A lighthouse shows successive one-second flashes of red, white, green, green, white, red. A second lighthouse does the same only with two-second flashes. The six-second sequence of the first lighthouse is repeated steadily, as is the twelve-second sequence of the other lighthouse. What fraction of the time do the two lights show the same color if the given sequences start at the same time?

The set A contains the integers 0,4,5,9,11,12,13,14,19,… The set B contains 1,2,3,6,7,8,10,15,16,17,18,… Place 20 and 21 in their proper sets.

Six grocers in a town each sell a different brand of tea in four ounce packets at 25 cents per packet. One of the grocers gives short weight, each packet of his brand weighing only 37a 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?

Very few people are aware of the growth pattern of Jack's beanstalk. On the first day it increased its height by 1/2, on the second day by 1/3, on the third day by 1/4, and so on. How long did it take to achieve its maximum height (100 times its original height)?

In the game "subtract-a-square," a positive integer is written down and two players alternately subtract squares from it with the restriction that the remainder must never be less than zero. The player who leaves zero wins. What square should the first player subtract if the original number is 29?

On a certain day, our parking lot contains 999 cars, no two of which have the same 3 digit license number. After 5:00 p.m. what is the probability that the license numbers of the first 4 cars to leave the parking lot are in increasing order of magnitude?

A hospital nursery contains only two baby boys; the girls have not yet been counted. At 2:00 p.m. a new baby is added to the nursery. A baby is then selected at random to be the first to have its footprint taken. It turns out to be a boy. What is the probability that the last addition to the nursery was a girl?

A gambler devised a game to be played with a friend. He bet 1/2 the money in his pocket on the toss of a coin; heads he won, tails he lost. The coin was tossed and the money handed over. The offer was repeated and the game continued. Each time the bet was for 1/2 the money then in his possession. Eventually the number of times he lost was equal to the number of times he won. Quickly now! Did he gain, lose, or break even?

A game of super-dominoes is played with pieces divided into three cells instead of the usual two, containing all combinations from triple blank to triple six, with no duplications. For example the set does not include both I 2 3 and 3 2 I since these are merely reversals of each other. (But, it does contain 1. 3 2.) How many pieces are there in a set?

A sharp operator makes the following deal. A player is 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 toss. The sucker pays ten dollars for this. How much can the operator expect to make if this is repeated a great many times?

There are three families, each with two sons and two daughters. In how many ways can all these young people be married?

In a carnival game 5 balls are tossed into a square box divided into 4 square cells, with baffles to insure that every ball has an equal chance of going in any cell. The player pays $ I and receives $1 for every cell which is empty after the 5 balls are thrown. How much does the operator expect to make per game?

Two men are walking towards each other at the side of a railway. A freight train overtakes one of them in 20 seconds and exactly ten minutes later meets the other man coming in the opposite direction. The train passes this man in l8 seconds. How long after the train has passed the second man will the two men meet? (Constant speeds are to be assumed throughout.)

If all 720 permutations of the digits 1 through 6 are arranged in numerical order, what is the 417th term?

There are nine cities which are served by two competing airlines. One or the other airline (but not both) has a flight between every pair of cities. What is the minimum number of triangular flights (i.e., trips from A to B to C and back to A on the same airline)?

A necklace consists of pearls which increase uniformly from a weight of 1 carat for the end pearls to a weight of 100 carats for the middle pearl. If the necklace weighs altogether 1650 carats and the clasp and string together weigh as much (in carats) as the total number of pearls, how many pearls does the necklace contain?

A pencil, eraser and notebook together cost $1.00. A notebook costs more than two pencils, and three pencils cost more than four erasers. If three erasers cost more than a notebook, how much does each cost?

Jai Alai balls come in boxes of 8 and 15; so that 38 balls (one small box and two large) can be bought without having to break open a box, but not 39. What is the maximum number of balls which cannot be bought without breaking boxes?

A new kind of atom smasher is to be composed of two tangents and a circular arc which is concave towards the point of intersection of the two tangents. Each tangent and the arc of the circle is I mile long. What is the radius of the circle?

