Compiled Littons Problematical Recreations PDF

Litton's Problematical Recreations What letter follows OTTFFSSE_? Chapter 1: For the Perceptive N Problems of a general nature whose solutions require insight as the principal ingredient...

Litton's Problematical Recreations What letter follows OTTFFSSE_? Chapter 1: For the Perceptive N Problems of a general nature whose solutions require insight as the principal ingredient The numbers are divided into three groups as follows: 0, 3, 6, 8, 9, … in the first group, 1, 4, 7, 11, 14, … in the Rufus T. Flypaper drives two miles to work every second group and 2, 5, 10, 12, 13, … in the third. In morning- Very precise, he knows he must average 30 which groups would 15, 16 and 17 be placed? mph to arrive on time. One morning a woman driver impedes him for the first mile, cutting his average to only 15 and 16 in the third group, 17 in the second l5 mph. He quickly calculated his proper speed for the group rest of his trip to arrive on time. Assume that his car could do 120 mph. Could he arrive on time? In a fast Major League baseball game, pitcher Hi N. Outside managed to get by with the minimum number No; he has already used 4 minutes, the time that of pitches possible. He played the entire game, which he has to go the whole 2 miles. was not called prior completion. How many pitches did he make? Stations A and B are 120 miles apart on a single-track railroad. At the same time that a train leaves A for B at 25 25 mph, a train leaves B for A at 15 mph. Just as the first train leaves A, a South American botfly flies from Determine the next three terms of the sequence 12, 1, the front of the engine straight toward the other train at 1, 1, … 100 mph. On meeting the second train it immediately turns back and flies straight for the first train. It continues 2, 1, and 3 to fly back and forth with undiminished speed until it is crushed in the eventual collision. How far had the fly [MAY 2017] What is the least number of links that must flown? be disengaged from a 23-link chain so that any number of links from 1 to 23 can be obtained by taking one or 300 miles more of the pieces? Maynard's Grandfather Clock is driven by two weights, Two (4th and 11th chain) one for the striking mechanism which strikes the hours only, the other for the time mechanism. When he hears The set A contains the integers 0, 4, 5, 9, 11, 12, 13, 14, the clock strike his bedtime, he immediately winds the 19, … The set B contains 1, 2, 3, 6, 7, 8, 10, 15, 16, 17, clock and retires. After winding, the weights are exactly 18, … Place the 20 and 21 in their proper sets. opposite each other. The weights are again opposite every six hours thereafter. What is Maynard's bedtime? 20 is in A, 21 is in B 9 p.m. or 3 a.m. How many colors are necessary for the squares of a chessboard in order to assure that a bishop cannot Two squares are removed from opposite corners of a move from one square to another of the same color? checkerboard leaving 62 squares. Can the checkerboard be filled with 31 dominoes, each domino 8 colors covering two adjacent squares? A neat computer programmer wears a clean shirt every No day. If he drops off his laundry and picks up the previous week’s load every Monday night, how many shirts must [MAY 2017] Assuming the sun rises at 6:00 a.m., sets he own to keep him going? at 6:00 p.m., and moves at a uniform rate, how can a lost boy scout determine south by means of a watch on 15 shirts a cloudless day? [MAY 2019] The passengers on an excursion bus Align the hour hand with the sun’s azimuth, and consisted of 14 married couples, 8 of whom brought no south will be midway between the hour hand and children, and 6 of whom brought 3 children apiece. 12. Counting the driver, the bus had 31 occupants. How is this possible? Included among the 18 children were 8 married couples Very few people are aware of the growth pattern of Chapter 2: For Sleuths Jack’s beanstalk. On the first day it increased its height Problems requiring careful analysis and logical by 1/2, on the second day by 1/3, on the third day by deduction 1/4, and so on. How long did it take to achieve its maximum height (100 times its original height)? On a certain campus liberal arts students always lie and engineers always tell the truth. A stranger meets 3 198 days students and asks the first if he is studying liberal arts. The first answers the question, but the stranger doesn't A jig-saw puzzle contains 100 pieces. A "move" consists hear him. The second student then says that the first of connecting two clusters (including "clusters" of just denied being a liberal arts student. Then the third one piece.) What is the minimum number of moves student says that the first is really a liberal arts student. required to complete the puzzle. How many are Iiberal arts students? Can we decide which? 99 moves 1, the first [NOV 2023] A novice librarian shelved a twelve-volume set of encyclopedias in the following order from left to Four boys, Alan, Brian, Charles, and Donald, and four right. Volume 8, 11, 5, 4, 9, 1, 7, 6, 10, 3, 12, and 2. girls, Eve, Fay, Gwen, and Helen are each in love with Using her system, where will the annual supplement, one of the others, and, sad to say, in no case is their Volume 13, go? love requited. Alan loves the girl who loves the man who loves Eve. Fay is loved by the man who is loved by the Between volumes 10 and 3 girl loved by Brian. Charles loves the girl who loves Donald. If Brian is not loved by Gwen, and the boy who With some sharp reasoning, you ought to be able to is loved by Helen does not love Gwen, who loves Alan? determine the last member of the sequence for which the first 20 members are: 11, 31, 71, 91, 