==> arithmetic/consecutive.sums.p <== Find all series of consecutive positive integers whose sum is exactly 10,000. ==> arithmetic/digits/all.ones.p <== Prove that some multiple of any integer ending in 3 contains all 1s. ==> arithmetic/digits/arabian.p <== What is the Arabian Nights factorial, the number x such that x! has 1001 digits? How about the prime x such that x! has exactly 1001 zeroes on the tail end. (Bonus question, what is the 'rightmost' non-zero digit in x!?) ==> arithmetic/digits/divisible.p <== Find the least number using 0-9 exactly once that is evenly divisible by each of these digits? ==> arithmetic/digits/equations/1992.p <== 1 = -1+9-9+2. Extend this list to 2 - 100 on the left side of the equals sign. ==> arithmetic/digits/extreme.products.p <== What are the extremal products of three three-digit numbers using digits 1-9? ==> arithmetic/digits/googol.p <== What digits does googol! start with? ==> arithmetic/digits/labels.p <== You have an arbitrary number of model kits (which you assemble for fun and profit). Each kit comes with twenty (20) stickers, two of which are labeled "0", two are labeled "1", ..., two are labeled "9". You decide to stick a serial number on each model you assemble starting with one. What is the first number you cannot stick. You may stockpile unused numbers on already assembled models, but you may not crack open a new model to get at its stickers. You complete assembling the current model before starting the next. ==> arithmetic/digits/nine.digits.p <== Form a number using 0-9 once with its first n digits divisible by n. ==> arithmetic/digits/palindrome.p <== Does the series formed by adding a number to its reversal always end in a palindrome? ==> arithmetic/digits/palintuples.p <== Find all numbers that are multiples of their reversals. ==> arithmetic/digits/prime.p <== What is the smallest number that cannot be made prime by changing a single digit? Are there infinitely many such numbers? ==> arithmetic/digits/rotate.p <== Find the smallest positive integer N such that if you move N's units digit to the front (in other words, a rotate right) the resulting number is 2N. (Assume that numbers are represented in decimal.) ==> arithmetic/digits/sesqui.p <== Find the least number where moving the first digit to the end multiplies by 1.5. ==> arithmetic/digits/squares/length.22.p <== Is it possible to form two numbers A and B from 22 digits such that A = B^2? Of course, leading digits must be non-zero. ==> arithmetic/digits/squares/length.9.p <== Is it possible to make a number and its square, using the digits from 1 through 9 exactly once? (It is "obviously" impossible to do it with 0 - 9 ...) ==> arithmetic/digits/squares/three.digits.p <== What squares consist entirely of three digits (e.g., 1, 4, and 9)? ==> arithmetic/digits/zeros.p <== How many zeros occur in the numbers from 1 to 1,000,000? ==> arithmetic/pell.p <== Find integer solutions to x^2 - 92y^2 = 1. ==> arithmetic/prime.series.p <== Is there an arithmetic series of 20 or more primes? ==> arithmetic/sequence.p <== Prove that all sets of n integers contain a subset whose sum is divisible by n. ==> arithmetic/sum.of.cubes.p <== Find two fractions whose cubes total 6. ==> arithmetic/tests.for.divisibility/seven.p <== What is the test to see if a number is divisible by 7? ==> arithmetic/tests.for.divisibility/three.p <== Prove that if a number is divisible by 3, the sum of its digits is likewise. ==> combinatorics/coinage/combinations.p <== How many ways are there to make change for a dollar? Count combinations of coins, not permuations. ==> combinatorics/coinage/impossible.p <== What is the smallest number of coins that you can't make a dollar with? I.e., for what N does there not exist a set of N coins adding up to a dollar? It is possible to make a dollar with 1 current U.S. coin (a Susan B. Anthony), 2 coins (2 fifty cent pieces), 3 coins (2 quarters and a fifty cent piece), etc. It is not possible to make exactly a dollar with 101 coins. ==> combinatorics/color.p <== An urn contains n balls of different colors. Randomly select a pair, repaint the first to match the second, and replace the pair in the urn. What is the expected time until the balls are all the same color? ==> combinatorics/full.p <== Consider a string that contains all substrings of length n. For example, for binary strings with n=2, a shortest string is 00110 -- it contains 00, 01, 10 and 11 as substrings. Find the shortest such strings for all n. ==> combinatorics/gossip.p <== n people each know a different piece of gossip. They can telephone each other and exchange all the information they know (so that after the call they both know anything that either of them knew before the call). What is the smallest number of calls needed so that everyone knows everything? ==> combinatorics/subsets.p <== Out of the set of integers 1,...,100 you are given ten different integers. From this set, A, of ten integers you can always find two disjoint subsets, S & T, such that the sum of elements in S equals the sum of elements in T. Note: S union T need not be all ten elements of A. Prove this. ==> cryptology/Beale.p <== What are the Beale ciphers? ==> cryptology/Feynman.p <== What are the Feynman ciphers? ==> cryptology/Voynich.p <== What are the Voynich ciphers? ==> cryptology/swiss.colony.p <== What are the 1987 Swiss Colony ciphers? ==> decision/allais.p <== The Allais Paradox involves the choice between two alternatives: A. 89% chance of an unknown amount 10% chance of $1 million 1% chance of $1 million B. 89% chance of an unknown amount (the same amount as in A) 10% chance of $2.5 million 1% chance of nothing What is the rational choice? Does this choice remain the same if the ==> decision/division.p <== N-Person Fair Division If two people want to divide a pie but do not trust each other, they can still ensure that each gets a fair share by using the