LITTON’S PROBLEMATICAL PROBLEMS
PROBLEM 1
A man walks one mile south, one mile west,
then one mile north ending where he began.
From how many points on the surface of the
earth can such a journey be made? (There
are more than 1)
Ans: Infinite
PROBLEM 2
Maynard’s Grandfather Clock is driven by
two weights, 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 weighs are exactly opposite each other.
The weighs are again opposite every six
hours thereafter. What is Maynard’s
bedtime?
Ans: 9 pm or 3 am
PROBLEM 3
Dr. Fubisher LaRouche, the noted
mathematician, was shoping at a hardware
store and asked the price of certain articles.
The salesman replied. “One would cost 10
cents, eight would ost 10 cents, seventeen
would cost 20 cents, one hundred and four
would cost 30 cents, seven hundred and
fifty-six would aslo cost 30 cents, and one
thousand and seventy-two would cost 40
cents.” What was Dr. LaRouche buying?
Ans: 10 cent/number
PROBLEM 4
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?
Ans: 25 pitches
PROBLEM 5
A chemist has three large test tubes and a
beaker with 54 c.c. of elixir. Using the test
tubes and ingenuity only, how can he retain
50 c.c. in the beaker?
Ans: 50
PROBLEM 6
How many colors are necessary for the
squares of a chessboard in order to assure
that a bishop cannot move from one square
to another of the same color?
Ans: 8 colors
PROBLEM 7
A neat computer programmer wears a clean
shirt every day. If he drops off his laundry
and piucks up the previous week’s load
every Monday night, how many shirts must
he own to keep him going?
Ans: 15
PROBLEM 8
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 what number to count by. If he is
second in the original counting order what
number should he choose?
Ans: 10
PROBLEM 9
PROBLEM 13
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?
Between Kroflite and Beeline are five other
towns. The seven towns are an integral
number of miles from each other along a
staight 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?
Ans: 4
PROBLEM 10
In a memorable game with the Podunk
Polecats, the Mudville Mets established a
record. They received the maximum number
of walks possible in one inning in which one
player (who happened to ne the Mighty
Casey) was up three times and accounted
for all three outs. How many walks did
Podunk allow in that tedious half-inning?
Ans: 30
PROBLEM 11
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?
Ans: 9
PROBLEM 12
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 3 ¾ 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?
Ans: 3.7 dollars
Ans: 25 miles
PROBLEM 14
On a certain day, our parking lot contains
999 cars, no two of which have the same 3digit 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?
Ans: 4! or 24
PROBLEM 15
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?
Ans: 2/5
PROBLEM 16
If two marbles are removed at random from
a bag containing lack 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 1/6. How many marbles
are there of each color?
Ans: 6 white, 4 black
PROBLEM 21
PROBLEM 17
There are three families, each with two sons
and two daughters. In how many ways can
all these young people be married?
Rigorously speaking, two men are “brothersin-law” if one is married to the full sister of
the other. How many men can there be with
each man a brother-in-law of every other
man?
Ans: 3 men
PROBLEM 18
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. How many days must elapse before his
location is completely indeterminate, i.e.,
when he could be in any one of the ten
cities?
Ans: 83
PROBLEM 19
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 in
the fraternity?
Ans: 15
PROBLEM 20
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 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?
Ans: 196
Ans: 80
PROBLEM 22
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 digit is zero.
Ans: 136
PROBLEM 23
Max and his wife Min each toss a pair of dice
to determine where they will spend their
vacation. If either of Mins dice displays the
same number of spots as iether of Max’s, she
wins and they go to Bermuda. Otherwise,
they go to Yellostone. What is the chance
they’ll see “ Old Faithful” this year?
Ans: 0.514
PROBLEM 24
There are four volumes of an encyclpedia on
a shelf, each volume containing 300 pages,
(that is, numbered 1 to 600), but these have
been placed n 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?
Ans: 500
PROBLEM 25
Venusian batfish come in three sexes, which
are indistinguishable (except by Venusian
batfish). Hwo 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?
