**Some Fascinating Puzzles**

Email me at “sprgene
at lycoming dot edu” if you
want to talk to me about any of these problems. Be sure to visit my Home Page.

**#1**
You are presented with five bags each containing
several coins. Four of the bags contain
all true gold coins, and one bag contains all counterfeit gold coins. You do not know which bag contains the
counterfeit gold, but you do know that each true gold coin weighs 1 gram, and
each counterfeit gold coin weighs 1.1 grams.
This very small difference in weight makes it absolutely impossible to
distinguish between true gold coins and counterfeit gold coins simply by
holding even several coins in your hand.
You will be permitted to make one __and only one__ weight measurement
on a scale. You must place whatever you
wish to weigh on the scale, and then press a button which causes the weight of
the objects on the scale to print on a card.
Once you have done this, you are not allowed to make any other weight
measurements. You must decide which bag
contains the counterfeit gold coins based only the weight printed on the card
from your one measurement.

(a) Suppose each bag
contains five coins. How will you
determine which bag contains the counterfeit gold?

(b) Suppose each bag
contains four coins. Could you determine
which bag contains the counterfeit gold coins? Why or why not?

(c) Suppose each bag
contains three coins. Could you
determine which bag contains the counterfeit gold coins? Why or why not?

**#2**
You are presented with five bags each containing
several coins. Each bag either contains
all true gold coins or contains all counterfeit gold coins; the number of bags
containing counterfeit gold could be zero, one, two, three, four, or five, but
you do not know anything about whether each bag contains true or counterfeit
gold coins. You know that each true gold
coin weighs 1 gram, and each counterfeit gold coin weighs 1.1 grams. This very small difference in weight makes it
absolutely impossible to distinguish between true gold coins and counterfeit
gold coins simply by holding even several coins in your hand. You will be
permitted to make one __and only one__ weight measurement on a scale. You
must place whatever you wish to weigh on the scale, and then press a button
which causes the weight of the objects on the scale to print on a card. Once you have done this, you are not allowed
to make any other weight measurements. You must decide which bags contain the
true gold coins and which bags contain the counterfeit gold coins based only
the weight printed on the card from your one measurement.

(a) Suppose each bag
contains 16 coins. How will you
determine which bags contain the true gold coins and which bags contain the counterfeit
gold coins?

(b) Suppose each bag
contains five coins? Could you determine
which bags contain the true gold coins and which bags contain the counterfeit
gold coins? Why or why not?

**#3**
You are given 20 coins and told to place them into 5
rows, each row consisting of 4 coins, and this of course is easy to do. Ten coins are then taken away from you,
leaving you with only ten coins. You are
told to place these ten coins into 5 rows, each row consisting of 4 coins. How can you do this?

**#4**
Suppose that 64 large buckets are each filled with
water. Exactly one bucket contains contaminated water while the other 63
buckets are filled with pure water, but the contaminated water and the pure
water look identical. The only way to tell whether water is contaminated is to
test it. The test requires only one drop of water but is time-consuming and
difficult to perform.

(a) How it possible
to identify the one bucket containing the contaminated water by performing only
6 tests?

(b) Suppose the
number of large buckets each filled with water is equal to *n.* What is
the formula for the smallest number of tests needed to identify the one bucket
containing the contaminated water?

**#5**
You are in a room with only one door from which you
can exit. This door has two doorknobs: one on the right and one on the left.
When one of these doorknobs is turned, the door opens, and you are set free; if
the other doorknob is turned, the room explodes. You do not know which doorknob
will set you free. In the room are two computers named A and B. Each computer
is programmed to answer questions, but one of the computers is programmed to
always tell the truth while the other is programmed to always lie. You do not
know which computer is programmed to always tell the truth. You are allowed to pick
one computer, and ask that computer one, and only one, question. Which computer
will you pick and what question will you ask to determine which doorknob will
set you free? (If you do not ask any question at all, you will eventually
starve to death in the locked room!)