technique that one person cuts and the other person chooses. Generalize this technique to more than two people. Take care to ensure that no one can be cheated by a coalition of the others. ==> decision/dowry.p <== Sultan's Dowry A sultan has granted a commoner a chance to marry one of his hundred daughters. The commoner will be presented the daughters one at a time. When a daughter is presented, the commoner will be told the daughter's dowry. The commoner has only one chance to accept or reject each daughter; he cannot return to a previously rejected daughter. The sultan's catch is that the commoner may only marry the daughter with the highest dowry. What is the commoner's best strategy assuming he knows nothing about the distribution of dowries? ==> decision/newcomb.p <== Newcomb's Problem A being put one thousand dollars in box A and either zero or one million dollars in box B and presents you with two choices: (1) Open box B only. (2) Open both box A and B. The being put money in box B only if it predicted you will choose option (1). The being put nothing in box B if it predicted you will do anything other than choose option (1) (including choosing option (2), flipping a coin, etc.). ==> decision/red.p <== I show you a shuffled deck of standard playing cards, one card at a time. At any point before I run out of cards, you must say "RED!". If the next card I show is red (i.e. diamonds or hearts), you win. We assume I the "dealer" don't have any control over what the order of cards is. The question is, what's the best strategy, and what is your probability of winning ? ==> decision/rotating.table.p <== Four glasses are placed upside down in the four corners of a square rotating table. You wish to turn them all right side up. You may do so by grasping any two glasses and, optionally, turning either over. There are two catches: you are blindfolded and the table is spun after each time you touch the glasses. How do you do it? ==> decision/stpetersburg.p <== What should you be willing to pay to play a game in which the payoff is calculated as follows: a coin is flipped until in comes up heads on the nth toss and the payoff is set at 2^n dollars? ==> decision/switch.p <== Switch? (The Monty Hall Problem) Two black marbles and a red marble are in a bag. You choose one marble from the bag without looking at it. Another person chooses a marble from the bag and it is black. You are given a chance to keep the marble you have or switch it with the one in the bag. If you want to end up with the red marble, is there an advantage to switching? What if the other person looked at the marbles remaining in the bag and purposefully selected a black one? ==> english/acronym.p <== What acronyms have become common words? ==> english/ambiguous.p <== What word in the English language is the most ambiguous? What is the greatest number of parts of speech that a single word can be used for? ==> english/capital.p <== What words change pronunciation when capitalized (e.g., polish -> Polish)? ==> english/charades.p <== A ....... surgeon was ....... to operate because he had ....... ==> english/contranym.p <== What words are their own antonym? ==> english/element.p <== The name of what element ends in "h"? ==> english/french.plural.p <== What English word, when spelled backwards, is its French plural? ==> english/frequency.p <== In the English language, what are the most frequently appearing: 1) letters overall? 2) letters BEGINNING words? 3) final letters? 4) digrams (ordered pairs of letters)? ==> english/gry.p <== Find three completely different words ending in "gry." ==> english/homographs.p <== List all homographs (words that are spelled the same but pronounced differently) Class A Homograph: all listed pronunciations different (not just accent); unrelated first definitions. Class B Homograph: variant pronunciations identical or differ only in accent; related definitions; foreign; proper; hyphenated; accented; optional spelling; not a common word; contrived; coined Constructions of the type prefix-word are hyphenated if the resultant word already exists and has another meaning: recreate -- to have fun ==> english/ladder.p <== Find the shortest word ladders stretching between the following pairs: hit - ace pig - sty four - five play - game green - grass wheat - bread order - chaos order - impel sixth - hubby ==> english/letter.rebus.p <== Define the letters of the alphabet using self-referential common phrases (e.g., "first of all" defines "a"). ==> english/lipograms.p <== What books have been written without specific letters, vowels, etc.? ==> english/near.palindrome.p <== What are some long near palindromes, i.e., words that except for one letter would be palindromes? ==> english/numbers.p <== Each equation below contains the initials of words that will make the phrase correct. Figure out the missing words. Lower case is used only to help the initials stand out better. Example: 26 = L. of the A. would be 26 = Letters of the Alphabet 1 = G. S. for M. K. 1 = S. C. in D. P. 1 = W. on a U. ==> english/palindromes.p <== What are some long palindromes? ==> english/pangram.p <== A "pangram" is a sentence containing all 26 letters. What is the shortest pangram (measured by number of letters or words)? What is the shortest word list using all 26 letters in alphabetical order? In reverse alphabetical order? ==> english/piglatin.p <== What words in pig latin also are words? ==> english/plurals/collision.p <== Two words, spelled and pronounced differently, have plurals spelled the same but pronounced differently. ==> english/plurals/doubtful.number.p <== A little word of doubtful number, a foe to rest and peaceful slumber. If you add an "s" to this, great is the metamorphosis. Plural is plural now no more, and sweet what bitter was before. What am I? ==> english/plurals/drop.s.p <== Drop an "s" and singular becomes plural, feminine becomes