Riddles Without Answers (Pages43)

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SYMBOLS

M

Needs math past

arithmetic and basic

probability.

CRequires knowing how to

play chess.

PPhysics knowledge is

helpful.

>=PI don't know the solution

to this problem myself.

CPURequires calculator/

computer power.

FORUM

RIDDLE INDEX

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med

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m$

cs

putnam

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microsoft

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putnam
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Stuck? Have compliments or

criticisms?

Want to test-drive a new riddle of

your own?

Who are we, and what do we do?

Visit the fantabulous riddle forum!

Thousands of posts by really clever

people.

RECENT ADDITIONS

Check out latest puzzles by

perusing the forumand the 10

most recent posts. Latest

additions to cover site can be

seen by clicking here.

relatively medium

GLOBETRAVERSAL

how many places are there on the earth that one could walk one

mile south, then one mile west, then one mile north and end up in

the same spot? to be precise, let's assume the earth is a solid

smooth sphere, so oceans and mountains and other such things do

not exist. you can start at any point on the sphere. also, the

rotation of the earth has nothing to do with the solution; you can

assume you're walking on a static sphere if that makes the

problem less complicated to you.

Hint 1: think you've figured it out? do you know that there's more

than one? in fact, there are more than two. also note that walking

north from the north pole (or south from the south pole) is illogical

and therefore does not enter into the problem. all normal

assumptions about directions will be used.

Hint 2: christopher columbus.

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GHETTO

ENCRYPTION I

You want to send a valuable object to a friend securely. You have a

box which can be fitted with multiple locks, and you have several

locks and their corresponding keys. However, your friend does not

have any keys to your locks, and if you send a key in an unlocked

box, the key could be copied en route. How can you send the object

securely?

Alternative, more precise phrasing: Andy and Grant are staying in

different rooms in the same hotel. Andy needs to give a gold

pendant to Grant, but spies are trying to assassinate Andy and

Grant so neither of them can leave their room. The only way they

can transfer objects is by using the bellhops. Both Andy and Grant

have a safe with a large clasp that can be secured with a padlock.

Both Andy and Grant have a padlock and a corresponding key. (So

1 gold pendant, 2 safes, 2 padlocks, and 2 keys.) But the bellhops

are thieves. Anything that is not padlocked in the safe will be

stolen by the bellhops - including any unlocked padlocks, the keys

or the pendant. How can Andy transfer the gold pendant to Grant

without it being stolen? (where both sides have encryption

capability, and where unsecured items are taken away rather than

just copied?)

GHETTO

ENCRYPTION II

Three coworkers would like to know their average salary. However

they are self-conscious and don't want to tell each other their own

salaries, for fear of either being ridiculed or getting their houses

robbed. How can they find their average salary, without disclosing

their own salaries?

ADJACENCY GRID

arrange the numbers 1 to 8 in the grid below such that adjacent

numbers are not in adjacent boxes (horizontally, vertically, or

diagonally).

1

6 4 3

2 7 5

8

the arrangement above, for example, is wrong because 3 and 4, 4



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and 5, 6 and 7, and 7 and 8 are adjacent.

FAUSTIAN

ROUND TABLE

COIN GAME

you die and the devil says he'll let you go to heaven if you beat him

in a game. the devil sits you down at a round table. he gives

himself and you a huge pile of quarters. he says "ok, we'll take

turns putting quarters down, no overlapping allowed, and the

quarters must rest on the table surface. the first guy who can't put

a quarter down loses." you guys are about to start playing, and the

devil says that he'll go first. however, at this point you immediately

interject, and ask if y o ucan go first instead. you make this

interjection because you are very smart, and you know that if you

go first, you can guarantee victory. explain how you can guarantee

victory.

DOMINOES ON A

CHESSBOARD

using 31 dominoes, where one domino covers exactly two squares,

can you cover all the empty squares on this chessboard (which has

only 62 spaces, since two opposite corner squares are removed). if

so, how? if not, why? prove your claim.

MANHOLES

Why are manholes round?

Note: This is a famous Microsoft question. Yet amusingly, the

Microsoft campus uses square manholes.

SQUARE

DIVISION

draw a square. divide it into four identical squares. remove the

bottom left hand square. now divide the resulting shape into four

identical shapes.



5/43


EQUILATERAL

TRIANGLE

DIVISION

draw an equilateral triangle (all sides same length). divide it into

four identical shapes. remove the bottom left hand shape. now

divide the resulting shape into four identical shapes.

LIGHT BULBS

AND SWITCHES

You are in a room with three light switches, each of which controls

one of three light bulbs in the next room. Your task is to determine

which switch controls which bulb. All lights are initially off, and

you can't see into one room from the other. You are allowed only

one chance to enter the room with the light bulbs. How can you

determine which lightswitch goes with which light bulb?

PUNCTUATION I

Add punctuation to the following phrase to make something

gramatically and logically coherent:

is is not not not is not is is is is not is not is it not

PAPER CUTTING

You have a 5x5 piece of paper. Two diagonally opposite corners of

this paper are truncated as shown in the diagram below. You also

have scissors. Show how to cut up the 5x5 paper into two pieces,

so that the two pieces can then be interlocked to form a 6x4

rectangle.



6/43


POPULATION OF

FUNKYTOWN

In the city of Funkytown, the following facts are true:

No two inhabitants have exactly the same number of hairs.

No inhabitant has exactly 483,207 hairs.

There are more inhabitants than there are hairs on the head

of any one inhabitant.

What is the largest possible number of inhabitants of Funkytown?

Note: I recently (11/24/2002 8:00PM) read this puzzle in a book

by Henry Ernest Dudeney, England's greatest puzzle creator. So

writing credits to him.

CHOCOLATE

MILK

You have two thermoses. The first contains a liter of milk, the

second contains a liter of pure chocolate syrup. You pour one cup

of milk out from the first thermos to the second one. Then, after

mixing that, you take one cup of the mixture from the second

thermos, and pour it back into the first thermos. After completing

these two operations, which thermos is more pure?