A farmer owned a square field measuring exactly 2261 yards on each side. 1898 yards from one corner and 1009 yards from an adjacent corner stood a beech tree. A neighbor offered to purchase a triangular portion of the field stipulating that a fence should be erected in a straight line from one side of the field to an adjacent side so that the beech tree was part of the fence. The farmer accepted the offer but made sure that the triangular portion was of minimum area. What was the area of the field the neighbor received, and how long was the fence?

A cowboy is five miles south ot a stream which flows due east. He is also 8 miles west and 6 miles north of his cabin. He wishes to water his horse at the stream and return home. What is the shortest distance he can travel and accomplish this?

Three hares are standing in a triangular field which is exactly 100 yards on each side. One hare stands at each corner; and simultaneously all three set off running. Each hare runs after the hare in the adjacent corner on his left, thus following a curved course which terminates in the middle of the field, all three hares arriving there together. The hares obviously ran at the same speed, but just how far did they run?

A coffee pot with a circular bottom tapers uniformly to a circular top with radius half that of the base. A mark halfway up the side says "2 cups." Where should the "3 cups" mark go?

A scalene triangle ABC which is not a right triangle has sides which are integers. If sin A = 5/13, find the smallest values for its sides, i.e., those values which make the perimeter a minimum.

A one-acre field in the shape of a right triangle has a post at the midpoint of each side. A sheep is tethered to each of the side posts and a goat to the post on the hypotenuse. The ropes are just long enough to let each animal reach the two adjacent vertices. What is the total area the two sheep have to themselves, i.e., the area the goat cannot reach?

A divided highway goes under a number of bridges, the arch over each lane being in the form of a semi-ellipse 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 can pass?

1960 and 1961 were bad years for ice cream sales but 1962 was very good. An accountant was looking at the tonnage sold in each year and noticed that the digital sum of the tonnage sold in 1962 was three times as much as the digital sum of the tonnage sold in 1961. Moreover, if the amount sold in 1960 (346 tons), was added to the 1961 tonnage, this total was less than the total tonnage sold in 1962 by the digital sum of the tonnage sold in that same year. Just how many more tons of ice cream were sold in 1962 than in the previous year?

A certain magic square contains nine consecutive 2-digit numbers. The sum of the numbers in any line is equal to one of the numbers in the square with the digits reversed. This is still the case if 7 is added to each entry. What is the number in the center square?

Three rectangles of integer sides have identical areas. The first rectangle is 278 feet longer than wide. The second rectangle is 96 feet longer than wide. The third rectangle is 542 feet longer than wide. Find the area and dimensions of the rectangles.

Every year an 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?

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?

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 mph, a train leaves B for A at l5 mph. Just as the first train leaves A, a South American botfly flies from the front of the engine straight toward the other train at IOO mph. On meeting the second train it immediately turns back and flies straight for the first train. It continues to fly back and forth with undiminished speed until it is crushed in the eventual collision. How far had the fly flown?

A jig-saw puzzle contains 100 pieces. A "move" consists of connecting two clusters (including "clusters" of just one piece.) What is the minimum number of moves required to complete the puzzle?

With some sharp reasoning, you ought to be able to determine the last member of the sequence for which the first 20 members are: 11, 31, 71, 91, 32, 92, 73, 73, 14, 34, 74, 35, 95, 16, 76, 17, 37, 97, 38, 98, ____?

The passengers on an excursion bus consisted of 14 married couples, 8 of whom brought no children, and 6 of whom brought 3 children apiece. Counting the driver, the bus had 31 occupants. How is this possible?

Between Kroflite and Beeline are five other towns. The seven towns are an integral number of miles from each other along a straight road. The towns are so spaced that if one knows the number of miles a person has traveled between any two towns he can determine the particular towns uniquely. What is the minimum distance between Kroflite and Beeline to make this possible?

You and a friend spot a loose $20 bill simultaneously and agree to an auction in which you write down your bids and compare them. High bidder gets the $20 and pays the other the amount of the higher bid. Tie bidders split the $20. How much do you bid?

If 2 marbles are removed at random from a bag containing black and white marbles, the chance that they are both white is 1/3. If 3 are removed at random, the chance that they all are white is l/6. How many marbles are there of each color?