32, 92, 13, 73, Gwen loves Alan 14, 34, 74, 35, 95, 16, 76, 17, 37, 97, 38, 98, __? A canoe is floating in a swimming pool. Which will raise 79 the level of the water in the pool higher, dropping a penny into the pool or into the canoe? Or does it make [NOV 2021] A lighthouse shows successive one-second any difference? flashes of red, white, green, green, white, red. A second lighthouse does the same only with two-second flashes. Dropping it into the canoe will raise the water level The six-second sequence of the first lighthouse is higher. repeated steadily, as is the twelve-second sequence of the other lighthouse. What fraction of the time do the Five suspects were rounded up in connection with the two lights show the same color if the given sequences famous "Cock Robin Murder.” Their statements were as start at the same time? follows: A: "C and D are lying.” B: "A and E are lying.” C: "B and D are lying.” D: "C and E are lying.” E: "A and B 1/6 are lying.” Who is lying? Four players played a hand of hearts at $1 a point A is lying (pairwise payoffs). Dave lost $10 to Arch, $12 to Bob, and $20 to Chuck. How many hearts did poor Dave Six grocers in a town each sell a different brand of tea take in? in four-ounce packets at 25 cents per packet. One of the grocers gives short weight, each packet of his brand 4 hearts and the queen of spades weighing only 3 3/4 ounces. If I can use a balance for only one weighing, what is the minimum amount I must spend to be sure of finding the grocer who gives short weight? 3.7 dollars Between Kroflite and Beeline are five other towns. The Chapter 3: For Gamblers seven towns are an integral number of miles from each Problems involving probability and determination other along a straight road. The towns are so spaced of the number of outcomes of various situations that if one knows the number of miles a person has traveled between any two towns, he can determine the A gambler devised a game to be played with a friend. particular towns uniquely. What is the minimum distance He bet 1/2 the money in his pocket on the toss of a coin; between Kroflite and Beeline to make this possible? heads he won, tails he lost. The coin was tossed, and the money handed over. The offer was repeated, and 25 miles the game continued. Each time the bet was for 1/2 the money then in his possession. Eventually the number of In the game "subtract-a-square," a positive integer is times he lost was equal to the number of times he won. written down and two players alternately subtract Quickly now! Did he gain, lose, or break even? squares from it with the restriction that the remainder must never be Iess than zero. The player who leaves He lost zero wins. What square should the first player subtract if the original number is 29? A prisoner is given 10 white balls, 10 black balls and two boxes. He is told that an executioner will draw one ball 9 from one of the two boxes. If it is white, the prisoner will go free; if it is black, he will die. How should the prisoner A lamp has three on-off buttons, all of which must be on arrange the balls in boxes to give himself the best for the lamp to light. A man wishes to turn on the lamp chance for survival? at a moment when only the second switch is off. He does not know this and proceeds to press the first 1 white ball in one box and the remaining balls in button. Getting no result, he presses the second and the other box (73.7%) eventually, on the seventh trial, (never repeating any on- off configuration), the lamp finally lights. In what order [NOV 2019] Smith and Jones, both 50% marksmen, did he press the buttons? decide to fight a duel in which they exchange alternate shots until one is hit. What are the odds in favor of the 1-2-3-2-1-2-3 man who shoots first? A game proceeds as follows: Smith starts with a 10-link 2/3 chain and removes any single link, presenting Jones either with a 9-link chain or two chains having a total of There are n points on a circle. A straight-line segment is 9 links. Jones removes any one link from each of the drawn between each pair of points. How many chains. Smith removes any one link from each of the intersections are there within the circle if no 3 lines are remaining chains and so on, until the winner removes collinear? the last link or links. What should Smith do first? nC4 or nP4 / 24 Leave two chains of 6 and 3 links On a certain day, our parking lot contains 999 cars, no [APR 2023] In Bristol 90% of the citizens drink tea; 80% two of which have the same 3-digit license number. After drink coffee; 70% drink whiskey; and 60% drink gin. No 5:00 p.m. what is the probability that the license one drinks all four beverages. What percent of Bristol’s numbers of the first 4 cars to leave the parking lot are in citizen drink liquor? increasing order of magnitude? 100% drink liquor 1/24 [MAY 2017] You and a friend spot a loose $20 bill [NOV 2016] Three marksmen simultaneously shoot at simultaneously and agree to an auction in which you and hit a rapidly spinning spherical target. What is the write down your bids and compare them. High bidder probability that the three points of impact are on the gets the $20 and pays the other the amount of the same. higher bid. Tie bidders split the $20. How much do you bid? 1 $9.99 A hospital nursery contains only two baby boys; the girls Three dart players threw simultaneously at a tic-tac-toe have not yet been counted. At 2:00 p.m. a new baby is board, each hitting a different square. What is the added to the nursery. A baby is then selected at random probability that the three hits constituted in a win at tic- to be the first to have its footprint taken. It turns out to tac-toe? be a boy. What is the