masculine. ==> english/plurals/endings.p <== List a plural ending with each letter of the alphabet. ==> english/plurals/man.p <== Words ending with "man" make their plurals by adding "s". ==> english/potable.color.p <== Find words that are both beverages and colors. ==> english/rare.trigraphs.p <== What trigraphs (three-letter combinations) occur in only one word? ==> english/records/pronunciation/silent.p <== What words have an exceptional number of silent letters? ==> english/records/pronunciation/syllable.p <== What words have an exceptional number of letters per syllable? ==> english/records/spelling/letter.choices/consonants.p <== What are some words containing consonants in unusual ways? ==> english/records/spelling/letter.choices/letter.dimension.p <== What are some words containing all narrow, tall, short, etc. letters? ==> english/records/spelling/letter.choices/lipogram.p <== What are some exceptional lipograms? ==> english/records/spelling/letter.choices/morse.p <== What are some words that have unusual properties in Morse code? ==> english/records/spelling/letter.choices/puzzle.word.p <== What are some words formed from chemical symbols, state abbreviations, etc.? ==> english/records/spelling/letter.choices/typewriter.p <== What are some words that are typed in unusual ways? ==> english/records/spelling/letter.choices/vowels.p <== What are some words containing vowels in unusual ways? ==> english/records/spelling/letter.count.p <== What are some exceptional isograms, polygrams, pyramid words, etc.? ==> english/records/spelling/letter.order.p <== What words have letters in various unusual orders (alphabetical, etc.)? ==> english/records/spelling/letter.pattern/entire.word.p <== What are some exceptional palindromes, tautonyms, head'n'tail words? ==> english/records/spelling/letter.pattern/subset.of.word.p <== What are some exceptional words containing doubled and repeated letters, etc.? ==> english/records/spelling/longest.p <== What is the longest word in the English language? ==> english/records/spelling/operations.on.words/deletion.p <== What exceptional words can turned into other words by deletion of letters? ==> english/records/spelling/operations.on.words/insertion.and.deletion.p <== What exceptional words can turned into other words by both insertion and deletion of letters? ==> english/records/spelling/operations.on.words/insertion.p <== What exceptional words can turned into other words by insertion of letters? ==> english/records/spelling/operations.on.words/movement.p <== What exceptional words can turned into other words by movement of letters? ==> english/records/spelling/operations.on.words/substitution.p <== What exceptional words can turned into other words by substitution of letters? ==> english/records/spelling/operations.on.words/transposition.p <== What exceptional words can turned into other words by transposition of letters? ==> english/records/spelling/operations.on.words/words.within.words.p <== What exceptional words contain other words? ==> english/records/spelling/sets.of.words/square.9x9.p <== Can nine nine-letter words be arranged in a solid nine by nine grid? ==> english/repeat.p <== What is a sentence containing the most repeated words, without: using quotation marks, using proper names, using a language other than English, anything else distasteful. ==> english/reverse.p <== What words, when a single letter is added, reverse their meanings? Exclude words that are obtained by adding an "a-" to the beginning. ==> english/rhyme.p <== What English words are hard to rhyme? "Rhyme is the identity in sound of an accented vowel in a word...and of all consonantal and vowel sounds following it; with a difference in the sound of the consonant immediately preceding the accented vowel." (From The Complete Rhyming Dictionary by Clement Wood). Appropriately Wood says a couple of pages later, "If a poet commences, 'October is the wildest month' he has estopped himself from any rhyme; since "month" has no rhyme in English." ==> english/self.ref.letters.p <== Construct a true sentence of the form: "This sentence contains _ a's, _ b's, _ c's, ...," where the numbers filling in the blanks are spelled out. ==> english/self.ref.numbers.p <== What true sentence has the form: "There are _ 0's, _ 1's, _ 2's, ..., in this sentence"? ==> english/sentence.p <== Find a sentence with words beginning with the letters of the alphabet, in order. ==> english/spoonerisms.p <== List some exceptional spoonerisms. ==> english/telegrams.p <== Since telegrams cost by the word, phonetically similar messages can be cheaper. See if you can decipher these extreme cases: UTICA CHANSON MIGRATE INVENTION ANNUAL KNOBBY SORRY IN FACTUAL BEEN CLOVER. WEED LICHEN ICE CHEST FOREARM OTHER DISGUISE DELIMIT. CANCEL MYOCARDIA ITS INFORMAL FUNCTION. YEARN AFFIX, LOST UKASE, UGANDA JAIL, CONSERVE TENURES YACHT APPEAL. ==> english/trivial.p <== Consider the free non-abelian group on the twenty-six letters of the alphabet with all relations of the form = , where and are homophones (i.e. they sound alike but are spelled differently). Show that every letter is trivial. For example, be = bee, so e is trivial. ==> english/weird.p <== Make a sentence containing only words that violate the "i before e" rule. ==> english/word.boundaries.p <== List some sentences that can be radically altered by changing word boundaries and punctuation. ==> english/word.torture.p <== What is the longest word all of whose contiguous subsequences are words? ==> games/chess.p <== How many different positions are there in the game tree of chess? ==> games/craps.p <== What are the odds in craps? ==> games/jeopardy.p <== What is the highest score a contestant can achieve during a single game of Jeopardy? What is the lowest score a contestant can achieve during a single game of Jeopard? What is the largest possible difference between two Jeopardy contestants' scores? ==> games/knight.tour.p <== For what board sizes is a knight's tour possible? ==> games/rubiks.clock.p <== How do you quickly solve Rubik's clock? ==> games/rubiks.magic.p <== How do you solve Rubik's Magic? ==> games/scrabble.p <== Make 14 7 letter words out of standard Scrabble tiles, not including blanks. ==> games/think.and.jump.p <== THINK & JUMP: FIRST THINK, THEN JUMP UNTIL YOU ARE LEFT WITH ONE PEG! O - O O - O / \ / \ / \ / \ O---O---O---O---O BOARD DESCRIPTION: To the right is a model of \ / \ / \ / \ / the Think & Jump board. The O---O---O---O---O---O O's represent holes which / \ / \ / \ / \ / \ / \ contain pegs. O---O---O---O---O---O---O \ / \ / \ / \ / \ / \ / O---O---O---O---O---O ==> games/tictactoe.p <== In random tic-tac-toe, what is the probability that the first mover wins? ==> geometry/bear.p <== If a hunter goes out his front door, goes 50 miles south, then goes 50 miles west, shoots a bear, goes 50 miles north and ends up in front of his house. What color was the bear? ==> geometry/calendar.p <== Build a calendar from two sets of cubes. On the first set, spell the months with a letter on each face of three cubes. Use lowercase three-letter abbreviations for the names of all twelve months (e.g., "jan", "feb", "mar"). On the second set, number the days with a digit on each face of two cubes (e.g., "01", "02", etc.). ==> geometry/coloring/cheese.cube.p <== A cube of cheese is divided into 27 subcubes. A mouse starts at one corner and eats through every subcube. Can it finish in the middle? ==> geometry/coloring/dominoes.p <== There is a chess board (of course with 64 squares). You are given 21 dominoes of size 3-by-1 (the size of an individual square on a chess board is 1-by-1). Which square on the chess board can you cut out so that the 21 dominoes exactly cover the remaining 63 squares? Or is it impossible? ==> geometry/cover.earth.p <== A thin membrane covers the surface of the earth. One square meter is added to the area of this membrane. How much is added to the radius and volume of this membrane? ==> geometry/dissections/circle.p <== Can a circle be cut into similar pieces without point symmetry about the midpoint? Can it be done with a finite number of pieces? ==> geometry/dissections/hexagon.p <== Divide the hexagon into: 1) 3 indentical rhombuses. 2) 6 indentical kites(?). 3) 4 indentical trapezoids. 4) 8 indentcal shapes (any shape). 5) 12 identical shapes (any shape). ==> geometry/dissections/square.70.p <== Since 1^2 + 2^2 + 3^2 + ... + 24^2 = 70^2, can a 70x70 sqaure be dissected into 24 squares of size 1x1, 2x2, 3x3, etc.? ==> geometry/dissections/square.five.p <== Can you dissect a square into 5 parts of equal area with just a straight edge? ==> geometry/four.triangles.p <== Can you construct 4 equilateral triangles with 6 toothpicks? ==> geometry/hike.p <== You are hiking in a half-planar woods, exactly 1 mile from the edge, when you suddenly trip and lose your sense of direction. What's the shortest path that's guaranteed to take you out of the woods? Assume that you can navigate perfectly relative to your current location and (unknown) heading. ==> geometry/hole.in.sphere.p <== Old Boniface he took his cheer, Then he bored a hole through a solid sphere, Clear through the center, straight and strong, And the hole was just six inches long. Now tell me, when the end was gained, What volume in the sphere remained? Sounds like I haven't told enough, But I have, and the answer isn't tough! ==> geometry/lattice/area.p <== Prove that the area of a triangle formed by three lattice points is integer/2. ==> geometry/lattice/equilateral.p <== Can an equlateral triangle have vertices at integer lattice points? ==> geometry/points.and.lines.p <== Arrange 10 points so that they form 5 rows of 4 each. ==> geometry/rotation.p <== What is the smallest rotation that returns an object to its original state? ==> geometry/tiling/rectangle.p <== A rectangular region R is divided into rectangular areas. Show that each of the rectangles in the region has at least one side with rational length if and only if the same can be said of R. ==> geometry/tiling/seven.cubes.p <== Consider 7 cubes of equal size arranged as follows. Place 5 cubes so that they form a Swiss cross or a + (plus). ( 4 cubes on the sides and 1 in the middle). Now place one cube on top of the middle cube and the seventh below the middle cube, to effectively form a 3-dimensional swiss cross. Can a number of such blocks (of 7 cubes each) be arranged so that they are able to completely fill up a big cube (say 10 times the size of the small cubes)? It is all right if these blocks project out of the big cube, but there should be no holes or gaps. ==> group/group.01.p <== AEFHIKLMNTVWXYZ BCDGJOPQRSU ==> group/group.01a.p <== 147 0235689 ==> group/group.02.p <== ABEHIKMNOPTXZ CDFGJLQRSUVWY ==> group/group.03.p <== BEJQXYZ DFGHLPRU KSTV CO AIW MN ==> group/group.04.p <== BDO P ACGJLMNQRSUVWZ EFTY HIKX ==> group/group.05.p <== CEFGHIJKLMNSTUVWXYZ ADOPQR B ==> group/group.06.p <== BCEGKMQSW DFHIJLNOPRTUVXYZ ==> induction/hanoi.p <== Is there an algorithom for solving the hanoi tower puzzle for any number of towers? Is there an equation for determining the minimum number of moves required to solve it, given a variable number of disks and towers? ==> induction/n-sphere.p <== With what odds do three random points on an n-sphere form an acute triangle? ==> induction/paradox.p <== What simple property holds for the first 10,000 integers, then fails? ==> induction/party.p <== You're at a party. Any two (different) people at the party have exactly one friend in common (the friend is also at the party). Prove that there is at least one person at the party who is a friend of everyone else. Assume that the friendship relation is symmetric and not reflexive. ==> induction/roll.p <== An ordinary die is thrown until the running total of the throws first exceeds 12. What is the