BROWN EYES

AND RED EYES

There is an island of monks where everyone has either brown eyes

or red eyes. Monks who have red eyes are cursed, and are

supposed to commit suicide at midnight. However, no one ever

talks about what color eyes they have, because the monks have a

vow of silence. Also, there are no reflective surfaces on the whole

island. Thus, no one knows their own eye color; they can only see

the eye colors of other people, and not say anything about them.

Life goes on, with brown-eyed monks and red-eyed monks living

happily together in peace, and no one ever committing suicide.

Then one day a tourist visits the island monastery, and, not

knowing that he's not supposed to talk about eyes, he states theobservation "At least one of you has red eyes." Having acquired

this new information, something dramatic happens among the

monks. What happens?

Hint: First consider the case where there are only a few monks on

the island, some with brown and some with red. Work through the

logic and find out what happens over time. Then generalize for the

case of M monks on the island, N of which have red eyes.



7/43


Update 10/15/2002 12:14AM: What happens if we change the

tourist's statement to each of the following?

1. "There are 10 Brown Eyed Monks"

2. "There are at lesat two Red Eyed Monks"

3. "There is an odd number of Red Eyed Monks"

4. "There is an even number of Red Eyed Monks"

5. "There is more than one Red Eyed Monk"

WHERE'S THE

FATHER?

The mother is 21 years older than the child. In 6 years from now,

the mother will be 5 times as old as the child. Question: Where's

the father?

.999 ...

Compare the numbers 0.99999... (infinitely many 9s) and 1. Which

of the following statements is true? Why?

0.99999 ... 1

Forum thread: Click here. Check out Icarus's (our local

mathematician) insightful exposition on this problem.

COLORED DISK

SPIN SENSORS

Imagine a disk spinning like a record player turn table. Half of the

disk is black and the other is white. Assume you have an unlimited

number of color sensors. How many sensors would you have toplace around the disk to determine the direction the disk is

spinning? Where would they be placed?

http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1027804564http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1027804564


8/43


GOLD CHAIN

a man has a gold chain with 7 links. he needs the service of a

laborer for 7 days at a fee of one gold link per day. however, each

day of work needs to be paid for separately. in other words, the

worker must be paid each day after working and if the laborer is

ever overpaid he will quit with the extra money. also he will never

allow himself to be owed a link. what is the fewest number of cuts

to the chain to facilitate this arrangement and how does that

guarantee payment?

DUMB CS JOKE

M

Why do computer programmers always get Christmas and

Halloween mixed up?

WATER BUCKETS

Using only a 5-gallon bucket and a 3-gallon bucket, put exactlyfour gallons of water in the 5-gallon bucket. (Assume you have an

infinite supply of water. No measurement markings on the

buckets.)

MASTERMIND I

This is the logic game of Mastermind. If you haven't played it

before, here's how it works. There is a board that is sectioned off

into many rows, each row having four slots in which pegs can be

inserted. There are six different colors of pegs: green, red, yellow,

brown, dark-blue, light-blue. There are two players, A and B. First,

A makes up some arrangement of four pegs along a row, the colors

and ordering of which are his or her choice. Then B spends the rest

of the game trying to guess what A's arrangement is. For every

guess that B makes, A will respond by putting some black and/or

white pegs right next to A's guess; the black and white pegs are

interpreted as follows:

Black keypeg = one of B's pegs is the correct color and in the

correct position

White keypeg = one of the B's pegs is the correct color but in

the wrong position

So if B manages to guess all four colors and positions correctly, A

will respond with four black keypegs, and the game is over. The

goal is to determine A's secret arrangement in the minimum



9/43


number of guesses. Below, we see a completed game of

Mastermind. Apparently the player was able to determine A's

arrangement by using only four guesses. What's is A's

arrangement?

TWO-CHILD

FAMILY I

In a two-child family, one child is a boy. What is the probability

that the other child is a girl?

TWO-CHILD

FAMILY II

In a two-child family, the older child is a boy. What is the

probability that the other child is a girl?

2 = 1

"Proof" that 2 = 1:

a = b

a2= ab

a2- b2= ab-b2

(a-b)(a+b) = b(a-b)

a+b = b

b+b = b

2b = b

2 = 1

Does this argument make sense?



10/43


BRIDGE

CROSSING

Four people, A, B, C, and D, are on one side of a bridge, and they all

want to cross the bridge. However, it's late at night, so you can't

cross without a flashlight. They only have one flashlight. Also, the

bridge is only strong enough to support the weight of two people

at once. The four people all walk at different speeds: A takes 1

minute to cross the bridge, B takes 2 minutes, C takes 5 minutes,

and D takes 10 minutes. When two people cross together, sharing

the flashlight, they walk at the slower person's rate. How quickly

can the four cross the bridge?

Note: Supposedly a classic Microsoft question.

CYCLOID

M

If you drew a dot on the edge of a wheel and traced the path of the

dot as the wheel rolled one complete revolution along a line, then

the path formed would be called a cycloid (shown in red below),

combining both forward and circular motion. What is the length of

the path formed by one complete revolution? Assume the wheel

has a radius of 1.

COCONUT

Ten people land on a deserted island. There they find lots of

coconuts and a monkey. During their first day they gather coconuts

and put them all in a community pile. After working all day they

decide to sleep and divide them into ten equal piles the next

morning. That night one castaway wakes up hungry and decides to

take his share early. After dividing up the coconuts he finds he is

one coconut short of ten equal piles. He also notices the monkey

holding one more coconut. So he tries to take the monkey's

coconut to have a total evenly divisible by 10. However when he

tries to take it the monkey conks him on the head with it and kills



11/43


MONKEY him. Later another castaway wakes up hungry and decides to take

his share early. On the way to the coconuts he finds the body of the

first castaway, which pleases him because he will now be entitled

to 1/9 of the total pile. After dividing them up into nine piles he is

again one coconut short and tries to take the monkey's coconut.