probability that the Iast addition to the nursery was a girl? 2/21 2/5 Four swimming pool builders submit sealed bids to a homeowner who is required by law to accept the last bid Assume that a single depth charge has a probability of that he sees, i.e., once he looks at a bid, he 1/2 of sinking a submarine, 1/4 of damage and 1/4 of automatically rejects all previous bids. He is not missing. Assume also that two damaging explosions required to open all the envelopes, of course. Assuming sink the sub. What is the probability that 4 depth that all four bids are different, what procedure will charges will sink the sub? maximize his chances of accepting the lowest bid. and what will be the probability of doing so? 251/256 11/24 What is the most likely distribution of the suits in a hand at Bridge? All the members of a fraternity play basketball while all but one play ice hockey; yet the number of possible 4-4-3-2 basketball teams (5 members) is the same as the number of possible ice hockey teams (6 members). If 2 marbles are removed at random from a bag Assuming there are enough members to form either containing black and white marbles, the chance that type of team, how many are in the fraternity? they are both white is 1/3. If 3 are removed at random, the chance that they all are white is 1/6. How many 15 marbles are there of each color? A game of super-dominoes is played with pieces divided w = 6; b = 4 into three cells instead of the usual two, containing all combinations from triple blank to triple six, with no An expert gives team A only a 40% chance to win the duplications. For example, the set does not include both World Series. Basing his calculation on this a gambler 1 2 3 and 3 2 1 since these are merely reversals of each offers 6 to 5 odds on team B to win the first game. Is his other. (But it does contain 1. 3 2.) How many pieces are judgment sound? there in a set? Yes 196 pieces A salesman visits ten cities arranged in the form of a Martian coins are 3-sided (heads, tails, and torsos), circle, spending a day in each. He proceeds clockwise each side coming up with equal probability. Three from one city to the next, except whenever leaving the Martians decided to go odd-man-out to determine who tenth city he may go to either the first or jump to the pays a dinner check. (If two coins come up the same second city- How many days must elapse before his and one different, the owner of the latter coin foots the location is completely indeterminate, i.e., when he could bill). What is the expected number of throws needed in be in any one of the ten cities? order to determine a loser? 83 days 1.5 throws A coin is so unbalanced that you are likely to get two There are three families, each with two sons and two heads in two successive throws as you are to get tails daughters. In how many ways can all these young in one. What is the probability of getting heads in a people be married? single throw? 80 ways 0.618 How many three-digit telephone area codes are In a carnival game, 12 white balls and 3 black balls are possible given that: (a) the first digit must not be zero or put in an opaque bottle, shaken up, and drawn out one one; (b) the second digit must be zero or one; (c) the at a time. The player gets 25 cents for each white ball third digit must not be zero; (d) the third digit may be one which emerges before the first black ball. If he pays one only if the second digit is zero. dollar to play, how much can he expect to win (or lose) on each game. 136 possible codes Average loss of a quarter a game Six men decide to play Russian roulette with a six-gun loaded with one cartridge. They draw for position, and In the binary system there are only two positive integers afterwards' the sixth man casually suggests that containing no digit more than once, namely 1 and 10. instead of letting the chamber rotate in sequence, each How many are there in base ten? man spin the chamber before shooting. How would this improve his chances? 8,677,690 integers 0.1 In the final seconds of the game, your favorite N.B.A. team is behind 117 to 118. Your center attempts a shot A long shot poker player draws two cards to the five and and is fouled for the 2nd time in the last 2 minutes as six of diamonds and the joker. What are his chances of the buzzer sounds. Three to make two in the penalty coming up with a pat hand? (straight or flush) situation. Optimistic? Note: the center is only a 50% free thrower. What are your team's overall chances of 0.168 winning? In Puevigi, the game of craps is played with a referee 11/16 (69%) calling the point by adding together the six faces (three on each die) visible from his vantage point. What is the One of a pair of dice is loaded so that the chance of a 1 probability of making 16 the hard way? (That is, by turning up is 1/5, the other faces being equally likely. Its throwing two eights.) mate is loaded so that the chance of a 6 turning up is 1/5, the other faces being equally likely. How much does 0 this loading increase the probability of throwing a 7 with the two dice? Max and his wife Min each toss a pair of dice to determine where they will spend their vacation. If either 1/750 of Min's dice displays the same number of spots as either of Max's, she wins, and they go to Bermuda. If all the 720 permutations of the digits 1 through 6 are Otherwise, they go to Yellowstone. What is the chance arrange in numerical order, what is the 417th term? they'll see "Old Faithful" this year? 432516 0.486 The local weather forecaster says, “no rain” and his There are four volumes of an encyclopedia on a shelf, record is 2/3 accuracy of prediction. But the Federal each volume containing 300 pages, (that is, numbered