LITTONS (LooksFam) RCE 2024

To God be the Glory

Using a steel tape that is too short to measure consecutive distances.

If an experiment is unaffected by experimental error, the results are accurate. Which of the following is an example of a systematic error?

The claim is inaccurate at both 95% and 99% confidence limit.

100 bearings were tested to failure. The average life was 1520 hours and the sample standard deviation was 120 hours. The manufacturer wants to claim an average 1600-hour life. Evaluate using confidence limits of 95% and 99%

Incremental stress

The discontinuity stress at an abrupt change in geometry is known as:

The metal is highly strain-hardening capacity.

True stress and true strain from a tensile test performed on a metal bar are plotted on a log-log graph in accordance with ASTM E646. If the slope of the curve is between 0.5 and 1.0, which conclusion can be drawn about the metal?

Allostropism

What property describes the variance of a metal's crystalline structure depending on temperature?

Anisotropic

What kind of material has properties that depend on direction?

Scleroscopic

According to ASTM standards, which test is utilized to test the hardness of metals?

d) All of the above

The modulus of rupture test yields a higher value of strength than a direct tensile test of splitting tensile test made on the same specimen because,

a) The assumed stress block shape does not match the real shape

b) Direct tensile tests are sensitive to any accidental eccentricity

c) The concrete is assumed to be perfectly elastic

d) All of the above

Mortar contains caustic lime

10. The primary difference between components (materials) used to produce Portland cement concrete and mortar for masonry is explained best by which of the following statements?

1, 18, 72, age of occupants

Street No.

1+18+72=91

Maynard the Census Taker visited a house and was told, "Three people live there. The product of their ages is 1296, and the sum of their ages is our house number." After an hour of cogitation Maynard returned for more information. The house owner said, "I forgot to tell you that my son and grandson live here with me." How old were the occupants and what was their street number?

A

A, B and C participate in a track meet, consisting of at least three events. A certain number of points are given for first place, a smaller number for second place, and a still smaller number for third place. A won the meet with a total score of 14 points; B and C are tied for second with 7 points apiece. B won first place in the high jump. Who won the pole vault assuming no ties occurred in any event?

x=2, y=5

Find the smallest number (x) of persons a boat may carry so that (n) married couples may cross a river in such a way that no woman ever remains in the company of any man unless her husband is present. Also find the least number of passages (y) needed from one bank to the other. Assume that the boat can be rowed by one person only.

1980pie or 3.36 furlongs

There is one flag at the entrance to a racetrack and another inside the track, half a mile from the first. A jockey notes that no matter where he is on the track, one flag is 3 times as far away as the other. How long is the track?

¼ that of a square or 12.25

The isosceles right triangle shown above has a vertex at the center of the square. What is the area of the common quadrilateral?

Will lie on the 3rd side or 0 distance

One side of the triangle is 10 feet longer than another and the angle between them is 60°. Two circles are drawn with these sides as diameters. One of the points of intersection of the two circles is the common vertex. How far from the third side is the other point of intersection?

x=175 degrees, y=185 degrees

A student beginning the study of Trigonometry came across an expression of the form sin (X + Y). He evaluated this as sin X + sin Y. Surprisingly he was correct. The values of X and Y differed by 10'; what were these values, assuming that 0" <X<Y<360°?

Top areas are equal insuring equal cake volume.

Side areas are equal insuring equal icing areas.

A hostess plans to serve a square cake with icing on top and sides. Upon determining how many guests want cake, what method should she use to insure that each guest will receive the same amount of cake and icing?

Place five at the vertices of a regular pentagon, the sixth at the center of the pentagon, and the seventh above the center at a distance equal to the radius of the pentagon.

How can seven points be placed, no three on the same line, so that every selection of three points constitutes the vertices of an isosceles triangle?

Ans: 2.5 in²

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?

One cut in the third link will allow two links to be swapped for a kiss and a link on the second transaction, and 3 links for a kiss and 2 links on the third and so on

A wizard in Numerical Analysis has a gold chain with 7 links. A Lady Programmer challenges him to use the to buy 7 kisses, each kiss to be paid for, separately, with one chain link. What is the smallest number of cuts he will have to make in the chain? What is his sequence of payments?