most likely final total that will be obtained? ==> induction/takeover.p <== After graduating from college, you have taken an important managing position in the prestigious financial firm of "Mary and Lee". You are responsable for all the decisions concerning take-over bids. Your immediate concern is whether to take over "Financial Data". There is no doubt that you will be successful if you are the first to bid and that this will be profitable for the firm and you in the long run. However, you know that there exist another n financial firms, similar to "Mary and Lee", that are also considering the possibility. Although you are likely to be the first one to move, you know that just after a take-over there is a lot of adjustment that needs to be ==> logic/29.p <== Three people check into a hotel. They pay $30 to the manager and go to their room. The manager finds out that the room rate is $25 and gives $5 to the bellboy to return. On the way to the room the bellboy reasons that $5 would be difficult to share among three people so he pockets $2 and gives $1 to each person. Now each person paid $10 and got back $1. So they paid $9 each, totalling $27. The waiter has $2, totalling $29. Where is the remaining dollar? ==> logic/ages.p <== 1) Ten years from now Tim will be twice as old as Jane was when Mary was nine times as old as Tim. 2) Eight years ago, Mary was half as old as Jane will be when Jane is one year older than Tim will be at the time when Mary will be five times as old as Tim will be two years from now. 3) When Tim was one year old, Mary was three years older than Tim will be when Jane is three times as old as Mary was six years before the time when Jane was half as old as Tim will be when Mary will be ten years older than Mary ==> logic/bookworm.p <== A bookworm eats from the first page of an encyclopedia to the last page. The bookworm eats in a straight line. The encyclopedia consists of ten 1000-page volumes. Not counting covers, title pages, etc., how many pages does the bookworm eat through? ==> logic/boxes.p <== Which Box Contains the Gold? Two boxes are labeled "A" and "B". A sign on box A says "The sign on box B is true and the gold is in box A". A sign on box B says "The sign on box A is false and the gold is in box A". Assuming there is gold in one of the boxes, which box contains the gold? ==> logic/centrifuge.p <== You are a biochemist, working with a 12-slot centrifuge. This is a gadget that has 12 equally spaced slots around a central axis, in which you can place chemical samples you want centrifuged. When the machine is turned on, the samples whirl around the central axis and do their thing. To ensure that the samples are evenly mixed, they must be distributed in the 12 slots such that the centrifuge is balanced evenly. For example, if you wanted to mix 4 samples, you could place them in slots 12, 3, 6 and 9 (assuming the slots are numbered from 1 to 12 like a clock). ==> logic/children.p <== A man walks into a bar, orders a drink, and starts chatting with the bartender. After a while, he learns that the bartender has three children. "How old are your children?" he asks. "Well," replies the bartender, "the product of their ages is 72." The man thinks for a moment and then says, "that's not enough information." "All right," continues the bartender, "if you go outside and look at the building number posted over the door to the bar, you'll see the sum of the ages." The man steps outside, and after a few moments he reenters and declares, "Still not enough!" The bartender smiles and says, "My youngest just loves strawberry ice cream." ==> logic/elimination.p <== 97 baseball teams participate in an annual state tournament. The way the champion is chosen for this tournament is by the same old elimination schedule. That is, the 97 teams are to be divided into pairs, and the two teams of each pair play against each other. After a team is eliminated from each pair, the winners would be again divided into pairs, etc. How many games must be played to determine a champion? ==> logic/family.p <== Suppose that it is equally likely for a pregnancy to deliver a baby boy as it is to deliver a baby girl. Suppose that for a large society of people, every family continues to have children until they have a boy, then they stop having children. After 1,000 generations of families, what is the ratio of males to females? ==> logic/flip.p <== How can a toss be called over the phone (without requiring trust)? ==> logic/friends.p <== Any group of 6 or more contains either 3 mutual friends or 3 mutual strangers. Prove it. ==> logic/hundred.p <== A sheet of paper has statements numbered from 1 to 100. Statement n says "exactly n of the statements on this sheet are false." Which statements are true and which are false? What if we replace "exactly" by "at least"? ==> logic/inverter.p <== Can a digital logic circuit with two inverters invert N independent inputs? The circuit may contain any number of AND or OR gates. ==> logic/josephine.p <== The recent expedition to the lost city of Atlantis discovered scrolls attributted to the great poet, scholar, philosopher Josephine. They number eight in all, and here is the first. THE KINGDOM OF MAMAJORCA, WAS RULED BY QUEEN HENRIETTA I. IN MAMAJORCA WOMEN HAVE TO PASS AN EXTENSIVE LOGIC EXAM BEFORE THEY ARE ALLOWED TO GET MARRIED. QUEENS DO NOT HAVE TO TAKE THIS EXAM. ALL THE WOMEN IN MAMAJORCA ARE LOYAL TO THEIR QUEEN AND DO WHATEVER SHE TELLS THEM TO. THE QUEENS OF MAMAJORCA ARE TRUTHFUL. ALL SHOTS FIRED IN MAMAJORCA CAN BE HEARD IN EVERY HOUSE. ALL ABOVE FACTS ARE KNOWN TO BE COMMON ==> logic/mixing.p <== Start with a half cup of tea and a half cup of coffee. Take one tablespoon of the tea and mix it in with the coffee. Take one tablespoon of this mixture and mix it back in with the tea. Which of the two cups contains more of its original contents? ==> logic/number.p <== S knows the sum