Again, the monkey conks the man on the head and kills him. One

by one each of the remaining castaways goes through the same

process, until the 10th person to wake up gets the entire pile for

himself. What is the smallest number of possible coconuts in the

pile, not counting the monkeys?

PRISONER

DILEMMA REDUX

You and your partner in crime are both arrested and questioned

separately. You are offered a chance to confess, in which you agree

to testify against you partner, in exchange for all charges being

dropped against you, unless he testifies against you also. Your

lawyer, whom you trust, says that the evidence against both of

you, if neither confesses, is scant and you could expect to take a

plea and each serve 3 years. If one implicates the other, the other

can expect to serve 20 years. If both implicate each other you

could each expect to serve 10 years. You assume the probability of

your partner confessing is p. Your highest priority is to keep

yourself out of the pokey, and your secondary motive is to keep

you partner out. Specifically you are indifferent to you serving x

years and your partner serving 2x years. At what value of p are youindifferent to confessing and not confessing?

Find the side length of the internal square and the radii of the

internal circles, in terms of a.



12/43


JAPANESE

TEMPLE

GEOMETRY I

Note: This geometry problem comes from the tradition of sangaku.

During the period between 1639 and 1854, many Japanese samura

apparently adopted geometry as a serious mental discipline! They

would solve many difficult geometric problems, and carefully

inscribe the solutions into beautiful wooden tablets, often with

color. Then they would hang their solutions under roofs for public

viewing, either to show respect for the elegance of math, or

perhaps just to show off their intelligence and challenge other

practitioners of sangaku -- a Japanese word that literally means

mathematical tablet. Sangaku problems are usually just Euclideangeometry, but others seem almost impossible to do without

cheating and using higher level math (e.g. calculus, affine

transformations). Also notable is a strong emphasis on ellipses and

circles, an emphasis not found in Western studies of geometry.

Note 2: For a hardcopy of the above image only, click hereand

print. Hardcopies are useful for drawing lines and labeling vertices

and what not.

http://www.ocf.berkeley.edu/~wwu/images/riddles/japtempgeo1-2.gifhttp://www.ocf.berkeley.edu/~wwu/images/riddles/japtempgeo1-2.gif


13/43


Problem source: Fukagawa, H. and D. Pedoe. Japanese Temple

Geometry problems. Charles Babbage Research Foundation:

Winnipeg, 1989.

JAPANESE

TEMPLE

GEOMETRY II

Find the radii of the internal circles, in terms of a.

Note: For a hardcopy of the above image only, click hereand print.

Hardcopies are useful for drawing lines and labeling vertices and

what not.

Note 2: Thanks a lot to Jasvir Nagra for sending me such a clear

solution, even with diagrams drawn with Postscript!



Winnipeg, 1989.



14/43


JAPANESE

TEMPLE

GEOMETRY III

Find the radii of the internal circles, in terms of a.

Note: For a hardcopy of the above image only, click hereand print.

Hardcopies are useful for drawing lines and labeling vertices and

what not.



Winnipeg, 1989.



15/43


MASTERMIND II

This is the logic game of Mastermind. If you haven't played it

before, here's how it works. There is a board that is sectioned off

into many rows, each row having four slots in which pegs can be

inserted. There are six different colors of pegs: green, red, yellow,

brown, dark-blue, light-blue. There are two players, A and B. First,

A makes up some arrangement of four pegs along a row, the colors

and ordering of which are his or her choice. Then B spends the rest

of the game trying to guess what A's arrangement is. For every

guess that B makes, A will respond by putting some black and/or

white pegs right next to A's guess; the black and white pegs are

interpreted as follows:

Black keypeg = one of B's pegs is the correct color and in the

correct position

White keypeg = one of the B's pegs is the correct color but in

the wrong position

So if B manages to guess all four colors and positions correctly, A

will respond with four black keypegs, and the game is over. The

goal is to determine A's secret arrangement in the minimum

number of guesses. Below, we see a completed game of

Mastermind. Apparently the player was able to determine A's

arrangement by using only four guesses. What's is A's

arrangement?



16/43


PUNCTUATION II

Add punctuation to the following phrase to make something

gramatically and logically coherent:

i had had had tom had had had had had had had had the praise of

the teacher

BROKEN CLOCK

My fancy new digital alarm clock is broken! The time 'jumps'

around.

When I reset it, it reads 12:00:00. Then it runs as it should, but

after 12:04:15 it resets back to 12:00:00. It counts up to 12:04:15

again and then it jumps to ... 12:08:32 ! Weird stuff. Do you know

what's wrong with my alarm clock?

Note: The contributor, Remco Hartog, actually constructed this

riddle himself!

WHAT

HAPPENED I

There's a man lying dead in a telephone booth beside a river. The

phone is off the hook, and there is smashed glass on the floor of

the phone booth. What happened?

Note: I've always felt uncomfortable with these "what happened"

riddles, because it seems like you could design a wide variety

fantastic explanations. Indeed, the official answer I've been e-

mailed for this riddle is quite fantastic.

BUS

PASSENGERS

The UC Berkeley bus had a minimal number of passengers. When it

arrived at Telegraph Avenue, 3/4 of the passengers got out, and 7

people got on. At the next two stops, Shattuck and Hearst, the

same thing happened. How many got off at Hearst?



17/43


SPRINKLER AND

PUMP

Take one of those spinning sprinklers everyone has seen before,

the kind that has two pipes sticking out with the ends slightly bent

to the side in opposite directions. Let's say the sprinkler normally

turns clockwise when shooting water. Hook a hose from the

sprinkler to a water pump. Submerge the sprinkler in water, so

that now when you turn on the pump, water is drawn through the

sprinkler's pipes. What happens to the sprinkler?

Note: From S u r e l y Yo u ' r e J o k i n g , M r . F ey n m a n ! by Richard

Feynman. Originally a problem hotly debated among Princeton

physicists in the 1940s. Feynman resolved the problem by

conducting an experiment that resulted in the shattering of a glass

tank and ultimately getting banned from a lab.