25

A pupil wrote on the blackboard a series of fractions having positive integral terms and connected by signs which were either all + or all x, 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?

since each v/w^2, w^3/x^4, x^5/y^6 and y^7/z^4 is a constant, therefore their product is also a constant

If v varies as w^2, w^3 as x^4, x^5 as y^6, and y^7 as z^4, show that the product v/z∙w/z∙x/z∙y/z does not vary at all.

4:26.853

At this moment, the hands of a clock in the course of normal operation describe a time somewhere between 4:00 and 5:00 on a standard clock face. Within one hour or less, the hands will have exactly exchanged positions; what time is it now?

60/143 minutes gain

Dr. Reed, arriving late at the lab one morning, pulled out his watch and the hour hand are exactly together every sixty-five minutes.” Does Dr. Reed’s watch gain or lose, and how much per hour?

97

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 = 9/4, B = 27/8

A student just beginning the study of logarithms was required to evaluate an expression of the form log A/log B . 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?

The two expressions are identically equal, respectively, to the smaller and the larger of the two numbers x and y.

Without performing any algebraic manipulation at all, Archimedes O’toole remarked that the sum and product of the two expressions (x+y-|x-y|)/2 and (x+y+|x-y|)/2 are respectively x+y and xy. Why was this obvious?

50 flags

Using the French Tricolor as a model, how many flags are possible with five available colors if two adjacent rows must not be colored the same?

33 pearls

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?

5562

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 18 seconds. How long after the train has passed the second man will the two men meet? (Constant speeds are to be assumed throughout.)

x = 0 or 2

Solve the equation √(x+√(x+√(x+…)) ) =√(x√(x√(x…)) ) where both members represent infinite expressions.

10

To stimulate his son is 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?

Pat = 8 errors, Mike = 2 errors

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?

Equal to 4

Which is greater: 3+4/(3+4/(3+…)) or 3+(3+(3+…)/4)/4 where the respective denominators and numerators continue indefinitely?

t2+1/2[(T2-T1)-(t2-t1)]

mathematician whose clock had stopped wound it but did not bother to set It correctly. Then he walked from his home to the home of a friend for an evening of hi-fi music. Afterwards, he walked back to his own home and set his clock exactly. How could he do this without knowing the time his trip took?

Using the rule that powers multiply by adding their exponents, a multiplicative magic square is easily obtained from the given square by substituting 2n (or kn where k > 1) for n in each block.

Represented above is a “magic square” in which the sum of each row, column, or main diagonal is the same. Using nine different integers, produce a “multiplicative” magic square, i.e., one in which the word “product” is substituted for “sum”.

1

Solve for real values of x: (7+4√3)^x-4(2+√3)^x=-1

25%

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?

764488

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?

Jones = 0.5 hours, Smith = 3.5 hours

A parking lot charges X for the first hour or fraction of an hour and 2/3 X for each hour or fraction thereafter. Smith parks 7 times as long as Jones, but pays only 3 times as much. How long did each park? (The time clock registers only in 5-minute intervals)

1.38 ft or 3.62 ft

A cubic box with sides ‘a’ feet long is placed flag against a wall. A ladder ‘p’ feet long is placed in such a way that it touches the wall as well as the free horizontal edge of the box. If a = 1 and p=√15, calculate at what height the ladder touches the wall.

29

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?

80 steps

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? CE 2017

1414

What is the millionth term of the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …. in which positive integer n occurs in blocks of n terms?

pencil = 26, eraser = 19, notebook = 55

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?

[CE BOARD APR 2023]

V1 = ¾ V2

Two snails start from the same point in opposite directions toward two bits of food. Each reaches his destination in one hour. If each snail had gone in direction the other took, the first snail would have reached his food 35 minutes after the second. How do their speeds compare?

50 miles/hour, 180 miles

There are four towns at the corners of a square. Four motorists set out, each driving in 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 204, and the second and fourth 212 (beeline) miles apart. How fast is the first car traveling and how far apart are the towns?

12

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)?

50 rupees

Citizens of Franistan pay as much income tax (percentage-wise) as they make rupees per week. What is the optimal salary in Franistan?

Half dollars = 1, Dimes = 39, Pennies = 60

There are 100 coins in a piggy bank totalling $5.00 in value, the coins consisting of pennies, dimes, and half dollars. How many of each are there?

Three hummingbirds were sharing the feeding station with cycles of 7, 11, and 13 minutes, respectively, in the order in which he first observed them.