Meteorological Service predicts rain, and their record is 1 to 600), but these have been placed on the shelf in 3/4. With no other data available, what is the chance of random order. A bookworm starts at the first page of Vol. rain? 1 and eats his way through to the last page of Vol. 4. What is the expected number of pages (excluding 3/5 covers) he has eaten through? A sharp operator makes the following deal. A player is 500 pages to toss a coin and receive 1, 4, 9, … n^2 dollars if the first head comes up on the first, second, third, … n-th Venusian batfish come in three sexes, which are toss. The sucker pays ten dollars for this. How much can indistinguishable (except by Venusian batfish). How the operator expect to make if this is repeated a great many live specimens must our astronauts bring home in many times? order for the odds to favor the presence of a “mated triple” with its promise of more little batfish to come? 6 Five specimens (50/81) In 1969, the World Series began in the stadium of Chapter 4: For the Algebraically Inclined American League pennant winner. Assume the Problems best attacked by means of variables and contenders are evenly matched. What is the probability equations that the series ended where it began? To stimulate his son in the pursuit of partial differential 5/8 equations, a math professor offered to pay him $8 for every equation correctly solved and to fine him $5 for In a carnival game 5 balls are tossed into a square box every incorrect solution. At the end of 26 problems, divided into 4 square cells, with baffles to insure that neither owed any money to the other. How many did the every ball has an equal chance of going in any cell. The boy solve correctly? player pays $1 and receives $1 for every cell which is empty after the 5 balls are thrown. How much does the 10 problems operator expect to make per game? An expert on transformer design relaxed one Saturday 5¢ by going to the races. At the end of the first race, he had doubled his money. He bet $30 on the second race and Having lost a checker game, a specialist in learning tripled his money. He bet $54 on the third race and programs threw one of the red checkers out the window. quadrupled his money. He bet $72 on the fourth race His wife reboxed the 12 black pieces and 11 red pieces and lost it, but still had $48 left. With how much money one at a time in random fashion. The number of black did he start? checkers in the box always exceeded the number of reds. What was the a priori probability of this 29$ occurrence. Between Sing-Sing and Tarry-Town, 1/23 I met my worthy friend, John Brown, And seven daughters, riding nags, And everyone had seven bags. In every bag were thirty cats, And every cat had forty rats, Besides a brood of fifty kittens. All but the nags were wearing mittens! Mittens, kittens - cats, rats - bag, nags - Browns, How many were met between the towns? 764,488 Dr. Reed, arriving late at the lab one morning, pulled out his watch and said, "I must have it seen to. I have noticed that the minute and the hour hand are exactly together every sixty-five minutes.” Does Dr. Reed's watch gain or lose, and how much per hour? 60/143 minutes At this moment, the hands of a clock in the course of normal operation describe a time somewhere between 4:00 and 5:00on a standard clock face. Within one hour or less, the hands will have exactly exchanged positions; what time is it now? 4:26.85 Two men are walking towards each other at the side of A necklace consists of pearls which increase uniformly a railway. A freight train overtakes one of them in 20 from a weight of 1 carat for the end pearls to a weight of seconds and exactly ten minutes later meets the other 100 carats for the middle pearl. If the necklace weighs man coming in the opposite direction. The train passes altogether 1650 carats and the clasps and string this man in l8 seconds. How long after the train has together weigh as much (in carats) as the total number passed the second man will the two men meet? of pearls, how many pearl does the necklace contain? (Constant speeds are to be assumed throughout.) 33 pearls 5562 seconds A pupil wrote on the blackboard a series of fractions [NOV 2023] Using the French Tricolor as a model, how having positive integral terms and connected by signs many flags are possible with five available colors if two which were either all * or all X, although they were so adjacent rows must not be colored the same? carelessly written it was impossible to tell which they were. It still wasn't clear even though he announced the 50 flags result of the operation at every step. The third fraction had denominator 19. What was the numerator? A cubic box with sides ‘a’ feet long is placed flat against a wall. A ladder ‘p’ feet long is placed in such a way that 25 it touches the wall as well as the free horizontal edge of the box. If a = 1 and p = sqrt(15), calculate at what height Jai Alai balls come in boxes of 8 and 15; so that 38 balls the ladder touches the wall from the floor. (one small box and two large) can be bought without having to break open a box, but not 39. What is the 3.63 or 1.38 feet maximum number of balls which cannot be bought without breaking boxes? [NOV 2017] Dr. Irving Weiman, who is always in a hurry, walks up an up-going escalator at the rate of one step 97 balls per second. Twenty steps bring him to the top. Next day he goes up at two steps per second, reaching the top in A parking lot charges X for the first hour or fraction of an 32 steps. How many steps are there in the escalator? hour and 2/3 of X for each hour or fraction thereafter. Smith parks 7 times as long as Jones but pays only 3 80 steps times as much. How long did each park? (The clock registers only 5-minute intervals.) Citizens of Franistan pay as much income tax (percentagewise) as they make rupees per week. What Jones parked for 0.5 hour and Smith for 3.5 hours. is the optimal salary in Franistan? Mr. Field, a speeder, travels on a busy highway having 50 rupees the same rate of traffic flow in each direction. Except for Mr. Field, the traffic is moving at the legal speed limit. There are nine cities which are served by two competing Mr. Field passes one car for every nine which he meets airlines. One or the other airline (but not both) has a from the opposite direction. By what percentage is he flight between every pair of cities. What is the minimum exceeding the speed limit? number of triangular flights (i.e., trips from A to B to C and back to A on the same airline)? 