LOOKS FAM part 1

Mr. Field, a speeder, travelers 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?

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

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

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?

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 town uniquely. What is the minimum distance between Kroflite and Beeline to make this possible?

The estimated cycle time for a wheel scraper is 6.5 min. Calculate the number of pushers required to serve a fleet of nine scrapers using single pushers. Determine the result for both backtrack and chain-loading methods.

Given the following information on a shovel/truck operation, calculate the number of trucks theoretically required and the production of this combination.

Shovel production
at 100% efficiency = 283 BCM/h

Job efficiency = 0.75

Truck capacity = 15.3 BCM

Truck cycle time,
excluding loading = 0.5h

Determine the design lateral force for the slab form 152 mm thick, 6.1 m wide, and 30.5 m long shown in Figure 5. The slab is to be poured in one pour. Assume concrete density is 2403 kg/m3 and that the formwork weighs 0.72 kPa.

A jaw crusher is producing 227 t/h of crushed gravel and discharging it onto a three-screen deck. The top screen in the deck is a 38-mm screen. The gradation of crusher output shows 100% passing 76 mm, 92% passing 38 mm, and 80% passing 19 mm. Material weight is 1842 kg/m3. Find the minimum size of the 38-mm screen to be used. Check both total screen load and screen passing capacity.

A copper wire has a nominal breaking strength of 300 MPa and a reduction of area of 77%. Calculate the true tensile strength (breaking load/true area).

An aluminum wire is stressed 34.5 MPa in tension. What temperature increase is required to change its length by the same amount?

The true density of a concrete to which an air-entraining agent was added is 2.80 g/cm3. However, a dry core (15.24 cm x 10.16 cm) of this concrete weighs only 2.982 kg. The same core weighs 3.08 kg when saturated with water. What are the percents of the open and closed pores?

The true density of a concrete to which an air-entraining agent was added is 2.80 g/cm3. However, a dry core (15.24 cm x 10.16 cm) of this concrete weighs only 2.982 kg. The same core weighs 3.08 kg when saturated with water. What are the bulk and apparent densities?

An electrical porcelain product has only 1 v/o porosity as sold. It had 27v/o porosity after presssing and drying, but before sintering. How much linear shrinkage occured during sintering?

A bus route on a city street is scheduled for 2-min headways. On the average, 7 passengers per bus board at a particular stop and 10 passengers alight. All boarding passengers use the front door and all alighting passengers use the back door. Fare collection is exact fare and farebox, but no single coin. Assume clearance time is 15 s, and that the maximum allowable probability of bus queuing is 1 percent. The stop is a near-side stop at an intersection with g/C ratio of 0.50. How many berths are required?

The number of hours daylight, D(t) at a particular time of the year can be approximated by D(t) = for t days and t = 0 corresponding to January 1. The constant K determines the total variation in day length and depends on the latitude of the locale. When is the day length the longest, assuming that it is NOT a leap year?

The number of hours daylight, D(t) at a particular time of the year can be approximated by D(t) = for t days and t = 0 corresponding to January 1. The constant K determines the total variation in day length and depends on the latitude of the locale. When is the day length the shortest, assuming that it is NOT a leap year?

Labor laws in Erewhon require factory owners to give every worker a holiday whenever one of them
has a birthday and to hire without discrimination on the grounds of birthdays. Except for these holidays,
they work a 365-day year. The owners want to maximize the expected total number of man-days
worked per year in a factory. How many workers do factories have in Erewhon?

A spider eats three flies a day. Until he fills his quota, he has an even chance of catching a fly that attempts to pass. A fly is about to make the attempt. What are the chances of survival, given that five flies have already made the attempt today?

In a little known work, the famous geometer of Skalenos proves the following theorem: “The square of the side opposite the Fandangle is equal to the sum of the squares of the other two sides added to the product of those two sides multiplied by the square root of two.” What is a Fandangle?

Assume that a single depth charge has a probability of ½ of sinking a submarine, ¼ of damage and ¼ missing. Assume also that two damaging explosions sink the sub. What is the probability that 4 depth charges sink the sub?

A billiard table is in the form of an ellipse with one axis two feet longer than the other. A ball is stuck from one focus and after bounding against two cushions returns to its starting point. At the halfway point in its trip, the ball is eight feet from the source. How big is the table?