Through binoculars a bird watcher observed a hummingbird feeder between one and two o’clock of an afternoon. He timed the visits and saw a ruby-throat take a drink at 1, 5, 6, 8, 15, 16, 19, 22, 27, 29, 32, 36, 38, 43, 45, 49, 50, 57, and 58 minutes after the hour of one. The last visit he saw took place at two, at which time he left in perplexity. He knew from experience that a hummer’s “feeding cycle” is remarkably stable and is generally between 5 and 15 minutes long. This one seemed rather erratic, to say the least. Can you advise him on what was going on?

36 years old

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

0 ft.

One side of the triangle is 10 feet longer than another and the angle between them is 60˚. Two circles are drawn with these sides as diameters. One of the points of intersection of the two circles is the common vertex. How far from the third side is the other point of intersection?

D = 20 dimes, Q = 23 quarters, N = 4 nickels

When little Willie had sold all his lemonade, he found he had $7.95 in nickels, dimes and quarters. There were 47 coins altogether and, having just started to study geometry, he noticed that the numbers of coins satisfied a triangle inequality, i.e., the sum of any two denominations was greater than the third. How many of each were there?

5 inches larger

Two wheels in the same plane are mounted on shafts 13 in. apart. A belt goes around both wheels to transmit power from one to the other. The radii of the two wheels and the length of the belt not in contact with the wheels at any moment are all integers. How much larger is one wheel than the other?

4.5 in.

Five points are located in or on the perimeter of an equilateral triangle with 9-inch sides. If d is the distance between the closest pair of points, what is the maximum possible value of d?

1154

The undergraduate of a School of Engineering wished to form ranks for a parade. In ranks of 3 abreasts, 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?

Area is 1,466,690 square feet, Rectangular dimensions are: 1080 x 1358; 1164 x 1260; 970 x 1512 feet

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

x=15, y=20, z=12

Find the simplest solution in integers for the equation 1/x^2 +1/y^2 =1/z^2

Monday

On what day of the week can the first day of a century fall? (The first day of the twenty-first century was Jan. 1, 2001)

free throw = 8, field goal = 11

In Byzantine basketball there are 35 scores which are impossible for a team to total, one of them being 58. Naturally a free throw is worth fewer points than a field goal. What is the point value of each?

Adam = 40 x 48, Brown = 32 x 60, Clark = 30 x 64

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?

The price was figured by adding the square of the sum of the digits of the previous price to the previous price.

A pet store offered a baby monkey for sale at $1.25. The monkey grew. Next week it was offered at $1.89, then $5.13, then 5.94, then $9.18 and on the sixth week a Ph.D in Aeronautics bought it for $12.42. How were the new prices figured?

8 years

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?

number of persons x = 2, number of passages y = 5

Find the smallest number (x) of persons a boat may carry so that (n) married couples may cross a river in such a way that no woman ever remains in the company of any man unless her husband is present. Also find the least number of passages (y) needed from one bank to the other. Assume that the boat can be rowed by one person only.

6786

If THAT = (AH)(HA), what is THAT?

Prove that the produce of 4 consecutive positive integers cannot be a perfect square

Let N be the smallest integer. The product is then N(N+ 1) (N + 2) (N + 3) = (N^2 + 3N) (N^2 + 3N + 2) = (N^2 + 3N + 1)^2 – 1. This is not a perfect square since 2 positive squares cannot differ 1.

438^14

A certain 3-digit number in base 10 with no repeated digits can be expressed in base R by reversing the digits. Find the smallest value of R.

6

What is the remainder upon dividing 5999,999 by 7?

1, 2, 3

Gherkin Gesundheit, a brilliant graduate mathematics student, was working on an assignment but, being a bit absent-minded, he forgot whether he was to add or to multiply the three different integers on his paper. He decided to do it both ways and, much to his surprise, the answer was the same. What were the three different integers?

84 1/3

The odd digits 1, 3, 5, 7,and 9 add up to 25, while the even figures, 2, 4, 6, and 8, only add up 20. Arrange these figures so that the odd ones and the even ones add up alike. Complex and improper fractions and recurring decimals are not allowed.

361 tons

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?

5.4 = 20

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?

4 girls, 2 boys

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?

First 8 integers

Starting with one, place each succeeding integer in one of two groups such that neither group contains three integers in arithmetic progression. How far can you get?

4736251

Find a permuation of the numbers one through seven with the property that when placed in both the first and third rows, the seven row totals will all be perfect squares.

a = 9/4, b = 27/8

Find unequal rational numbers, a, b, (other than 2 and 4) such that ab = ba

100 miles

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?