25% 12 triangular flights What is the millionth term of the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, … in which each positive integer n occurs in blocks of n terms? Two snails start from the same point in opposite directions toward two bits of food. Each reaches his 1,414 destination in one hour. If each snail had gone in the direction the other took, the first snail would have The teacher marked the quiz on the following basis: one reached his food 35 minutes after the second. How do point for each correct answer, one point off for each their speeds compare? question left blank and two points off for each question answered incorrectly. Pat made four times as many The first snail traveled at three-fourths the speed errors as Mike, but Mike left nine more questions blank. of the second. If they both got the same score, how many errors did each make? Pat made eight errors and Mike made two errors A student, just beginning the study of logarithms, was Chapter 5: For the Geometrically Inclined required to evaluate an expression of the form of (log Problems involving plane geometry and A)/(log B). He proceeded to cancel common factors in trigonometry both numerator and denominator, (including the “factor” log), and arrived at the result 2/3. Surprisingly, this was A forgetful physicist forgot his watch one day and asked correct. What were the values of A and B? an E.E. on the staff what time it was. The E.E. Iooked at his watch and said: “The hour, minute, and sweep A = 9/4, B = 27/8 second hands are as close to trisecting the face as they ever come. This happens only twice in every 12 hours, [MAY 2017] There are four towns at the corners of a but since you probably haven't forgotten whether you square. Four motorists set out, each driving to the next ate lunch, you should be able to calculate the time.” (clockwise) town, and each man but the fourth going 8 What time was it to the nearest second? mph faster than the car ahead – thus the first car travels 24 mph faster than the fourth. At the end of one hour the 2:54:35 or 9:5:25 first and third cars are 204, and the second and fourth 212 (beeline) miles apart. How fast is the first car The faces of a solid figure are all triangles. The figure traveling and how far apart are the towns? has nine vertices. At each of six of these vertices, four faces meet, and at each of the other three vertices, six 50 mph, 180 miles faces meet. How many faces does the figure have? Solve the equation sqrt(x + sqrt(x + sqrt(x + … = sqrt(x 14 faces sqrt(x sqrt x … where both members represent infinite expressions. [NOV 2017] A new kind of atom smasher is to be composed of two tangents and a circular arc which is x = 0 or 2 concave towards the point of intersection of the two tangents. Each tangent and the arc of the circle is 1 mile [APR 2023] A pencil, eraser and notebook together cost long. What is the radius of the circle? $1.00. A notebook costs more than two pencils, and three pencils cost more than four erasers. If three 1437.45 feet erasers cost more than a notebook, how much does each cost? A spider and a fly are located at opposite vertices of a room of dimensions 1, 2 and 3 units. Assuming that the P = $26, E = $19, N = $55 fly is too terrified to move, find the minimum distance the spider must crawl to reach the fly. Sqrt(18) units [NOV 2017] In a room 40 feet long, 20 feet wide, and 20 feet high, a bug sits on an end wall at a point one foot from the floor, midway between the sidewalls. He decides to go on a journey to a point on the other end wall which is one foot from the ceiling midway between the sidewalls. Having no wings, the bug must make this trip by sticking to the surfaces of the room. What is the shortest route that the bug can take? 58 feet A circle of radius I inch is inscribed in an equilateral triangle. A smaller circle is inscribed at each vertex, tangent to the circle and two sides of the triangle. The process is continued with progressively smaller circles. what is the sum of the circumference of all circles. 