Data over the last 5 years indicate that for each fatality, there are 40 nonfatal injury accidents (\$15,000 present cost each) and 300 property
damage accidents (\$2000 present cost each). What is the breakeven value of a statistical life (VSL) needed to justify a highway project if i is 8%? The death rate on a particular three-lane road is 8 per
100 million vehicle-miles. Adding a lane would reduce this to 5 per 100 million, and other accidents would be reduced proportionately. The lane would cost \$1.5M per mile to build, and annual maintenance would be 3% of the first cost. Assume that the lane would last 40 years. The road carries 10,000 vehicles per day

Rosenberg Engineering has offices in northern California, where a heat pump can be used for cooling in the summer and heating in the winter.
Replacing their current system will cost \$1500 in May and \$500 in June. Starting in July, it will save
them \$200 per month for the summer months (June–August), \$100 for the fall and spring months,
and \$150 for the winter months (November–March). What is the payback period for the heat pump?

Six men decide to play Russian roulette with a six gun loaded with one cartridge.
They draw for position, and afterwards, the sixth man casually suggests that instead of letting the chamber rotate in sequence, each man spin the chamber before shooting. How would this improve his
chances?

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?

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?

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

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 occurence?

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?

Four boys, Alan, Brian, Charles, and Donald, and four girls, Eve, Fay, Gwen, and Helen are each in love with one of the others, and, sad to say, in no case is their love requited. Alan loves the girl who loves the man who loves Eve. Fay is loved by the man who is loved by the girl loved by Brian. Charles loves the girl who loves Donald. Brian is not loved by Gwen, and the boy who is loved by Helen does not love Gwen, who loves Alan?

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?

Assuming the sun rises at 6:00 a.m., sets 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 a cloudless day? (past board exam)

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? (past board exam)

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?

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 \$1 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?

A hospital nursery contains only two baby boys; the girls have not yet been counted. At 2:00 pm, 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?

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?

In 1969, the World Series began in the stadium of the American League pennant winner. Assume the contenders are evenly matched. What is the probability that the series ended where it began?

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 1/4, and so on. How long did it take to achieve its maximum height (100 times its original height)?

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?

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 grocers in a twon 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 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?

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?

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 shots first?

In the final seconds of the game, your favorite NBA 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 50% free-thrower. What are your team's overall chances of winning?

A hunter wished to take his one-piece rifle on a train but the conductor refused to permit it in the coach and the baggage man could not take any article whose greatest dimension exceeded 1 yard. The length of the rifle was 1.7 yards. What could the hunter do?

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 second group and 2, 5, 10, 12, 13, ... in the third. In which groups would 15, 16, and 17 be placed?

Tom, Dick, and Harry played a round of golf, each ending with a total of 72 strokes. Each pair competed against each other in match play (most holes won). Tom beat Dick, and Dick beat Harry. Does it follow that Tom beat Harry?

A gambler devised a game to be played with a friend. He bet ½ 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 he bet was for ½ 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?

Determine the next three terms of the sequence 12, 1, 1, 1, ...

What letters follow OTTFFSSE_?

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

One of a pair of dice is loaded so that the chance of a 1 turning up is 1/5, the other faces being equally likely. Its mate is loaded so that the chance of a 6 turning up is 1/5, the other faces being equally likely. How much does this loading increase the probability of throwing a 7 with the
two dice?

There are four volumes on 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?

Dr. Furbisher LaRouche, the noted mathematician, was shopping at a hardware store and asked the price of certain articles. The salesman replied, "One would cost 10 cents, eight would cost 10 cents, seventeen would cost 20 cents, one hundred and four would cost 30 cents, seven hundred and fifty six would also cost 30 cents, and one thousand and seventy two would cost 40 cents." What was Dr. LaRouche buying?

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 to form either type of team, how many are there in the fraternity?

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?

Rufus T. Flypaper drives two miles to work every morning. Very precise, he knows he must average 30 mph to arrive on time. One morning a woman driver impedes him for the first mile, cutting his average to only 15 mph. He quickly calculated his proper speed for the rest of his trip to arrive on time. Assume that his car could do 120 mph. Could he arrive on time?

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 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?