87912

Find the only number consisting of five different digits which is a factor of its reversal.

938

The above alphametic involving Roman numerals is correct. It will still be correct if the proper Arabic numerals are substituted. Each letter denotes the same digit throughout and no 2 letters stand for the same digit. Find the unique solution.

p = 2, q = 2

The sum and difference of two squares may be primes: 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? If so, for how many different primes is this possible?

11

No factorial can end in five zeros. What is the next smallest number of zeros in which a factorial cannot end?

 3 x 10 or 4 x 6

A rectangular picture, each of whose dimensions is an integral number of inches, has an ordinary rectangular frame 1 inch wide. Find the dimensions of the picture if the area of the picture and the area of the frame are equal.

20 winners

In a lottery the total prize money available was a million dollars, paid out in prizes which were powers of $11 viz., $1, $11, $121, etc. Noe more than 6 people received the same prize. How many prize winners were there, and how was the money distributed?

the 2nd and 3rd numbers are not divisible by 7

The numbers of 6,227,020,800; 6,227,028,000 and 6,227,280,000 are all large and roughly in the same ball park. But only one is equal to 13! Find it without use of tables, desk calculators, or hard work.

49,999 is prime, and the factorization is complete

Using a desk calculator, a student was asked to obtain the complete factorization of 24,949,501. Dividing by successively increasing primes, he found the smallest prime divisor to be 499 with quotient 49,999. At this point, he quit. Why didn’t he carry the factorization to completion?

11

In the arithmetic of Puevigi, 14 is a factor of 41. What is the base of the number system?

Tuesday

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)

 255 – 286 are missing pages

Barnie Bookworm bought a thriller – found to his dismay, Just before the denouement a fascicle astray. Instead of counting one through ten, a standard cure for rages. He totalled up the number of the missing sheaf of pages. The total was eight thousand and six hundred fifty-six. What were the missing pages? Try to find them just for kicks

three fingers on each hand

It is rumored that the above inscription appears on the purple moon boulder, a fragment of which was brought home by our Apollo 11 astronauts. If the visitors whos inscribed it were humanoid, and if the plausible inference is made that it represents an addition in a place notation system, can one make a further inference as to the number of fingers these visitors had?

29

Among those numbers whose literal representations in capitals consist of straight line segments only (e.g. FIVE), only one is “orthonymic”, i.e., is equal to the number of segments which comprise it. Find the number.

 A=319,600; B=318,801

Solve for A and B, both triangular numbers: 7993 = A2 – B2

 house number = 204, no. of houses = 208

My house is on a road where the numbers run 1, 2, 3, 4… consecutively. My number is a three digit one and, by a curious coincidence, the sum of all house numbers less than mine is the same as the sum of all house numbers greater than mine. What is my number and how many houses are there on my road?

W=1, A=0, H=9

If you solve the alphametic WATER – HEAT = ICE, you will have the solution to this double riddle: “This bird’s assured of his breakfast/and these before steeds cause a wreck fast.” Curiously, 70243 is the answer to both riddles!

13

The first expedition to Mars found only the ruins of a civilization. The explorers were able to translate a Martian equation as follows:

 This was strange mathematics. The value x = 5 seemed legitimate enough but x = 8 required some explanation. If the Martian number system developed in a manner similar to ours, how many fingers would you say the Martians had?

9842 wives

The Sultan arranged his wives in order of increasing seniority and presented each with a golden ring. Next, every 3rd wide, starting with the 2nd, was given a 2nd ring; of these every 3rd one starting with the 2nd received a 3rd ring, etc. His first and most cherished wife was the only one to receive 10 rings. How many wives had the Sultan?         

9^99

Assume the universe is a billion billion light years in diameter and is packed solidly with matter weighing a billion billion tons per cubic inch and each gram of this matter contains a billion billion atoms. Also, every second during the past billion billion years, a billion billion similar universes were created. Without using any symbols and restricting yourself to a total of three digits, write a number that far exceeds the total atoms of all these universes.

19^12

Find a five-digit number whose first two digits, central digit, and last two digits are perfect squares and whose square root is a prime palindrome.

3^32• 2^2

What is the largest number which can be obtained as the product of positive integers which add up to 100?

First 8 integers

A certain 6-digit number is a square in both the scale of 5 and the scale of 10. What is it?