Hard Knox College is a member of a six school basketball league in which every pair of schools plays each other twice. The other five schools ended the season with respective league records of .200, .300, .500, .600, and .800. How did Hard Knox make out?

A coin is so unbalanced that you are likely to get two heads in two successive throws as you are to get tails in one. What is the probability of getting heads in a single throw?

All the members of a fraternity play basketball while all but one play ice hockey; yet the number of possible basketball teams (5 members) is the same as the number of possible ice hockey teams (6 members). Assuming there are enough members tb form either type of team, how many are in the fraternity?

The local weather forecaster says "no rain" and his record is 2/3 accuracy of prediction. But the Federal Meteorological Service predicts rain and their record is 3/4. With no other data available, what is the chance of rain?

Martian coins are 3-sided (heads, tails, and torsos), each side coming up with equal probability. Three Martians decided to go odd--man-out to determine who Pays a dinner check. (If two coins come up the same and one different, the owner of the latter coin foots the bill). what is the expected number of throws needed in order to determine a loser?

Max and his wife Min each toss a pair of dice to determine where they will spend their vacation. If either of Min's dice displays the same number of spots as either of Max's, she wins and they go to Bermuda. Otherwise, they go to Yellowstone. What is the chance they'll see "Old Faithful" this year?

To stimulate his son in the pursuit of partial differential equations, a math professor offered to pay him $8 for every equation correctly solved and to fine him $5 for every incorrect solution. At the end of 26 problems, neither owed any money to the other. How many did the boy solve correctly?

Venusian batfish come in three sexes, which are indistinguishable (except by Venusian batfish). How many live specimens must our astronauts bring home in order for the odds to favor the presence of a "mated triple" with its promise of more little batfish to come?

A forgetful physicist forgot his watch one day and asked an E.E. on the staff what time it was. The E.E. Iooked at his watch and said: "The hour, minute, and sweep second hands are as close to trisecting the face as they ever come. This happens only twice in every 12 hours, but since you probably haven't forgotten whether you ate lunch, you should be able to calculate the time.” What time was it to the nearest second?

Mr. Field, a speeder, travels on a busy highway having the same rate of traffic flow in each direction. Except for Mr. Field, the traffic is moving at the legal speed limit. Mr. Field passes one car for every nine which he meets from the opposite direction. By what percentage is he exceeding the speed limit?

The teacher marked the quiz on the following basis: one point for each correct answer, one point off for each question left blank and two points off for each question answered incorrectly. Pat made four times as many errors as Mike, but Mike left nine more questions blank. If they both got the same score, how many errors did each make?

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?

There are four towns at the corners of a square. Four motorists set out, each driving to the next (clockwise) town, and each man but the fourth going 8 mi./hr. faster than the car ahead-thus the first car travels 24 mi./hr. faster than the fourth. At the end of one hour the first and third cars are 2O4, and the second and fourth 212 (beeline) miles apart. How fast is the first car traveling and how far apart are the towns?

A man leaves from the point where the prime meridian crosses the equator and moves forty-five degrees northeast by geographic compass which always points toward the north geographic pole. He constantly corrects his route. Assuming that he walks with equal facility on land and sea, where does he end up and how far will he have travelled when he gets there?

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 wlll it take them to reach the railroad?

If the equal sides of an isosceles triangle are given, what length of the third side will provide maximum area? (No calculus, please.)

Three farmers, Adams, Brown and Clark all have farms containing the same number of acres. Adams' farm is most nearly square, the length being only 8 miles longer than the width. Clark has the most oblong farm, the length being 34 miles longer than the width. Brown's farm is intermediate between these two, the length being 28 miles longer than the width. If all the dimensions are in exact miles, what is the size of each farm?

A diaper is in the shape of a triangle with sides 24, 20 and 20 inches. The long side is wrapped around the baby’s waist and overlapped two inches. The third point is brought up to the center of the overlap and pinned in place. The pin is to go through three thicknesses of material. What is the area in which the pin may be placed?