x+y, P knows the product x*y, and they say the following: P: I cannot determine x and y. S: I knew you couldn't before you told me. P: In that case I know what they are. S: Me too. ==> logic/race.p <== An Arab sheikh tells his two sons that are to race their camels to a distant city to see who will inherit his fortune. The one who arrives last will win. The brothers, after wandering aimlessly for days, ask a wiseman for advise. After hearing the advice they jump on the camels and race as fast as they can to their destination. ==> logic/riddle.p <== Jed's List of Situation Puzzles (btw, my name is Jed Hartman, but I used to use "Lord Tattersall" as a net.name; I currently use "Zorn of Zorna." Same management, different name.) History: original compilation 11/28/87 major revision 08/09/89 further additions 08/23/89 - 10/21/90 ==> logic/river.crossing.p <== Three humans, one big monkey and two small monkeys are to cross a river: a) Only humans and the big monkey can row the boat. b) At all times, the number of human on either side of the river must be GREATER OR EQUAL to the number of monkeys on THAT side. ( Or else the humans will be eaten by the monkeys!) ==> logic/self.ref.p <== Find a number ABCDEFGHIJ such that A is the count of how many 0's are in the number, B is the number of 1's, and so on. ==> logic/smullyan/fork.three.men.p <== Three men stand at a fork in the road. One fork leads to Someplaceorother; the other fork leads to Nowheresville. One of these people always answers the truth to any yes/no question which is asked of him. The other always lies when asked any yes/no question. You have no information about the third person; he may be a liar, he may be a truthteller, or he may answer according to some other pattern rather than always lying or always telling the truth. You of course cannot distinguish who is who. You can pick one person and ask a yes/no question. After receiving an answer to this question you can then ask one more question of one of the three. On the basis of these two questions, can you pick the road ==> logic/smullyan/fork.two.men.p <== Two men stand at a fork in the road. One fork leads to Someplaceorother; the other fork leads to Nowheresville. One of these people always answers the truth to any yes/no question which is asked of him. The other always lies when asked any yes/no question. By asking one yes/no question, can you determine the road to Someplaceorother? ==> logic/smullyan/stamps.p <== The moderator takes a set of 8 stamps, 4 red and 4 green, known to the logicians, and loosely affixes two to the forehead of each logician so that each logician can see all the other stamps except those 2 in the moderator's pocket and the two on her own head. He asks them in turn if they know the colors of their own stamps: A: "No" B: "No" C: "No" A: "No B: "Yes" ==> logic/smullyan/three.men.p <== Of three men, one always lies, only never lies, and one sometimes lies. They each know which is which. You must determine the identity of each man by asking three yes-or-no questions. ==> logic/timezone.p <== Two people are talking long distance on the phone; one is in an East- Coast state, the other is in a West-Coast state. The first asks the other "What time is it?", hears the answer, and says, "That's funny. It's the same time here!" ==> logic/unexpected.p <== Swedish civil defense authorities announced that a civil defense drill would be held one day the following week, but the actual day would be a surprise. However, we can prove by induction that the drill cannot be held. Clearly, they cannot wait until Friday, since everyone will know it will be held that day. But if it cannot be held on Friday, then by induction it cannot be held on Thursday, Wednesday, or indeed on any day. What is wrong with this proof? ==> logic/weighing/balance.p <== You are given N balls and a balance scale and told that one ball is slightly heavier or lighter than the other identical ones. The scale lets you put the same number of balls on each side and observe which side (if either) is heavier. 1. What's the minimum # of weighings X (and way of doing them) that will always find the unique ball and whether it's heavy or light? 2. If you are told the unique ball is, in fact, heavier than the others, what's the minimum # of weighings Y to find it? ==> logic/weighing/box.p <== You have ten boxes; each contains nine balls. The balls in one box weigh 0.9 kg; the rest weigh 1.0 kg. You have one weighing on a scale to find the box containing the light balls. How do you do it? ==> logic/weighing/weighings.p <== Some of the supervisors of Scandalvania's n mints are producing bogus coins. It would be easy to determine which mints are producing bogus coins but, alas, the only scale in the known world is located in Nastyville, which isn't on very friendly terms with Scandalville. In fact, Nastyville's king will only let you use the scale twice. Your job? You must determine which of the n mints are producing the bogus coins using only two weighings and the minimum number of coins (your king is rather parsimonious, to put it nicely). This is a true scale, i.e. it will tell you the weight of whatever you put on it. Good coins are known to have a weight of 1 ounce and it is also known that all the bogus mints (if any) produce coins that are ==> logic/zoo.p <== I took some nephews and nieces to the Zoo, and we halted at a cage marked Tovus Slithius, male and female. Beregovus Mimsius, male and female. Rathus Momus, male and female. Jabberwockius Vulgaris, male and female. The eight animals were asleep in a row, and the children began to guess which was which. "That one at the end is Mr Tove." "No, no! It's Mrs Jabberwock," and so on. I suggested that they should each write down ==> physics/balloon.p <== A helium-filled balloon is tied to the floor of a car that makes a sharp right turn. Does the balloon tilt while the turn is made? If