**#6**
"I have three nieces, and two of them are twins. If you can guess their
ages, I will buy you dinner," Jeanne says to Marty. "Well, at least
give me some kind of hint," Marty replies to Jeanne. "Oh, Okay. The
sum of their ages is 13, and the product of their ages is an even
integer," Jeanne sighs. "Will you tell me whether the twin nieces are
younger or older than the third niece?" Marty begs. "No, I won't tell
you that, because then you wouldn't have to guess their ages, you would know
their ages!" exclaims Jeanne. "Aha!" cries Marty, "But now
I do know their ages!" Can you find the ages of the three nieces the way
Marty has?

**#7**
Let us agree to represent integers by *abcd**...*,
where *a* is the first digit, *b* is the
second digit, *c* is the third digit, etc.

(a) Find a four‑digit
number *abcd*, where *a*
is the number of digits equal to 0 (zero), *b* is the number of
digits equal to 1 (one), *c* is the number of digits equal to 2 (two), and
*d* is the number of digits equal to 3 (three). There are two possible answers. (Hint: It
will be helpful to realize that *a* + *b* +* c* + *d* must be equal to the total number of digits,
which is four.)

(b) Prove that there
is no three-digit number *abc*,
where *a* is the number of digits equal to 0 (zero), *b* is the
number of digits equal to 1 (one), and *c* is the number of digits equal
to 2 (two).

(c) Prove that there
is no two-digit number *ab*, where *a* is
the number of digits equal to 0 (zero), and *b* is the number of digits
equal to 1 (one).

(d) Find a
five-digit number *abcde*, where *a* is the number of digits equal to 0 (zero), *b*
is the number of digits equal to 1 (one), *c* is the number of digits
equal to 2 (two), *d* is the number of digits equal to 3 (three), and *e*
is the number of digits equal to 4 (four). There is only one possible answer.

**#8**
Jane writes a positive integer on a blank card. She then adds one to the
integer and writes the answer on another blank card. Simon and Jason do not know what numbers Jane
has written on the cards, but Jane does tell them that positive integers
differing by one are written on the cards.
Jane gives Simon one of the cards to hold up so Jason can see the number
Simon is holding, but Simon cannot see the number he himself is holding. Jane then gives Jason the other card to hold
up so Simon can see the number Jason is holding, but Jason cannot see the
number he himself is holding. The two
men stand there for a moment staring at the number that the other is
holding. Consider each of the three
following possibilities for what might happen next:

(a) Suppose Jason
suddenly says, "I know what integer is written on my card!". How can Jason
possibly know what integer is written on his card, and what must this integer
be equal to?

(b) Suppose Jason
calmly says, "I don't know what integer is written on my card", after
which Simon replies, "I don't know what number is written on my
card", after which Jason says, "I know what integer is written on my
card!".
How can Jason possibly know what integer is written on his card, and
what must this integer be equal to?

(c) Suppose Jason
calmly says, "I don't know what integer is written on my card", after
which Simon replies, "I don't know what number is written on my
card". What is described in the
previous sentence occurs 12 more times.
Then Jason says, "I know what integer is written on my card!". How can Jason
possibly know what integer is written on his card, and what must this integer
be equal to?

**#9**
Two people are talking long distance on the phone; one
is in an East-Coast state of the US, the other is in a West-Coast state of the
US. One person asks the other "What time is it?" After the second
person answers, the first person exclaims, "That's funny. It's the same
time here!" How can this be possible?

**#10**
The integers 1, 2, 3, ..., 100 are randomly arranged
in a square consisting of 10 rows and 10 columns. Suppose you find the smallest integer in each
row, after which you find the largest of these integers, and you call this
integer *R*. Suppose you then find
the largest integer in each column, after which you find the smallest of these
integers, and you call this integer *C*.
Which integer will be larger, *R* or *C*?

**#11**
Secret agents Jones and Smith can never see each other. They just leave messages for each other in
locked boxes placed under the sink in the men's room of Sam's Diner. Jones goes to the men's room between 12:00
noon and 1:00 pm every day, and Smith goes to the men's room between 12:00 midnight
and 1:00 am. Jones and Smith each have
exactly two padlocks, the one and only key to the two padlocks, and exactly one
box with a lid which can be locked with one or more padlocks. A box placed in the men's room by either
agent must always have at least one padlock on it to insure that enemy agents
can never get their hands on the contents or even on an unlocked empty box
(thereby insuring that enemy agents can never even fully figure out how these
boxes are constructed). If the padlock
is tampered with in any way, the padlock, the box, and the contents of the box
explode. Under the sink in the men's
room of Sam's Diner is the only place where Jones and Smith make
exchanges. How does one agent get a
message to the other without ever having an agent’s key leave his possession
and with the certainty that no one else can ever see the message?