Forum thread: click here

SOLDER CUBE

Say you have some bendable wires (any number, any length).

What is the minimum number of solder connections needed to

make a cube? Prove it. Also, what is the minimum number of wires

necessary? Prove it.

ANALOG CLOCK

III

Consider an analog clock face on which the hour and minute hands

move smoothly. If we swap the position of the hour and minute

hands, we usually end up with an invalid clock face which never

occurs during the day (the hour hand pointing directly at 12 and

the minute hand at 3, for instance). But there are some times when

these switched hands *do* tell a valid time: specifically, when the

two hands are pointing in the same direction. Are there any other

such times?

Note: From John Leen, one of my CS170 TAs from Spring 2002 atUC Berkeley. An excellent teacher.

http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1028067360http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1028067360


18/43


PARTY

HANDSHAKES

The master of a college and his wife has decided to throw a party

and invited N guest and their spouses. On the night of the party, all

guests turned up with their spouse, and they all had a great time.

When the party was concluding, the master requested all his

guests (including his wife, but not himself) to write down the

number of persons they shook hands with, and to put the numbers

in a box. When the box was opened, he was surprised to find all

integers from 0 to 2N inclusive.

Assuming that a person never shake hands with their own spouse

and that no one lied, how many hands did the master shake?

RANDOMIZED

ICE CREAM

A vendor is handing out free ice cream cones in alphabetical order

of flavor, each cone being a different flavor. Kids are lined up at

the ice cream truck, and you're first in line! The vendor will hand

you ice cream cones one at a time, and you must decide whether to

keep the cone or pass it on to the next kid in line. The first cone is

guaranteed to be chocolate.

You like all flavors equally, so you want to randomly select a cone

with each flavor having an equal chance of being chosen.

Unfortunately, you don't know the total number of flavors, but

being the little hipster that you are, you are carrying a pocket

calculator which can generate random numbers from 1 to X, where

X is a value you punch in. How can you decide which flavor to

keep?

Note: Problem made by Yosen Lin and William Wu! 2002. Special

thanks to the forum regulars for proofreading it.

ANSWER

Consider an answering machine with remote inquiry facility, whereyou can call the answering machine and enter a 4 digit passcode

into your telephone keypad, so you can listen to the messages from

anywhere you like. Many of these machines will let you in if you

enter the correct consecutive sequence of digits, regardless of

what preceded that sequence.



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MACHINE

HACKING

Example: Passcode is 1234.

if you feed the machine 1234, you're in




To brute-force hack the machine, you could try all numbers from

0000 to 9999, sending 40000 sounds across the wire. But since you

are a smart hacker, you see that there's room for optimization.

What is the shortest series of digits you have to send to the

answering machine in order to break the code in any case?

LADDER RUNGS

You are presented with a ladder. At each stage, you may choose to

advance either one rung or two rungs. How many different paths

are there to climb to any particular rung; i.e. how many unique

ways can you climb to rung "n"?

After you've solved that, generalize. At each stage, you can

advance any number of rungs from 1 to K. How many ways are

there to climb to rung "n"?

Note: the non-generalized question was asked at a M$ interview.

RACETRACK LAPS

You are at a track day at your local racecourse in your new

Porsche. Because it's a crowded day at the track, you are only

allowed to do two laps. You haven't driven your car at the track

yet, so you took the first lap easy, at 30 miles per hour. But you do

want to see what your ridiculous sports car can do. How fast do

you have to go on the second lap to end the day with an average

speed of 60 miles per hour?



20/43


IMPATIENT

PATIENTS

All patients at the d'Oralian National Hospital require constant

care. At least one nurse must be beside a patient's bed at any time

and a nurse can only care for one patient simultaniously.

The nurses won't work more that eight hours within any 24 hour

period, but don't care when their shift starts. Their shifts look like

this: work five hours, one hour break, work another three hours.

How many nurses do you need for one patient? How many nurses

do you need for n patients?

Note: Designed by the contributor, Remco Hartog!

E VS PI

W i t h o u t c o m p u t i n g t h e i r a c t u a l v a l u e s , which is greater, e^(pi) or

(pi)^e?

COIN FLIP GAME

WORTH II

Game: You continue flipping a coin until you get a tails. I then

award you prize money equal to $2^(number of heads).

How much are you willing to pay me to play this game?

COIN FLIP GAME

WORTH III

Game: You continue flipping a coin until the number of heads

equals the number of tails. I then award you prize money equal to

the number of flips you conducted.

How much are you willing to pay me to play this game?

271Write 271 as the sum of positive real numbers so as to maximize

their product.



21/43


LADDER INTO

TOWER

Given a cylindrical tower with a diameter of D and a door of height

H from the ground, what is the longest ladder of length L that you

could move into the tower and completely enclose? Ignore the

thickness of the ladder.

Note: While the contributor was serving in the German military, a

superior rank tried to outsmart him with this riddle. It didn't

work :)

6 PERSON

ACQUAINTANCE

Prove that in any group of 6 people, at least 3 must be either

mutually acquainted with each other, or mutually unacquainted

with each other.

XXIf x is a positive rational number, prove that x^x is irrational

unless x is an integer.

WILLYWUTANG

HAS STDS?

While willywutang was trying to be clever by using only two

condomsto satisfy three women, increased friction between

overlapped latex layers produces a tear and a dangerous accident!

He then becomes very worried about having contracted an STD,

especially since one of the women was really skanky looking.

Although a random person has only probability 0.001 of having an

STD, poor willywutang just can't sleep over those odds. Frantically

he hustles to the nearest drugstore, to purchase the ACME All-

Purpose STD Checker. The packaging boasts a 0.93 correctness

probability. That is, if the user has an STD, the ACME STD Checker

will return positive 93% of the time; if the user does not have an

STD, it will return negative 93% of the time. Willywutang returns

home and uses the checker in his bathroom.