The sum of the digits on the odometer in my car (which reads up to 99999.9 miles) has never been higher than it is now, but it was the same 900 miles ago. How many miles must I drive before it is higher than it is now.

In European countries the decimal point is often written a little above the line. An American, seeing a number written this way, with one digit on each side of the decimal point, assumed the numbers were to be multiplied. He obtained a two-digit number as a result, but was 14.6 off. What was the original number?

Dad and his son have the same birthday. On the last one, Dad was twice as old as Junior. Uncle observed that this was the ninth occasion on which Dad's birthday age had been an integer multiple of Junior's. How old is Junior?

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?

The undergraduates of a School. of Engineering wished to form ranks for a parade. In ranks of 3 abreast, 2 men were left over; in ranks of 5, 4 over; in 7's, 6 over; and 11's, 10 over. What is the least number of marchers there must have been?

A novice librarian shelved a twelve volume set of encyclopedias in the following order from left to right. Volumes 8, 11, 5, 4, 9, 1, 7, 6, 10, 3, 12, and 2. Using her system, where will the annual supplement, Volume 13, go?

Assuming the sun rises at 6:00 a.m., sets at 6:00p.m., and moves at a uniform rate. how can a lost boy scout determine south by means of a watch on a cloudless day?

Four players played a hand of hearts at $1 a point (pairwise payoffs). Dave lost $10 to Arch, $12 to Bob, and $20 to Chuck. How many hearts did Poor Dave take in?

What is the least number of links that must be disengaged from a 23-link chain so that any number of links from 1 to 23 can be obtained by taking one or more of the pieces?

Mary Ann Moore's father has a yacht and so has each of his four friends: Colonel Downing, Mr. Hall, Sir Barnacle Hood, and Dr. Parker. Each of the five also has one daughter and each has named his yacht after a daughter of one of the others. Sir Barnacle's yacht is the Gabrielle, Mr. Moore owns the Lorna; Mr. Hall the Rosalind. The Melissa, owned by Colonel Downing, is named after Sir Barnacle's daughter. Gabrielle's father owns the yacht which is named after Dr. Parker's daughter. Who is Lorna's father?

In Bristol 90% of the citizens drink tea; 80% drink coffee; 7O% drink whiskey; and 6O% drink gin. No one drinks all four beverages. What percent of Bristol's citizens drink liquor?

Three marksmen simultaneously shoot at and hit a rapidly spinning spherical target. What is the probability that the three points of impact are on the same hemisphere?

A teenager wants to go out 2 consecutive nights out of a 3-day weekend. Permission for each night is obtained (or denied) by asking either Father or Mother. Father is known to be more likely to grant permission. However, if the same parent is asked on 2 consecutive days the answers are never the same 2 days running. Whom should he ask first?

Assume that a single depth charge has a probability of 1/2 of sinking a submarine, l/4 of damage and l/4 of missing. Assume also that two damaging explosions sink the sub. What is the probability that 4 depth charges will sink the sub?

A list contains 1968 statements, numbered in serial order' For each k, the kth statement is: "This list contains exactly k false statements." Determine the truth or falsity of each statement?

How many three digit telephone area codes are possible given that: (a) the first digit must not be zero or one; (b) the second digit must be zero or one; (c) the third digit must not be zero; (d) the third digit may be one only if the second is zero.

Having lost a checker game, a specialist in learning programs threw one of the red checkers out the window. His wife reboxed the 12 black pieces and 11 red pieces one at a time in random fashion. The number of black checkers in the box always exceeded the number of reds. What was the a priori probability of this occurrence?

A long shot poker player draws two cards to the five and six of diamonds and the joker. What are his chances of coming up with a pat hand? (straight or flush).

There are four volumes of an encyclopedia on a shelf, each volume containing 300 pages, (that is, numbered 1 to 600), but these have been placed on the shelf in random order. A bookworm starts at the first page of Vol. 1 and eats his way through to the last page of Vol. 4. What is the expected number of pages (excluding covers) he has eaten through?

An expert on transformer design relaxed one Saturday 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 tripled his money. He bet $54 on the third race and quadrupled his money. He bet $72 on the fourth race and lost it, but still had $48 left. With how much money did he start?