**#12**
I have an envelope containing several dollar bills, each of which has a value
more than one dollar. All of the bills
are ten-dollar bills except two, all of the bills are twenty-dollar bills except
two, and all of the bills are fifty-dollar bills except two. What is the dollar value of the money in my
envelope?

**#13**
Three people check into a hotel. Together they pay $30
to the manager and go to their room. After
the manager finds out that the room rate is $25, he gives $5 to the bellboy to
return to three people. On the way to
the room the bellboy reasons that $5 would be difficult to share among three
people so he pockets $2 and gives $1 to each person. Now each person paid $10
and got back $1. So, the three people
paid $9 each, totaling $27, and the bellboy has $2, totaling $29. Where is the
remaining dollar?

**#14**
Fifty miles from home, an Arab sheikh gives the elder of his two sons a red
camel and gives the younger son an orange camel. The sheikh then says, "If the red camel
reaches home before the orange camel, my younger son will inherit my entire
fortune, but if the orange camel reaches home before the red camel, my older
son will inherit my entire fortune."
Each brother stands next to his camel and just stares at the other
brother for several hours. A wise man
then walks up to the two brothers and says two words. After hearing these two words, the brothers
quickly jump on the camels and race as fast as they can toward home. What did the wise man say?

**#15**
Joe's Diner gives each customer a ticket numbered with a positive integer. Any customer who brings in tickets with
integers summing to 100 wins $100,000.
However, being a devious person, Joe doesn't want anybody to actually be
able to win the contest. He realizes
that if he just prints tickets with the 49 integers from 51 to 99, nobody can
possibly win, but it will quickly become obvious that nobody can win when
people realize that there are never any ticket numbers below 51. However, Joe figures out a way that he can
print 49 ticket numbers which will never give a sum equal to 100, and 16 of
these 49 ticket numbers are below 51.
Which 49 integers did Joe choose?

**#16**
A shipping clerk has five different boxes each weighing between 50 and 100
pounds. Unfortunately, the only scale he
is allowed to use will only measure weights over 100 pounds. After weighing all
possible pairs of the five boxes, he records these ten weights as 110, 112,
113, 114, 115, 116, 117, 118, 120, and 121 pounds. What were the weights of the five individual
boxes?

**#17**
At the beginning of World War I, the German army did
not have metal helmets. At one point
during the war, the German army began using metal helmets in order to reduce
the high number of head injuries that were occurring. However, even though the intensity of the
fighting did not change after the helmets were introduced, the number of head
soldiers hospitalized for head injuries increased tremendously. Why did this happen?

**#18**
Two trains on the same track are traveling toward each
other, each at a constant speed of 50 miles per hour. When the trains are exactly 100 miles apart,
Superman lands on the front of one train and begins flying straight to the
front of the other train instantly turning around and flying straight back and
forth again, always at a constant speed of 75 miles per hour, and repeating
this until the two trains crashed. What
was the total distance Superman traveled from the time he first landed on the
front of a train until the trains crashed?

**#19**
You are shown three sealed boxes, a red box, a yellow
box, and a blue box. You are told that
only one of the following three statements is true:

The $1000 is not in the
yellow box. |

The $1000 is not in the
blue box. |

The $1000 is in the yellow
box. |

Which box contains the $1000?

**#20**
You have 64 coins, only 63 of which are real; the fake
coin weighs one half of an ounce more than a real coin. This very small difference in weight makes it
absolutely impossible to distinguish between true gold coins and counterfeit
gold coins simply by holding even several coins in your hand. You also have a very accurate balance scale
(a scale with two places to put objects, after which the scale either indicates
that the weight in each place is exactly the same or indicates which side is
heavier).

(a) How can you find
the fake by only using the balance scale four times?

(b) What is the
largest number of coins for which the method used in part (a) will work by only
using the balance scale four times?