To his dismay, the results are positive.

Assuming that willywutang's promiscuity on average is identical to

that of a randomly chosen person, what is the probability that

willywutang has STDs?

http://www.ocf.berkeley.edu/~wwu/riddles/easy.shtml#twoCondomsThreeWomenhttp://www.ocf.berkeley.edu/~wwu/riddles/easy.shtml#twoCondomsThreeWomenhttp://www.ocf.berkeley.edu/~wwu/riddles/easy.shtml#twoCondomsThreeWomenhttp://www.ocf.berkeley.edu/~wwu/riddles/easy.shtml#twoCondomsThreeWomen


22/43


KABOUTERS

Kabouters: very small (up to 10 cm) people living in the forests, in

mushrooms and roots. Wearing old-fashioned clothes AND they all

wear a pointed hat. Those are kabouters...

In the forests of the Netherlands live a large number of kabouters

wearing two different colors of pointy hats. Kabouters consider it

very rude to talk about the color of the hat they are wearing, so

much so that they don't even know the color of their own hat. They

are able to look and distinguish the color of the hats on other

kabouters. (They just won't talk about it.) Now every year all

kabouters need to be counted and traditionally they present

themselves in two groups divided by the color of their hats. How do

they do this without talking or communicating the color of their

hats to any of the other kabouters?

CIRCUIT

FAILURE

ANALYSIS

Consider the circuit shown below. If all components along a path

from In to Out are working, then the whole system is considered to

be working.

Through appropriate experimentation and modelling, it has been

determined that at time T, component Ckfails with probability pk.

Also assume that component failure are independent.

a. W h a t i s t h e p r o b a b i l i t y t h e s y s t e m h a s f a i le d a t t i m e T?

b. Y o u r m a i n t e n a n c e t e ch n i c ia n r e p o r t s t h a t a t t i m e T,

c o m p o n e n t C kh a s f a i l ed , b u t s h e h a s n ' t b e e n a b l e t o c h e c k

a n y o t h e r c o m p o n e n t s y e t . W h a t i s t h e p r o b a b i l i t y t h e sy s t e m



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i s w o r k i n g ? R ep e a t f o r k = 1 , 2 , 3 , 4 .

RUMOR

SPREADING

You are part of a group of 10 people. All 10 people are in their

respective and SEPERATE houses, waiting by the phone. You have a

rumor. You pick up the phone to start spreading it and you call

someone, beginning a chain such that the nth person to be called

will then call and tell it to one other person, except that he will not

call himself nor would he (naturally) call the person who just told

him (assume, for simplicity, that he WOULD call someone who has

told him the rumor before, as long as it was not the person who

just called him).

So you make the first call. What is the probability that the 5th call

is to you?

Now, can you rework the problem such that a person will NEVER

call anyone who has ever told him the rumor before (more

realistic, figuring they already know the rumor so why call)?

Finally, can you generalize the formula to calculate the probability

that the n-th call is to you?

TREES FOR

WILLYWUTANG

So, Willywutang has (somehow) managed to get himself a nice big

mansion. The mansion has a nice huge yard in front. However, the

yard is completely flat and boring, so Willy decides it'd look nice

with a few trees in front. So, he has a landscaper come in to put in

some trees. Being the puzzlemeister that he is, Willy decides to

give the landscaper a riddle: Plant 9 trees in the yard, so that there

are 10 rows of three trees each. Help the poor landscaper decide

how to place the trees.



24/43


FORCEFIELD

DETAINMENT

A group of prisoners are trapped in a forcefield. These prisoners

are perfectly brave, meaning that they would attempt an escape on

any positive probability of success. The prisoners are monitored by

a guard who has only one bullet in his gun, but who also has

perfect marksmanship skills (he never misses). A maintenance

technician needs to tune up the forcefield generator, and so for one

second, the forcefield is released. How can the guard still keep all

the prisoners detained?

WHAT

HAPPENED II

A black man, dressed in black, is crossing a road. He's blind and

deaf. A truck is speeding towards him, with its lights turned off.

The street lamps are also off and there's no moonlight. When the

truck is about to hit the man, the driver hits the brakes and

manages to stop just a few centimetres from him. How did the

driver see the man?

WHAT

HAPPENED III

A man walks into a bar and says "I want a coffee and a glass of

water. But make sure the coffee is boiling hot and the water is ice-

cold." The barman says "Sure thing, mr. fireman." The barman had

never met him before. How did he know the man was a fireman?

ONE REAL

SOLUTION

Mathematics departments at some south-western universities

received Mr. H.N.s sly letters asking for the one real solution x of

the following two equations:

18 = ((1 + x)18/x)

17 = ((1 + x)17/x)

Professor A.S. at one of those departments sent Mr. H.N. the

following brief solution:

18/17 = ((1 + x)18/x) / ((1 + x)17/x) = 1 + x , so x = 1/17

Are there any other real solutions x? Why?



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PROOF THAT Y=0

Here is a "proof" that y = 0:

Define f(x,y) := (x+y)^2. Now substitute x = u-v, and y=u+v.

Taking partial derivatives, we see that

f/ x =

f/ y = 2(x

+y)

x/ v = -1 y/ v = +1

Now the Chain Rule implies

f/ v = ( f/ x)( x/ v) + ( f/ y)( y/ v) = 2(x+y)(-1) + 2(x+y

(1) = 0

But the definition of f(u, v) = (u + v)^2 implies also that f/ v = 2

(u+v) = 2y. Therefore y = 0. Thus we should never divide by

variables named "y" while manipulating equations, because y = 0.

COINS SAME

WEIGHT

M

Suppose an odd number (at least three) of coins have the property

that, if any one coin is removed, the rest can be partitioned into

two groups each with the same number of coins and also the same

total weight. Show that all the coins must have the same weight.

Note: Linear algebra can be useful here, but I don't know if it's

absolutely necessary.