An expert gives team A only 40% chance to win the World Series. Basing his calculation on this a gambler offers 6 to 5 odds on team B to win the first game. Which of the following gives equation to find if his judgment is sound?

In Puevigi, the game of craps is played with a referee calling the point by adding together the six faces (three on each die) visible from his vantage point. What is the probability of making 16 the hard way? (That is, by throwing two eights.)

Maynard's Grandfather Clock is driven by two weighs, one for the striking mechanism which strikes the hours only, the other for the time mechanism. When he hears the clock strike his bedtime, he immediately winds the clock and retires. After winding, the weights are exactly opposite each other. The weights are again opposite every six hours thereafter. What is Maynard's bedtime?

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?

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

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

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

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).

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 must not be zero; (d) the third digit may be one only if the second digit is zero.

A prisoner is given 10 white balls, 10 black balls and two boxes. He is told that an executioner will draw one ball 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 arrange the balls in the boxes to give himself the best chance for survival?

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 15 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 100 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?

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 are all white is 1/6. How many marbles are there of each color?

Four swimming pool builders submit sealed bids to a homeowner who is required by law to accept the last bid that he sees, i.e., once he looks at a bid, he automatically rejects all previous bids. He is not required to open all the envelopes, of course. Assuming that all four bids are
different, what procedure will maximize his chances of accepting the lowest bid, and what will be the probability of doing so?

In a fast Major League baseball game, pitcher Hi N. Outside managed to get by with the minimum number of pitches possible. He played the entire game, which was not called prior to completion. How many pitches did he make?

A drawer contains red socks and black socks. When two socks are drawn at random, the probability that both are red is 1/2.

(a) How small can the number of socks in the drawer be?

A drawer contains red socks and black socks. When two socks are drawn at random, the probability that both are red is 1/2.

(b) How small if the number of black socks is even?

To encourage Elmer’s promising tennis career, his father offers him a prize if he wins (at least) two tennis sets in a row in a three-set series to be played with his father and the club champion alternately: father-champion-father or champion-father-champion, according to Elmer’s choice. The champion is a better player than Elmer’s father. Which series should Elmer choose?

A three-man jury has two members each of whom independently has probability p of making the correct decision and a third member who flips a coin for each decision (majority rules). A one-man jury has a probability p of making the correct decision. Which jury has the better probability of making the correct decision?

On the average, how many times must a die be thrown until one gets a 6?

In a common carnival game, a player tosses a penny from a distance of about 5 feet onto the surface of a table ruled in 1-inch squares. If the penny (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does not get his penny back; otherwise, he loses his penny. If the penny lands on the table, what is his chance to win? If the 1-inch square is made smaller by thickening the lines to 1/16 in wide, what is his chance to win?

Chuck-a-Luck is a gambling game often played at carnivals and gabling houses. A player may bet on any one of the numbers 1, 2, 3, 4, 5, 6. Three dice are rolled. If the player’s number appears on one, two, or three of the dice, he receives respectively one, two, or three times his original stake plus his own money back; otherwise, he loses his stake. What is the player’s expected loss per unit stake? (Actually, the player may distribute stakes on several numbers, but each such stake can be regarded as a separate bet.)

Mr. Brown always bets a dollar on the number 13 at a roulette against the advice of Kind Friend. To help cure Mr. Brown of playing roulette, Kind Friend always bets Brown \$20 at even money that Brown will be behind at the end of 36 plays. How is the cure working? (Most American roulette wheels have 38 equally likely numbers. If the player’s number comes up, he is paid 35 times his stake and gets his original stake back; otherwise, he loses his stake.)

We often read of someone who has been dealt 13 spades at bridge. With a well-shuffled pack of cards, what is the chance that you are dealt a perfect hand (13 of one suit)?
(Bridge is played with an ordinary pack of 52 cards, 13 in each of 4 suits, and each of 4 players is dealt 13.)

The game of craps, played with two dice, is one of America’s fastest and most popular gambling games. Calculating the odds associated with it is an instructive exercise.The rules are these. Only totals for the two dice count. The player throws the dice and wins at once if the total for the first throw is 7 or 11, loses at once if it is 2, 3, or 12. Any other throw is called his “point.” If the first throw is a point, the player throws the dice repeatedly until he either wins by throwing his point again or loses by throwing 7. What is
the player’s chance to win?