so, which way? The windows are closed so there is no connection with the outside air. ==> physics/bicycle.p <== A boy, a girl and a dog go for a 10 mile walk. The boy and girl can walk 2 mph and the dog can trot at 4 mph. They also have bicycle which only one of them can use at a time. When riding, the boy and girl can travel at 12 mph while the dog can peddle at 16 mph. What is the shortest time in which all three can complete the trip? ==> physics/boy.girl.dog.p <== A boy, a girl and a dog are standing together on a long, straight road. Simulataneously, they all start walking in the same direction: The boy at 4 mph, the girl at 3 mph, and the dog trots back and forth between them at 10 mph. Assume all reversals of direction instantaneous. In one hour, where is the dog and in which direction is he facing? ==> physics/cannonball.p <== A person in a boat drops a cannonball overboard; does the water level change? ==> physics/dog.p <== A body of soldiers form a 50m-by-50m square ABCD on the parade ground. In a unit of time, they march forward 50m in formation to take up the position DCEF. The army's mascot, a small dog, is standing next to its handler at location A. When the B----C----E soldiers start marching, the dog | | | forward--> begins to run around the moving A----D----F body in a clockwise direction, keeping as close to it as possible. When one unit of time has elapsed, the dog has made one complete circuit and has got back to its handler, who is now at location D. (We ==> physics/milk.and.coffee.p <== You are just served a hot cup of coffee and want it to be as hot as possible when you drink it some number of minutes later. Do you add milk when you get the cup or just before you drink it? ==> physics/mirror.p <== Why does a mirror appear to invert the left and right directions, but not up down? ==> physics/monkey.p <== Hanging over a pulley, there is a rope, with a weight at one end. At the other end hangs a monkey of equal weight. The rope weighs 4 ounces per foot. The combined ages of the monkey and it's mother is 4 years. The weight of the monkey is as many pounds as the mother is years old. The mother is twice as old as the monkey was when the mother was half as old as the monkey will be when the monkey is 3 times as old as the mother was when she was 3 times as old as the monkey. The weight of the rope and the weight is one-half as much again as the difference between the weight of the weight and the weight of the weight ==> physics/particle.p <== What is the longest time that a particle can take in travelling between two points if it never increases its acceleration along the way? ==> physics/resistors.p <== What are the resistances between lattices of resistors in the shape of a: 1. Cube 2. Platonic solid 3. Hypercube 4. Plane sheet ==> physics/sail.p <== A sailor is in a sailboat on a river. The water (current) is flowing downriver at a velocity of 3 knots with respect to the land. The wind (air velocity) is zero, with respect to the land. The sailor wants to proceed downriver as quickly as possible, maximizing his downstream speed with respect to the land. Should he raise the sail, or not? ==> physics/wind.p <== Is a round-trip by airplane longer or shorter if there is wind blowing? ==> probability/amoeba.p <== A jar begins with one amoeba. Every minute, every amoeba turns into 0, 1, 2, or 3 amoebae with probability 25% for each case ( dies, does nothing, splits into 2, or splits into 3). What is the probability that the amoeba population eventually dies out? ==> probability/apriori.p <== An urn contains one hundred white and black balls. You sample one hundred balls with replacement and they are all white. What is the probability that all the balls are white? ==> probability/cab.p <== A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. Here is some data: a) Although the two companies are equal in size, 85% of cab accidents in the city involve Green cabs and 15% involve Blue cabs. b) A witness identified the cab in this particular accident as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed ==> probability/coincidence.p <== Name some amazing coincidences. ==> probability/coupon.p <== There is a free gift in my breakfast cereal. The manufacturers say that the gift comes in four different colours, and encourage one to collect all four (& so eat lots of their cereal). Assuming there is an equal chance of getting any one of the colours, what is the expected number of packets I must plough through to get all four? Can you generalise to n colours and/or unequal probabilities? ==> probability/darts.p <== Peter throws two darts at a dartboard, aiming for the center. The second dart lands farther from the center than the first. If Peter now throws another dart at the board, aiming for the center, what is the probability that this third throw is also worse (i.e., farther from the center) than his first? Assume Peter's skilfulness is constant. ==> probability/envelope.p <== I show you two sealed envelopes and say that one envelope has x dollars and the other has 2x but don't say what x is. It is not possible to determine which envelope has the larger amount without opening them. I hand you one envelope (at random - whatever that means) and say that you have two options: (1) Open the envelope that you have and keep whatever you find. (2) Exchange envelopes with me, open the envelope that you receive from the exchange, and keep the amount you find. End of game. You analyze: ==> probability/flips.p <== Consider a run of coin tosses: HHTHTTHTTTHTTTTHHHTHHHHHTHTTHT Define a success as a run of one H or T (as in THT or HTH). Use two different methods of sampling. The first method would consist of sampling the above data by taking 7 groups of three. This translates into the following sequences HHT, HTT, HTT, THT, TTT, HHH, and THH. In this sample there was one success, THT. The second method of sampling could be gotten by taking the groups in a serial sequence. Another way of explaining the method would be to take the tosses 1, 2, and 3 as the first set then tosses 2, 3, and 4 as the second set and ==> probability/flush.p <== You are dealt four cards, face down. A person peeks at the hand and reports a fact from the following list: 1) I looked at one of your cards, and it's an ace. 2) I looked at all of your cards, and at least one is an ace. 3) I looked at one of your cards, and it's the ace of spades. 