**#21**
You have a large and very expensive bottle of medicine
(costing $1000). You are supposed to drink exactly one ounce of medicine mixed
with one ounce of water each morning. If you take more or less than exactly one
ounce of the medicine, you will suffer from severe headaches all day. On one
morning, you fill a one-ounce glass with the medicine, and place it on a table.
You then go over to the sink, and while you are filling another one-ounce glass
with water, the phone rings. While you are talking to a friend on the phone,
you place the one-ounce glass with water on the table, but you forget that you
already put a one-ounce glass with the medicine on table and don't it, because
you are so involved in your phone conversation. While still talking to your
friend on the phone, you fill a third one-ounce glass with the medicine, and
place it on a table. When the phone conversation finally ends, you go over to
the table and see the three identical one-ounce glasses. Since the medicine
looks and tastes exactly like water, you are not sure which glass contains the
water and which two contain the medicine. You don't want to throw the medicine
away because it is so expensive, but you also don't want to take the wrong
dosage because of the severe consequences. How can you take the right dosage
without wasting any medicine?

**#22**
Amanda's dad buys a bag of 30 chocolate bars on his way home from work, but he
eats some of the chocolate bars before getting to home. When he gets home, he
puts the bag on the living room table, and tells Amanda and her mom that the
chocolate bars are to be divided equally among the three of them, but no one
may eat any until after dinner. Thirty minutes before dinner, Amanda's dad
sneaks into the living room and counts the number of chocolate bars in the bag.
He then realizes that the number of bars is one more than a multiple of three.
So he eats one bar and removes one-third of the bars remaining in the bag,
while thinking to himself, "I'll take my share now, but I'll wait until
after dinner to eat any of these." Twenty minutes before dinner, Amanda's
mom sneaks into the living room and counts the number of chocolate bars in the
bag. She then realizes that the number of bars is one more than a multiple of three.
So she eats one bar and removes one-third of the bars remaining in the bag,
while thinking to herself, "I'll take my share now, but I'll wait until
after dinner to eat any of these." Ten minutes before dinner, Amanda
sneaks into the living room and counts the number of chocolate bars in the bag.
She then realizes that the number of bars is one more than a multiple of three.
So she eats one bar and removes one-third of the bars remaining in the bag,
while thinking to herself, "I'll take my share now, but I'll wait until
after dinner to eat any of these." How many chocolate bars did Amanda's
dad eat from the bag of 30 chocolate bars, before he brought the bag home?

**#23**
I hope to add some more problems real soon!!!!!

**#Special Problem #1** It is Friday afternoon, and
high school student Amanda is sitting in her last class of the day. As soon as
she (gleefully) notices that there are five minutes of class left, the teacher,
Mr. Grumble, announces that a quiz will be administered one day next week, but
the day on which the quiz will occur will be a surprise.

"Oh, Mr. Grumble," Amanda chirps
sweetly after raising her hand, "Don't you realize that it is impossible
to have a surprise quiz one day next week. We all know you won't have it on
Friday, because if the quiz has not occurred by Thursday, then everyone would
know that Friday has to be the day for the quiz, and it won't be surprise. This
means that the quiz has to occur on Monday, Tuesday, Wednesday, or Thursday. We
all know you won't have it on Thursday, because if the quiz has not occurred by
Wednesday, then everyone would know that Thursday has to be the day for the
quiz. This means that the quiz has to occur on Monday, Tuesday, or Wednesday.
We all know you won't have it on Wednesday, because if the quiz has not
occurred by Tuesday, then everyone would know that Wednesday has to be the day
for the quiz. This means that the quiz has to occur on Monday or Tuesday. If
you don't have the quiz on Monday, then we will all know it is coming on Tuesday,
but that leaves Monday as the only choice for the quiz, and it certainly will
not be a surprise if we all know it's Monday."

Mr. Grumble then stands in awe of Amanda's
logic, scratching his head and searching his brain for an answer. Is this #%$(insert some choice words here)@& kid's logic
correct? If not, where is the flaw?

**#Special Problem #2** If a triangle having all three sides equal in length
is inscribed inside a circle of radius 1 (one), the length of each side will be
the square root of 3. (This is easy to prove.)
If a chord of the circle is selected "at random" (say by
dropping a long, straight piece of thin wire at random over the circle), what
is the probability that the length of the chord will be smaller than the square
root of 3? (HINT: no matter what your
answer is, it's wrong!!!)