PAPER FOLD

RATIO

A rectangular sheet of paper is folded so that two diagonally

opposite corners come together. The crease thus formed is as long

as the longer side of the rectangle. What is the ratio of the longer

side of the rectangle to the shorter?

FARMER'S

ENCLOSURE

A farmer has four straight pieces of fencing: 1, 2, 3, and 4 yards in

length. What is the maximum area he can enclose by connecting

the pieces?



26/43


GOOD AND BAD

COMPUTERS

You have N computers on a space station. An accident happens,

and some of the computers are damaged, but you know the

number of good (undamaged) computers is greater than the

number of bad (damaged) ones. Your goal is to find *one*

computer that's still good.

Your only method of testing is the following: Use one computer

(say, X) to test another (Y). If X is a good computer, it tells you

correctly the status of Y. If X is bad, it may or may not give the

correct status of Y; assume it will give whatever answer is least

useful to your testing strategy.

In worst-case, how many tests must you use to find one computer

that's still good? (in terms of N)

You're permitted any combination of tests, though keep in mind

the bad machines may not be consistent in the results they give

you.

CHESS STUDY I

C

White to move. What is optimal play for White?

Note: A very famous study by Richard Reti, dating all the way back

to 1921. Reti was a professional in both mathematics and chess; he

found the latter more interesting because he could force people to

believe the truth of his ideas. He became quite famous in 1924 for



27/43


being the first to defeat Capablanca in a decade.

CHESS STUDY II

C

White to move. What is optimal play for White?

Note: Another Richard Reti classic study involving the endgame

and the surprising power of Kings.

15-14 SLIDE

PUZZLES

Many of us have had experience with slide puzzles, in which there

are tiles numbered from 1 through 15, and you have to slide the

tiles such that the numbers are in increasing sequence when read

from left to right, top row to bottom. Other variants have

fragments of a picture printed on the tiles.

Can you solve the following slide puzzles? If so, list a sequence of

moves that produce a solution; if not, explain rigorously. We want

the first puzzle to read "1 2 3 4 5 6 7 8 9 10 11 12 13 14 15", and

we want the second puzzle to read "Rate Your Mind Pal."



28/43


29/43


OBTUSE

TRIANGLES

An obtuse triangle has one of its angles greater than 90 degrees.

An acute triangle has all of its angles less than 90 degrees. (A right

triangle is neither acute nor obtuse.)

Either find a way to cut up any obtuse triangle into pieces, each of

which is an acute triangle, or prove that it can't be done.

Note: "This one is from an old Martin Gardner book. Gardner claims

professional mathematicians have been known to get it wrong.

Then again, professional mathematicians get the Monty Hall

problem wrong sometimes too." - Tim Mann

LOADED CRAPSDICE

Craps is a 1-player dice game that is played as follows: Roll two 6-sided dice; their sum becomes your "initial" roll. If this initial roll is

2, 3, or 12, you lose. If the initial roll is 7 or 11, you win.

Otherwise, keep rolling the dice until you reroll you initial number

(and win) or until you roll a 7 (and lose).

You're betting that your adversary is going to lose his game of

craps, which should be a favorable bet for you. But you receive an

anonymous tip that he's secretly loaded one of the dice, so that it

will always come up 5. This increases his chances of winning to

2/3.

Having learned of his evil deed, you're going to secretly load his

other die so as to minimize his chance of winning. With what

probability should you load each of the six faces? And how does

that change his probability of winning?

Note: Writing credits to Matt Lahut.



30/43


PENNY TRACK

GAME

There are three one-dimensional tracks, of length 12, 7, and 5

spaces respectively. You start with pennies in the first space of

each track; your opponent starts with pennies in the last space of

each track. On your turn, you may move any one of your pennies

any number of spaces in either direction along a track (as a chess

rook), however you are not permitted to bypass the other player's

penny or occupy its space. If a player has no legal move, he loses.

What should your first move be?

Note: Hardcore puzzlers will probably solve this immediately, but

newer people might take quite some time. For the newer people,

the solution to this is a useful trick to have in your random-puzzle-

solving arsenal. As a sub-problem, you may want to try solving it

without the 5-length track.

POSITIVE

MATRIX SUMS

You have an m x n grid with some real numbers in each cell. You

can multiply any row by -1, or any column by -1. Show that by

making multiplications of this kind, the sum of the numbers in

every row, and the sum of the numbers in every column, can all be

made non-negative simultaneously.

TETRAMINOES

ON A

CHECKERBOARD

Shown to the below-left is a checkerboard mutilated by the

removal of two squares from each of two opposite corners. Shown

to the below-right are two T-shaped tiles, each of which can cover

four squares of the checkerboard exactly.



31/43


If tiles of both kinds are abundant, and if tiles may be rotated, can

the mutilated checkerboard be covered exactly with

nonoverlapping tiles that match the colors of covered squares?

Why?

ALWAYS AHEAD

In an presidential election between Willywutang and Billybutane,

the winning candidate Willywutang has received n+k votes,

whereas Billybutane has received n votes. (n and k are positive

integers.) If ballots are counted in a random order, what is the

probability that Willywutang's accumulating count will always lead

his opponent's, and why?

Note: Cute eh? Also a practical calculation ... well, maybe.

CALLING A

CARD'S COLOR

A deck of 26 red and 26 black cards is shuffled into random order

and placed face down. Then the cards are turned up one by one

and observed by a guesser. He gets one guess: At a moment of his

choice he may assert that the next card will turn up red. After this

card is turned up the game ends and he wins if his assertion was

correct, loses otherwise. And if he doesn't guess at all by the time

all cards have been dealt, he loses by default. What guessing policy

chosen in advance maximizes his chance of winning?

COMPLEX

MULTIPLICATION

Consider computing the product of two complex numbers (a + bi)

and (c + di). By foiling the polynomials as we learned in grade

school, we get

a + bi

c + di

----------

adi - bd

ca + cbi

----------------

(ca - bd) + (ad + cb)i

Note that this standard method uses 4 multiplications and 2

additionsto compute the product. (The plus sign in between (ca -



32/43


bd) and (ad + cb)i does not count as an addition. Think of a

complex number as simply a 2-tuple.)