5π [NOV 2023] A farmer owned a square field measuring A one-acre field in the shape of a right triangle has a exactly 2261 yards on each side. 1898 yards from one post at the midpoint of each side. A sheep is tethered to corner and 1009 yards from an adjacent corner stood a each of the side posts and a goat to the post on the beech tree. A neighbor offered to purchase a triangular hypotenuse. The ropes are just long enough to let each portion of the field stipulating that a fence should be animal reach the two adjacent vertices. What is the total erected in a straight line from one side of the field to an area the two sheep have to themselves, i.e., the area adjacent side so that the beech tree was part of the the goat cannot reach? fence. The farmer accepted the offer but made sure that the triangular portion was of minimum area. What was 1 acre the area of the field the neighbor received, and how long was the fence? A divided highway goes under a number of bridges, the arch over each lane being in the form of a semi-ellipse 939,120 square yards, 2018 yards with the height equal to the width. A truck is 6 ft. wide and 12 ft. high. What is the lowest bridge under which it A man leaves from the point where the prime meridian can pass? crosses the equator and moves forty-five degrees northeast by geographic compass which always points 13 ft. 5 in. high toward the north geographic pole. He constantly corrects his route. Assuming that he walks with equal A cowboy is five miles south of a stream which flows due facility on land and sea, where does he end up and how east. He is also 8 miles west and 6 miles north of his far will he have travelled when he gets there? cabin. He wishes to water his horse at the stream and return home. What is the shortest distance he can travel 10^7 sqrt(2) meter and accomplish this? Near the town of Lunch, Nebraska there is a large 17.9 miles triangular plot of land bounded by three straight roads which are 855, 870, and 975 yards long respectively. While still at a sizable distance from the Pentagon The owner of the land, a friend of mine, told me that he building, a man first catches sight of it. Is he more likely had decided to sell half the plot to a neighbor, but that to be able to see two sides or three? the buyer had stipulated that the seller of the land should erect the fence which was to be a straight one. Equal probability of 1/2 The cost of fences being high, my friend naturally wanted the fence to be as short as possible. What is the A pirate buried his treasure on an island, a conspicuous minimum length the fence can be? landmark of which were three palm trees, each one 100 feet from the other two. Two of these trees were in a N- 600 yards S line. The directions for finding the treasure read: “Proceed from southernmost tree 15 feet due north, Three hares are standing in a triangular field which is then 26 feet due west.” Is the treasure buried within the exactly 100 yards on each side. One hare stands at triangle formed by the trees? each corner; and simultaneously all three set off running. Each hare runs after the hare in the adjacent The treasure lies outside corner on his left, thus following a curved course which terminates in the middle of the field, all three hares An Origami expert started making a Nani-des-ka by arriving there together. The hares obviously ran at the folding the top left corner of a sheet of paper until it same speed, but just how far did they run? touched the right edge and the crease passed through the bottom left corner. He then did the same with the 100 yards lower right corner, thus making two slanting parallel lines. The paper was 25 inches long and the distance Scalene triangle ABC, which is not a right triangle, has between the parallel lines was exactly 7/40 of the width. sides which are integers. If sin A = 5/13, find the smallest How wide was the sheet of paper? values for its sides, i.e., those values which make the perimeter a minimum. 24 in a = 25, b = 16, c = 39 The Ben Azouli are camped at an oasis 45 miles west of Taqaba. They decide to dynamite the Trans- Hadramaut railroad joining Taqaba to Maqaba, 60 miles north of the oasis. If the Azouli can cover 18 miles a day, how long will it take them to reach the railroad? Two days [NOV 2017] A cross section through the center of a Chapter 5: For Fans of Diophantus football is a circle x inches in circumference. The football Problems involving Diophantine equations, those is x – 8 inches long from the tip to tip and each seam is for which only whole number solutions are sought an arc of a circle ¾ if x inches in diameter. Find x. Find the smallest number (x) of persons a boat may 20.69 in carry so that (n) married couples may cross a river in such a way that no woman ever remains in the company A yang, ying, and yung is constructed by dividing a of any man unless her husband is present. Also find the diameter of a circle, AB, into three parts by points C and least number of passages (y) needed from one bank to D, then describing on one side of AB semicircles having the other. Assume that the boat can be rowed by one AC and AD as diameters and on the other side of AB person only. semicircles having BD and BC as diameters. Which is larger, the central portion or one of the outside pieces? n = 2, x = 2, y = 5 All three are the same size A, B, and C participate in a track meet, consisting of at least three events. A certain number of points are given A diaper is in the shape of a triangle with sides 24, 20 for first place, a smaller number for second place, and a and 20 inches. The long side is wrapped around the still smaller number for third place. A won the meet with baby’s waist and overlapped two inches. The third point a total score of 14 points; B and C are tied for second is brought up to the center of the overlap and pinned in with 7 points apiece. B won first place in the high jump. place. The pin is to go through three thicknesses of Who won the pole vault assuming no ties occurred in material. What is the area in which the pin may be any event? placed? A won the pole vault 2.5 square in Find the simplest solution in integers for the equation [NOV 2016] A coffee pot with a circular bottom tapers (1/x^(2)) + (1/y^(2)) = (1/z^(2)). uniformly to a circular top with radius half that of the base. A mark halfway up the side says “2 cups.” Where x = 15, y = 20, z = 12 should the “3 cups” mark go? What is the capacity of the pot? Maynard the Census Taker visited a house and was told, “Three people live