4) I looked at all of your cards, and one is the ace of spades. Which fact improves your odds of having a flush? ==> probability/icos.p <== The "house" rolls two 20-sided dice and the "player" rolls one 20-sided die. If the player rolls a number on his die between the two numbers the house rolled, then the player wins. Otherwise, the house wins (including ties). What are the probabilities of the player winning? ==> probability/intervals.p <== Given two random points x and y on the interval 0..1, what is the average size of the smallest of the three resulting intervals? ==> probability/lights.p <== Waldo and Basil are exactly m blocks west and n blocks north from Central Park, and always go with the green light until they run out of options. Assuming that the probability of the light being green is 1/2 in each direction and that if the light is green in one direction it is red in the other, find the expected number of red lights that Waldo and Basil will encounter. ==> probability/lottery.p <== There n tickets in the lottery, k winners and m allowing you to pick another ticket. The problem is to determine the probability of winning the lottery when you start by picking 1 (one) ticket. A lottery has N balls in all, and you as a player can choose m numbers on each card, and the lottery authorities then choose n balls, define L(N,n,m,k) as the minimum number of cards you must purchase to ensure that at least one of your cards will have at least k numbers in common with the balls chosen in the lottery. ==> probability/particle.in.box.p <== A particle is bouncing randomly in a two-dimensional box. How far does it travel between bounces, on avergae? Suppose the particle is initially at some random position in the box and is traveling in a straight line in a random direction and rebounds normally at the edges. ==> probability/pi.p <== Are the digits of pi random (i.e., can you make money betting on them)? ==> probability/reactor.p <== There is a reactor in which a reaction is to take place. This reaction stops if an electron is present in the reactor. The reaction is started with 18 positrons; the idea being that one of these positrons would combine with any incoming electron (thus destroying both). Every second, exactly one particle enters the reactor. The probablity that this particle is an electron is 0.49 and that it is a positron is 0.51. What is the probablity that the reaction would go on for ever?? Note: Once the reaction stops, it cannot restart. ==> probability/unfair.p <== Generate even odds from an unfair coin. For example, if you thought a coin was biased toward heads, how could you get the equivalent of a fair coin with several tosses of the unfair coin? ==> series/series.01.p <== M, N, B, D, P ? ==> series/series.02.p <== H, H, L, B, B, C, N, O, F ? ==> series/series.03.p <== W, A, J, M, M, A, J? ==> series/series.03a.p <== G, J, T, J, J, J, A, M, W, J, J, Z, M, F, J, ? ==> series/series.03b.p <== A, J, B, C, G, T, C, V, J, T, D, F, K, B, H, ? ==> series/series.03c.p <== M, A, M, D, E, L, R, H, ? ==> series/series.04.p <== A, E, H, I, K, L, ? ==> series/series.05.p <== A B C D E F G H? ==> series/series.06.p <== Z, O, T, T, F, F, S, S, E, N? ==> series/series.06a.p <== F, S, T, F, F, S, ? ==> series/series.07.p <== 1, 1 1, 2 1, 1 2 1 1, ... What is the pattern and asymptotics of this series? ==> series/series.08a.p <== G, L, M, B, C, L, M, C, F, S, ? ==> series/series.08b.p <== A, V, R, R, C, C, L, L, L, E, ? ==> series/series.09a.p <== S, M, S, S, S, C, P, P, P, ? ==> series/series.09b.p <== M, S, C, P, P, P, S, S, S, ? ==> series/series.10.p <== D, P, N, G, C, M, M, S, ? ==> series/series.11.p <== R O Y G B ? ==> series/series.12.p <== A, T, G, C, L, ? ==> series/series.13.p <== M, V, E, M, J, S, ? ==> series/series.14.p <== A, B, D, O, P, ? ==> series/series.14a.p <== A, B, D, E, G, O, P, ? ==> series/series.15.p <== A, E, F, H, I, ? ==> series/series.16.p <== A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, X, Y? ==> series/series.17.p <== T, P, O, F, O, F, N, T, S, F, T, F, E, N, S, N? ==> series/series.18.p <== 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, ___ , 100, 121, 10000 ==> series/series.19.p <== 1 01 01011 0101101011011 0101101011011010110101101101011011 etc. Each string is formed from the previous string by substituting '01' for '1' and '011' for '0' simultaneously at each occurance. Notice that each string is an initial substring of the previous string so that we may consider them all as initial substrings of an infinite string. The puzzle then is, given n, determine if the nth digit is 0 or 1 without having to construct all the previous digits. That is, give a non-recursive formula for the nth digit. ==> series/series.20.p <== 1 2 5 16 64 312 1812 12288 ==> series/series.21.p <== 5, 6, 5, 6, 5, 5, 7, 5, ? ==> series/series.22.p <== 3 1 1 0 3 7 5 5 2 ? ==> series/series.23.p <== 22 22 30 13 13 16 16 28 28 11 ? ==> series/series.24.p <== What is the next letter in the sequence: W, I, T, N, L, I, T? ==> series/series.25.p <== 1 3 4 9 10 12 13 27 28 30 31 36 37 39 40 ? ==> series/series.26.p <== 1 3 2 6 7 5 4 12 13 15 14 10 11 9 8 24 25 27 26 ? ==> series/series.27.p <== 0 1 1 2 1 2 1 3 2 2 1 3 1 2 2 4 1 3 1 3 2 2 1 4 2 ? ==> series/series.28.p <== 0 2 3 4 5 5 7 6 6 7 11 7 13 9 8 8 17 8 19 9 10 13 23 9 10 ? ==> series/series.29.p <== 1 1 2 1 2 2 3 1 2 2 3 2 3 3 4 1 2 2 3 2 3 3 4 2 3 3 4 3 4 ?