It is actually possible to compute this complex product using only 3

multiplications and 3 additions. From a logic design perspective,

this is preferable since multiplications are more expensive to

implement than additions. Can you figure out how to do this?

CRAZYSPRINKLER

A little while ago, Willy Wutang was thinking about the Sprinkler

and Pumppuzzle. Willy, having a creative mind (or at least a well-

developed sense of mischief) had a flash of inspiration, and came

up with a very novel sprinkler design. However, before he can

patent his new sprinkler, Willy must figure out what it does--his

lawyer says it's a necessary part of the patent process.

1. First of all, does the sprinkler turn? If so, in which direction

does it turn?

2. Second, what do the jets of water do? In the picture above,

the sprinkler is shown at the instant just before the jets of

water collide. However, in the picture the sprinkler is being

held still, so that even if it wants to turn, it cannot.

3. Third, does the behaviour of the jets of water depend on

whether or not the sprinkler is allowed to rotate? Does it

maybe depend on how fast the sprinkler rotates?

Note: Writing credits to James Fingas!



33/43


CONWAY

SEQUENCE

What row of numbers comes next?

1

11

21

1211

111221

312211

13112221

Note: Mathematician John Conway spent considerable time

studying this sequence.

CRYPTIC

ADDRESS

Willy Wutang is writing to his overseas sweetheart. He writes a

steamy love letter, seals the envelope, and then inscribes the

following cryptic address on the front:

Wood, S

April

England

Strangely enough, the letter reaches its intended recipient. What is

Willy's sweetheart's address?

Note: Writing credits to James Fingas :)

Willy Wutang is making a robot, and he has bought himself a set of

24 finger gauges. For those of you who have never seen a set of

these, each gauge is a strip of metal about 1/2" wide and about

one foot long. Each is a different thickness (0.001" up to 0.024").

Each has a hole in one end, and they are fastened together with a

nut and bolt, in order of thickness.

Willy's calculations indicate that, with the flame thrower

attachment extended, he must adjust the spark plug to be 0.086"



34/43


35/43


CAKE DIVISION

Willywutang is holding a birthday party. Either p or q people are

going to show up, where p and q are relatively prime numbers.

Willy wants to cut the cake beforehand, so his guests will not

waste time waiting for slices to be carved out. What is the smallest

number of slices Willy must cut out such that every guest can be

given the same amount of cake, regardless of whether p or q

people show up? The slices do not have to be equally sized, and

each guest could receive more than one slice. Also, Willy is not

allowed to eat the whole cake and give nothing to all the guests.

Also, the entire cake must be distributed amongst the guests --

there can be no leftovers.

Note 1: Two numbers are relatively prime if they share no common

factors. So for example, the numbers 4 and 9 are relatively prime,

but the numbers 10 and 15 are not because they share the number

5 as a common factor.

Note 2: (5/27/2003 2:17PM) Edited to specify that the entire cake

must be given away.

LEMMINGDROWNINGS

1/27/2003 8:30PM

Somewhere in Northern Eurasia, a group of 20 lemmings is

planning a special group suicide this year. Each of the lemmings

will be placed in a random position along a thin, 100 meter long

plank of wood which is floating in the sea. Each lemming is equally

likely to be facing either end of the plank. At time t=0, all the

lemmings walk forward at a slow speed of 1 meter per minute. If a

lemming bumps into another lemming, the two both reverse

directions. If a lemming falls off the plank, he drowns. What is the

longest time that must elapse till all the lemmings have drowned?

Author: William Wu

Note 1: If you make a certain observation, the calculations

required become very trivial!



36/43


Note 2: Yes, I know Lemmings don't really commit group suicides.

Problem and storyline was inspired by the Lemmings video game.

RESISTOR CUBE

1/27/2003 8:30PM

Imagine a cube where each edge is a 1 ohm resistor. Find the

resistance between opposite corners of the cube.

There are many ways to solve this problem, but some ways are

more clever than others.

ASTROWRENCH

P

1/27/2003 8:30PM

Two astronauts are standing on a spinning space station shaped

like a disk. They are the same radial distance away from the disk's

center, and standing opposite to each other across from the center

(e.g., if you draw a line connecting the two astronauts, the line

crosses the disk's center.) One astronaut wants to toss a wrench to

the other. Among the infinitude of trajectories which will

accomplish this goal, characterize one of the trajectories without

writing a single equation.

Note: Be wary grasshopper. There are several common wrong

answers to this problem!

SLASH AND

DASH SEQUENCE

2/2/2003 3:31AM

What are the next few terms in the sequence?

-

/-\

/-\-

//-\\

/-\--

/-\/-\

/-\---

///-\\\



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//-\\-

/-\-/-\

Hint 1: Maybe it'll go easier if you replace - with 0, and the / and \

with ( and ). Or not.

Hint 2: These are just the numbers 1 through 10, expressed

differently. And despite the presence of zero, you can't express

zero this way.

Hint 3: All your base are belong to prime numbers.

JEWEL THIEVES

2/2/2003 3:31AM

Two thieves conspire to steal a valuable necklace made of

diamonds and rubies (evenly spaced, but not necessarily

alternating or symmetric). After they take it home, they decide that

the only way to divide the booty fairly is to physically cut the

necklace in half.

Prove that, if there is an even number of diamonds and an even

number of rubies, it's possible to cut the necklace into two pieces,

each of which contains half the diamonds and half the rubies.

WIRECUFFS

3/9/2003 4:11AM

In a cliche effort to illustrate the importance of teamwork-oriented

problem solving, the Boss has chained Dilbert to Carol The

Secretary via wire wrapped around their wrists, as shown in the

following snapshot:

The goal is for Dilbert and Carol to unlink themselves from each



38/43


other; considering what a horrible woman Carol is, Dilbert wouldn't

have it any other way. The wire is unbreakable, and as much as

Dilbert would like to saw off Carol's limbs, that's against company

policy. How can Dilbert and Carol get away from each other?