there, The product of their ages 2% of the way down from the top, 3 1/37 cups is 1296, and the sum of their ages is our house number.” After an hour of cogitation Maynard returned for more [MAY 2018] An icicle forming from a dripping gutter is in information. The house owner said, “I forgot to tell you the shape of a cone five times as long as it is wide (at that my son and grandson live here with me.” How old the top). A few hours later it has doubled in length and were the occupants and what was their street number? the generating angle has also doubled. How does its present weight compare with its previous weight? Ages: 1, 18, and 72 Street number: 91 Almost 33 times In Byzantine basketball there are 35 scores which are There is one flag at the entrance to a racetrack and impossible for a team to total, one of them being 58. another inside the track, half a mile from the first. A Naturally a free throw is worth fewer points than a field jockey notes that no matter where he is on the track, goal. What is the point value of each? one flag is 3 times as far away as the other. How long is the track? Free throw: 8 Field goal: 11 1980π Gherkin Gesundheit, a brilliant graduate mathematics What is the longest 6' wide shuffleboard court which will student, was working on an assignment but, being a bit fit in a 20’ x 30’ rectangular room? absent-minded, he forgot whether he was to add or to multiply the three different integers on his paper" He 30.875 decided to do it both ways and, much to his surprise, the answer was the same. What were the three different integers? 1, 2, 3 [NOV 2022] Three farmers, Adams, Brown and Clark all In European countries the decimal point is often written have farms containing the same number of acres. a little above the line. An American, seeing a number Adams' farm is most nearly square, the length being written this way, with one digit on each side of the only 8 miles longer than the width. Clark has the most decimal point, assumed the numbers were to be oblong farm, the length being 34 miles longer than the multiplied. He obtained a two-digit number as a result width. Brown's farm is intermediate between these two, but was 14.6 off. What was the original number? the length being 28 miles longer than the width. If all the dimensions are in exact miles, what is the size of each 5.4 farm? Two wheels in the same plane are mounted on shafts 40x48, 32x60, 30x64 13 in. apart. A belt goes around both wheels to transmit power from one to the other. The radii of the two wheels 1960 and 1961 were bad years for ice cream sales but and the length of the belt not in contact with the wheels 1962 was very good. An accountant was looking at the at any moment are all integers - How much larger is one tonnage sold in each year and noticed that the digital wheel than the other? sum of the tonnage sold in 1962 was three times as much as the digital sum of the tonnage sold in 1961. 5 in Moreover, if the amount sold in 1960 (346 tons), was added to the 1961 tonnage, this total was less than the Five points are located in or on the perimeter of an total tonnage sold in 1962 by the digital sum of the equilateral triangle with 9-inch sides. If d is the distance tonnage sold in that same year. Just how many more between the closest pair of points, what is the maximum tons of ice cream were sold in 1962 than in the previous possible value of d? year? 4.5 in 361 tons If THAT = (AH)(HA), what is THAT? Three rectangles of integer sides have identical areas. The first rectangle is 278 feet longer than wide. The 6786 second rectangle is 96 feet longer than wide. The third rectangle is 542 feet longer than wide. Find the area and A group of hippies are pondering whether to move to dimensions of the rectangles. Patria, where polygamy is practiced but polyandry and spinsterhood are prohibited, or Matria, where polyandry Area: 1,466,640 ft2 is permitted, and polygamy and bachelorhood are Dimensions: 1080x1358, 1164x1260, 970x1512 proscribed. In either event the possible number of “arrangements” is the same. The girls outnumber the When little Willie had sold all his lemonade, he found he boys. How many are there? had $7.95 in nickels, dimes, and quarters. There were 47 coins altogether and, having just started to study 4 girls and 2 boys geometry, he noticed that the numbers of coins satisfied a triangle inequality, i.e., the sum of any two Dad and his son have the same birthday. On the last denominations was greater than the third. How many of one, Dad was twice as old as Junior. Uncle observed each were there? that this was the ninth occasion on which Dad's birthday age had been an integer multiple of Junior's. How old is D = 20, Q = 32, N = 4 Junior? There are 100 coins in a piggy bank totaling $5.00 in Junior: 36 value, the coins consisting of pennies, dimes, and half Dad: 72 dollars. How many of each are there? P = 60, D = 39, H = 1 Every year engineering consultant pays a bonus of $300 to his most industrious assistant, and $75 each to the rest of his staff. After how many years would his outlay be exactly $6,000 if all but two of his staff had merited the $300 bonus, but none of them more than twice? Years: 8 Staff: 7  Chapter 7: For Number Theorists My house is on a road where the numbers run 1, 2, 3, Problems involving properties of integers 4., - consecutively. My number is a three digit one and, by a curious coincidence, the sum of all house numbers The undergraduates of a School. of Engineering wished less than mine is the same as the sum of all house to form ranks for a parade. In ranks of 3 abreast, 2 men numbers greater than mine. What is my number and were left over; in ranks of 5, 4 over; in 7's,6 over; and how many houses are there on my road? 11's, 10 over. What is the least number of marchers there must have been? House number: 204 Number of houses: 288 1154 marchers The sum and difference of two squares may be primes: What is the remainder upon dividing 5^999,999 by 7? 