Three prisoners, A, B, and C, with apparently equally good records have applied for parole. The parole board has decided to release two of the three, and the prisoners know this but not which two. A warder friend of prisoner A knows who are to be released. Prisoner A realizes that it would be unethical to ask the warder if he, A, is to be released, but thinks of asking for the name of one prisoner
other than himself who is to be released He thinks that before he asks, his chances of release are 2/3. He thinks that if the warder says, "B will be released," his own chances have now gone down to 1/2, because either A and B or B and C are to be released. And so, A decides not to reduce his chances by asking. However, A is mistaken in his calculations Explain.

Coupons in cereal boxes are numbered 1 to 5, and a set of one of each is required for a prize. With one coupon per box, how many boxes on average are required to make a complete set?

Eight eligible bachelors and seven beautiful models happen randomly to have purchased single seats in the same 15 seat row of a theater. On the average, how many pairs of adjacent seats are ticketed for marriageable couples?

A tennis tournament has 8 players. The number a player draws from a hat decides his first-round rung in the tournament ladder. Suppose that the best player always defeats the next best and that the latter always defeats all the rest. The loser of the finals gets the runner-up cup. What is the chance that the second-best player wins the runner-up cup?

(a) Suppose King Arthur holds. a jousting tournament where the jousts are in pairs as in a tennis tournament. See Problem 16 for tournament ladder. The 8 knights in the tournament are evenly matched, and they include the twin knights Balin and Balan. What is the chance that the twins meet in a match during the tournament?

(b) Replace 8 by 2^n in the above problem. Now what is the chance that they meet?

Pepys wrote Newton to ask which of three events is more likely: that a person get

A, B, and C are to fight a three cornered pistol duel. All know that A’s chance of hitting his target is 0.3, C’s is 0.5, and B never misses. They are to fire at their choice of target in succession in the order of A, B, C, cyclically (but a hit man loses further turns and is no longer shot at) until only one man is left unhit. What should A’s strategy be?

In an election, two candidates, Albert, and Benjamin, have in a ballot box a and b votes respectively, a > b, for example, 3 and 2. If ballots are randomly drawn and tallied, what is the chance that at least once after the first tally the candidates have the same number of tallies

Players A and B match pennies N times. They keep a tally of their gains and losses. After the first toss, what is the chance that at no time during the game will they be even.

If a chord is selected at random on a fixed circle what is the probability that its length exceeds the radius of the circle?

Duels in the town of Discretion are rarely fatal. There, each contestant comes at a random moment between 5 A.M. and 6 A.M. on the appointed day and leaves exactly 5 minutes later, honor served, unless his opponent arrives within the time interval and then they fight. What fraction of duels lead to violence?

The king’s minter boxes his coins 100 to a box. In each box he puts 1 false coin. The king suspects the minter and from each of 100 boxes draws a random coin and has it tested. What is the chance the minter’s peculations go undetected?

A bread salesman sells on the average 20 cakes on a round of his route. What is the chance that he sells an even number of cakes?

What is the least number of persons required if the probability exceeds 1/2 that two or more of them have the same birthday? (Year of birth need not match.)

You want to find someone whose birthday matches yours. What is the least number of strangers whose birthdays you need to ask about to have a 50-50 chance?

If r persons compare birthdays in the pairings problem, the probability is $P_R$ that at least 2 have the same birthday. What should n be in the personal birthmate problem to make your probability success approximately $P_R$?

Labor laws in Erewhon require factory owners to give every worker a holiday whenever one of them has a birthday and to hire without discrimination on grounds of birthdays. Except for these holidays they work a 365-day year. The owners want to maximize the expected total number of man-days worked per year in a factory. How many workers do factories have in Erewhon?

From where he stands, one step toward the cliff would send the drunken man over the edge. He takes random steps, either toward or away from the cliff. At any step his probability of taking a step away is 2/3, if a step toward the cliff 1/3. What is his chance of escaping the cliff?

Player M has \$1, and Player N has \$2. Each play gives one of the players \$1 from the other. Player M is enough better than Player N that he wins 2/3 of the plays. They play until one is bankrupt. What is the chance that Player M wins?