Ans: 4
PROBLEM 29
PROBLEM 26
Citizens of Franistan pay as much income
tax (percentage-wise) as they make rupees
per week. What is the optimal salary in
Franistan?
A sharp operator makes the following deal.
A player is to toss a coin and receive 1, 4, 9,
….. n2 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
repeated a great many times?
Ans: $4 per game
Ans: 50 rupees
PROBLEM 30
PROBLEM 27
In a carnival game, 12 white balls and 3
black balls are put in an opaque bottle,
shaken up, and drawn out one at a time.
Thee plaer gets 25 cents for each white ball
which emerges before the first black ball. If
he pays one dollar to play, how much can be
he expect to win )or lose) on each game?
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?
Ans: Loss of 25 cents/game
Ans: 29
PROBLEM 28
PROBLEM 31
In the final seconds of the game, your
favorite N.B.A team is behind 117 to 118.
Your center attemps 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?
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?
Ans: 69%
Ans: Gains 60/143 minutes
PROBLEM 32
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?
Ans: 4:26.853
PROBLEM 33
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.)
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?
Ans: 33 pearls
PROBLEM 36
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?
Ans: Numerator = 25
Ans: 5562
PROBLEM 34
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?
Ans: V1 = 3/4V2
PROBLEM 37
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?
Ans: 25%
PROBLEM 38
PROBLEM 35
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?
Ans: Pat = 8 errors, Mike = 2 errors
PROBLEM 39
A forgetful physicist forgot his watch one
day and asked an E.E. on the staff what time
it was. The E.E. looked 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?
Ans: 2:54:35 and 9:05:25
PROBLEM 40
PROBLEM 42
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˚?
Ans: x = 175˚, y = 185˚
PROBLEM 43
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?
Ans: Junior = 36 years old, Dad = 72 years
old
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.
Ans: 18
PROBLEM 44
PROBLEM 41
A circle of radius 1 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?
Ans: 5π
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?
Ans: Adam = 40 x 48, Brown = 32 x 60, Clark
= 30 x 64
PROBLEM 45
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?
Ans: 361 tons
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 rectangels.
Ans: Area is 1,466,690 square feet
Rectangular dimensions are: 1080 x 1358;
1164 x 1260; 970 x 1512 feet
PROBLEM 48
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?
Ans: 5.4 = 20
PROBLEM 49
PROBLEM 46
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?
Ans: 5 inches larger
PROBLEM 47
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?
Ans: 4 girls, 2 boys
PROBLEM 50
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 traveled when he gets there?
home. What is the shortest distance he can
travel and accomplish this?
Ans: North Pole, meters sqrt 2 x 10^7
An Origami expert started making a Nanides-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?
PROBLEM 51
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 arrriving
there together. The hares obviously ran at
the same speed, but just how far did they
run?
Ans: 100 yards
PROBLEM 52
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?
Ans: one acre
PROBLEM 53
A cowboy is five miles south of 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
Ans: 17.9 miles
PROBLEM 54
Ans: 24 inches
PROBLEM 55
The Ben Azouli are camped at an oasis 45
miles west of Taqaba. They decided 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?
Ans: 2 days
PROBLEM 56
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.
Ans: 3 x 10 or 4 x 6
PROBLEM 57
PROBLEM 60
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?
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?
Ans: 2.5 in^2
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.
PROBLEM 58
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?
Ans: 17
PROBLEM 59
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?
Ans: age of occupants = 1 and 18, street
number = 72
Ans: free throw = 8, field goal = 11
PROBLEM 60
Ans: number of persons x = 2, number of
passages y = 5
PROBLEM 61
The undergraduate of a School of
Engineering wished to form ranks for a
parade. In ranks of 3 abreasts, 2 m2n 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?
Ans: 1154
PROBLEM 62
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?
Ans: 100
PROBLEM 63
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?
Ans: house number = 204, no. of houses =
208
PROBLEM 64
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?
Ans: 20 winners
PROBLEM 65
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?
Ans: 9842 wives