4 – 1 = 3 and 4 + 1 = 5; 9 – 4 = 5 and 9 + 4 = 13, etc. Can the sum and difference of two primes be squares? 6 If so, how many different primes is this possible? A pet store offered a baby monkey for sale at $1.25. The p = 2, q = 2 monkey grew. Next week it was offered at $1.89, then $5.13, then $5.94, then $9.18. What will be the price of In what days of the week can the first day of a century the monkey on the sixth week? fall? (The first day of the twentieth century was Jan. 1, 1901) $12.42 Monday, Tuesday, Thursday, and Saturday only Assume the universe is a billion billion light years in The 21st century will start on a Monday diameter and is packed solidly with matter weighing a billion billion tons per cubic inch and each gram of this In the arithmetic of Puevigi, 14 is a factor of 41. What is matter contains a billion billion atoms. Also, every the base of the number system? second during the past billion billion years, a billion billion similar universes were created- Without using 11 any symbols and restricting yourself to a total of three digits, write a number that far exceeds the total atoms No factorial can end in five zeros. What is the next of all these universes. smallest number of zeros in which a factorial cannot end? 9^(9^(9)) 11 The sum of the digits on the odometer in my car (which reads up to 99999.9 miles) has never been higher than Barnie Bookworm bought a thriller - found to his dismay, Just before the denouement a fascicle astray. it is now, but it was the same 900 miles ago. How many Instead of counting one through ten, a standard cure for rages, miles must I drive before it is higher than it is now. He totaled up the number of the missing sheal of pages. The total was eight thousand and six hundred fifty-six. 100 miles What were the missing pages? Try to find them just lor kicks The first expedition to Mars found only the ruins of a Page 225-286 civilization. The explorers were able to translate a Martian equation as follows: 5x^2 – 50x + 125 = 0, x = The Sultan arranged his wives in order of increasing 5, x = 8. This was strange mathematics. The value x = seniority and presented each with a golden ring. Next, 5 seemed legitimate enough but x = 8 required some every 3rd wife, starting the 2nd, was given a 2nd ring; explanation. If the Martian number system developed in of every 3rd one starting with the 2nd received a 3rd a manner similar to ours, how many fingers would you ring, etc. His first and most cherished wife was the only say the Martians had? one to receive 10 rings. How many wives had the Sultan. 13 fingers 9,842 wives A rectangular picture, each of whose dimensions is an integral number of inches, has an ordinary rectangular The alphametic LIX + LVI = CXV and X^2 = C involving frame 1 inch wide. Find the dimensions of the picture if Roman numerals is correct. It will still be correct if the the area of the picture and the area of the frame are proper Arabic numerals are substituted. Each letter equal? denotes the same digit throughout, and no 2 letters stand for the same digit. Find the unique solution. 3x10 or 4x6 938  Chapter 8: More Advanced Problems The price per cubic inch for plantinum trays is the same A collection of problems from various fields which as that per square inch for platinum sheets. A metal require calculus or other sophisticated supply house has a square of platinum which will yield mathematical tools the same amount whether sold as a sheet or fashioned into a tray of maximum volume with the four cut-out A ball is dropped from a height of 10 feet. It rebounds corners sold as sheets. How big is the square? one-half the distance in each bounce. What is the total distance it travels? 1 foot 30 feet A rectangular box without a top is to be made from a sheet of metal in the manner familiar to all calculus students, i.e., by cutting out squares from the corners and bending up the sides. The finished product is to have maximum volume and its dimensions are to be all integers" How will these dimensions compare if the metal cutout amounts to 10% of the original sheet? 6:3:1 A student studying series starts with the familiar 1 + 1/2 + 1/4 + 1/8 + … and inserts terms midway between these, obtaining 1 + 3/4 + 1/2 + 3/8 + ¼ + … He divides this by 2, since, as he explains it. “There are now twice as many terms as before.” He repeats the process, interpolating terms between those already placed, again dividing by 2. If he continues this indefinitely, what limit will the series approach? 3/2 Mr. X veers to the right when he walks. The curvature of his path is proportional to his latitude. He starts walking North from point A on the equator, in the area of a large level plain, and finds he Is proceeding East when he is one mile north of the equator. He continues walking and arrives back at the equator at point B. What is the straight-line distance from A to B? 2x = a trifle less than 1.2 miles Which contains more terms: the general polynomial of tenth degree in six variables or the general polynomial of sixth degree in ten variables? Both are equal (8,008 terms) A boat owner agrees to take a group on an outing at $4.50 apiece if the number of passengers is equal to or less than his break-even point. For each person above this he reduces the fare for all passengers 3 cents per person. If he has on board now the number of passengers that maximizes the total collected, what is the boat owner's profit? Zero