BUFFON'S

NEEDLE

M

4/7/2003 12:47AM

While recovering from wounds in the American Civil War, a Captain

Fox threw needles at a surface ruled with parallel lines, and was

thus able to experimentally infer the value of pi. True story! To find

out exactly how pi plays into this scenario, solve the following

problem: A surface is ruled with parallel lines. The lines are at

distance D apart from each other. Suppose we throw a needle of

length L on the surface at random. What is the probability that the

needle will intersect one of the lines?

Note 1: A famous problem posed and solved in 1777 by French

naturalist Buffon. It has long since fascinated scientists, and marks

the origin of geometrical probability -- the analysis of geometrical

configurations of randomly placed objects.

Note 2: These 19th century experiments began development of the

Monte Carlo method, which uses repeated simulation to

approximate true statistics.

Note 3: This isn't much of a riddle ... more like just an interesting

mathematical exercise. The most challenging part is setting up the

problem.

4/7/2003 12:47AM

Samwise Gamgee has a square plot of land, each side being 1 unit.

One day, Sam finds out that the dark Lord Sauron has a telephone

line that he uses to speak with a traitor amongst the hobbits.

Gandalf informs him that the telephone line runs in a straight line

parallel to the ground and passes beneath the square plot of land,

but he does not know its location. Sam decides to dig up around



39/43


SAMWISE AND

GANDALF

the perimeter of his land to discover the telephone line, but

Gandalf says it is not necessary to dig around the entire length of 4

units.

Sam brightens up, and says "I know what you mean. I can just dig

3 sides and still discover it. For even if the phone line runs along

the fourth side, I will still detect it at the end points ! "

Gandalf shakes his head. "No, Sam. You are on the right track, but

you can do better than that."

What solution does Gandalf have in mind for the optimum length of

the "digging curve" ?

RETROGRADE

CHESS I

C

4/7/2003 12:47AM

What was the last move made?

Note: From Karl Fabel's 1955 book Rund um das Schachbrett.



40/43


PATH COLLISION

5/11/2003 10:44PM

Consider an N x N grid. Denote one corner as point A, and the

opposite corner as point B. George is walking from A to B, and

Lennie is walking from B to A. All paths are equally likely, as long

as they follow the grid and never move away from the destination.

(Hence George's path can never move down or left, and Lennie's

path can never move up or right.)

What is the probability that George and Lennie collide? If George runs and thus moves three times faster than

Lennie, what is the probability of collision?

ROPE AROUND

THE EARTH

5/11/2003 10:44PM

Assume the Earth is a perfect sphere of radius r and suppose a

rope of zero elasticity is tied tightly around it. One metre is nowadded to the rope's length. If the rope is now pulled at one point as

high as possible above the Earth's surface, what height will be

reached?

MOLINA'S URNS

5/11/2003 10:44PM

Two urns contain the same total numbers of balls, some blacks and

some whites in each. From each urn are drawn n balls withreplacement, where n >= 3. Find the number of drawings and the

composition of the two urns so that the probability that all white

balls are drawn from the first urn is equal to the probability that

the drawing from the second is either all whites or all blacks.

Author: E.C. Molina.



41/43


42/43


After that, consider solving the problem when a two-thirds or

three-fourths majority is required. Try to generalize the result.

FAIR COIN

GAMBLING

Contributor: William Wu 9:00 PM 8/19/200

Starting with 0 dollars, you repeatedly flip a fair coin, earning $1

each time heads appears, and losing $1 each time tails appears.

When your net cash reaches either A or -B, you stop gambling.

1. As a function of A and B, compute the expected number of

flips until the game stops.

2. Now consider the unstopped version of this process, in which

you gamble indefinitely regardless of your current profit or

debt. Prove that the expected time till your net cash is +$1 is

infinite. Likewise, the expected time till your net cash is -$1

is also infinite. And yet one of these two events must occur

upon the first coin flip!

Forum threads: More Heads Than Tailsand Lazy hunter catch the

rabbit.

JUMPING FROG

ON LADDER

Contributor: William Wu 9:00 PM 8/19/200

Imagine a upright ladder with n rungs. At each time step, a frog

jumps to one of the n rungs uniformly at random. (The frog could

jump in place.) Assume time starts counting from 1, and the initial

rung position is also chosen uniformly at random.

As the number of rungs tends to infinity, what is the expected time

till the frog first jumps downwards? Prove it.

Author: William Wu

http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1068612055;start=0http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_hard;action=display;num=1088356565;start=23http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_hard;action=display;num=1088356565;start=23http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_hard;action=display;num=1088356565;start=23http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_hard;action=display;num=1088356565;start=23http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1068612055;start=0


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COLORING THE

RATIONAL PLANE

Contributor: Eigenray 9:00 PM 8/19/200

Let Q denote the set of rationals. How many colors are required to

color every point of Q^k in such a way that no two points a unit

distance apart are the same color?

1. Easy: k = 1

2. Medium: k=2,3

3. Hard: k>3

RANDOM

POINTS ONSPHERE

Contributor: Joc Koelman 9:00 PM 8/19/200

Two points on the surface of a sphere are drawn uniformly at

random. What is the maximum likelihood estimate of the distance

between these two points? The answer may be shocking at first.

After getting the answer, try to explain it intuitively.

NUMBER OF

MIDPOINTS

Contributor: Aryabhatta 9:00 PM 8/19/200

You are given n>=2 points in the 2-D plane. For each pair of points

(P,Q) from the n points, mark the midpoint of PQ red. Show that

there are at least 2n-3 distinct red points. For each n, show an

arrangement of n points for which there are exactly 2n-3 distinct

red points.

Source: Iranian Mathematical Olympiad

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Riddles Without Answers (Pages43)