A pupil wrote on the blackboard a series of fractions having positive integral terms and connected by signs which were either all + or all ×, although they were so carelessly written it was impossible to tell which they were. It still wasn't clear even though he announced the result of the operation at every step. The third fraction had denominator 19. What was the numerator?

The faces of a solid figure are all triangles. The figure has nine vertices. At each of six of these vertices, four faces meet, and at each of the other three vertices, six faces meet. How many faces does the figure have?

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?

A spider and a fly are located at opposite vertices of a room of dimensions 1, 2 and 3 units. Assuming that the fly is too terrified to move, find the minimum distance the spider must crawl to reach the fly.

A student, just beginning the study of logarithms, was required to evaluate an expression of the form logA/logB. He proceeded to cancel common factors in both numerator and denominator, (including the "factor" log), and arrived at the result 2/3. Surprisingly, this was correct. What were the values of A and B?

An Origami expert started making a Nani des-ka by folding the top left corner of a sheet of paper until it touched the right edge and the crease passed through the bottom left corner, He then did the same with the lower right corner, thus making two slanting parallel lines. The paper was 25 inches long and the distance between the parallel lines was exactly 7/40 of the width. How wide was the sheet of paper?

Near the town of Lunch, Nebraska there is a large triangular plot of land bounded by three straight roads which are 855, 870, and 975 yards long respectively. The owner of the land, a friend of mine, told me that he had decided to sell half the plot to a neighbor, but that the buyer had stipulated that the seller of the land should erect the fence which was to be a straight one' The cost of fences being high, my friend naturally wanted the fence to be as short as possible. What is the minimum length the fence can be?

A cross section through the center of a football is a circle x inches in circumference. The football is x-8 inches long from tip to tip and each seam is an arc of a circle 3/4 x inches in diameter. Find x.

An icicle forming from a dripping gutter is in the shape of a cone five times as long as it is wide (at the top). A few hours later it has doubled in length and the generating angle has also doubled. How does its present weight compare with its previous weight?

On what days of the week can the first day of a century fall? (The first day of the twentieth century was Jan. 1, 1901)?

A ball is dropped from a height of 10 feet. It rebounds one-half the distance on each bounce. What is the total distance it travels?

A group of hippies are pondering whether to move to Patria, where polygamy is practiced but polyandry and spinsterhood are prohibited, or Matria, where polyandry is permitted and polygamy and bachelorhood are proscribed. In either event the possible number of "arrangements” is the same. The girls outnumber the boys. How many are there?

A neat computer programmer wears a clean shirt every day. If he drops off his laundry and picks up the previous week's load every Monday night, how many shirts must he own to keep him going?

Six boys on a hockey team pick a captain by forming a circle and counting out until only one remains. Joe is given the option of deciding that number to count by. If he is second in the original counting order what number should he choose?

A salesman visits ten cities arranged in the form of a circle, spending a day in each. He proceeds clockwise from one city to the next, except whenever leaving the tenth city he may go to either the first or jump to the second city- How many days must elapse before his location is completely indeterminate, i.e., when he could be in any one of the ten cities?

Three dart players threw simultaneously at a tic-tac-toe board, each hitting a different square. What is the probability that the three hits constituted a win at tic-tac-toe?

Smith and Jones, both 50% marksmen, decide to fight a duel in which they exchange alternate shots until one is hit. What are the odds in favor of the man who shoots first?

There are n points on a circle. A straight line segment is drawn between each pair of points. How many intersections are there within the circle if no 3 lines are collinear?

In the final seconds of the game, your favorite N.B,A. team is behind 117 to 118. Your center attempts a shot and is fouled for the 2nd time in the last 2 minutes as the buzzer sounds. Three to make two in the penalty situation. Optimistic? Note: the center is only a 5O% free-thrower. What are your team's overall chances of winning?

Dr. Irving Weiman, who is always in a hurry, walks up an upgoing escalator at the rate of one step per second. Twenty steps bring him to the top. Next day he goes up at two steps per second, reaching the top in 32 steps. How many steps are there in the escalator?

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 circle?

RCE