A cable, 16 meters in
length, hangs between two pillars that are both 15 meters high. The ends of the
cable are attached to the tops of the pillars. At its lowest point, the cable
hangs 7 meters above the ground. How far are the two pillars apart?
Note that it is a kind of trick question: the pillars stand next to each other.
Which means that the cable goes 8 meters straight down and 8 meters straight up.
Conclusion: The distance between the pillars is zero meters..
From a book, a number of
pages are missing. The sum of the page numbers of these pages is 9808. Which
pages are missing?
Let the number of missing pages be n and the first missing page p+1. Then the
pages p+1 up to and including p+n are missing, and n times the average of the
numbers of the missing pages must be equal to 9808:
n×( ((p+1)+(p+n))/2 ) = 9808
In other words:
n×(2×p+n+1)/2 = 2×2×2×2×613
So:
n×(2×p+n+1) = 2×2×2×2×2×613
One of the two terms n and 2×p+n+1 must be even, and the other one must be
odd. Moreover, the term n must be smaller than the term 2×p+n+1. It follows
that there are only two solutions:
n=1 and 2×p+n+1=2×2×2×2×2×613, so n=1 and p=9808,
so only page 9808 is missing. n=2×2×2×2×2 and
2×p+n+1=613, so n=32 and p=290, so the pages 291 up to and including 322
are missing.
Because it is asked which pages (plural) are missing, the solution is: the pages
291 up to and including 322 are missing.
In front of you are 10 bags,
filled with marbles. The number of marbles in each bag differs, but all bags
contain ten marbles or more. Nine of the ten bags only contain marbles of 10
grams each. One bag only contains marbles of 9 grams. In addition, you have a
balance which can weigh in grams accurate, and you are allowed to use it only
once (i.e. weigh a single time). How can you find out in one weighing, which bag
contains the marbles of 9 grams?
Number the ten bags from 1 up to and including 10. Then take one marble from bag
1, two marbles from bag 2, three marbles from bag 3, etc. Place all 55 marbles
that you selected from the bags together on the balance. The number of grams
that the total weight of these 55 marbles differs from 550 grams, is equal to
the number of marbles of 9 grams that are among those 55 marbles, and that is
equal to the number of the bag which contains the marbles of 9 grams.
A snail is at the bottom of
a 20 meters deep pit. Every day the snail climbs 5 meters upwards, but at night
it slides 4 meters back downwards. How many days does it take before the snail
reaches the top of the pit?
On the first day, the snail reaches a height of 5 meters and slides down 4
meters at night, and thus ends at a height of 1 meter. On the second day, he
reaches 6 m., but slides back to 2 m. On the third day, he reaches 7 m., and
slides back to 3 m. ... On the fifteenth day, he reaches 19 m., and slides back
to 15 m. On the sixteenth day, he reaches 20 m., so now he is at the top of the
pit! Conclusion: The snail reaches the top of the pit on the 16th day!... .
William lives in a street
with house-numbers 8 up to and including 100. Lisa wants to know at which number
William lives. She asks him: "Is your number larger than 50?" William answers,
but lies. Upon this Lisa asks: "Is your number a multiple of 4?" William
answers, but lies again. Then Lisa asks: "Is your number a square?" William
answers truthfully. Upon this Lisa says: "I know your number if you tell me
whether the first digit is a 3." William answers, but now we don't know whether
he lies or speaks the truth. Thereupon Lisa says at which number she thinks
William lives, but (of course) she is wrong. What is Williams real house-number?
Note that Lisa does not know that William sometimes lies. Lisa reasons as if
William speaks the truth. Because Lisa says after her third question, that she
knows his number if he tells her whether the first digit is a 3, we can conclude
that after her first three questions, Lisa still needs to choose between two
numbers, one of which starts with a 3. A number that starts with a 3, must in
this case be smaller than 50, so William's (lied) answer to Lisa's first
question was "No". Now there are four possibilities: number is a multiple of 4 :
(16, 36 number is a square) : 8, 12, 20, and more number is not a square number
is not a multiple of 4 : (9, 25, 49 number is a square) : 10, 11, 13, and more
number is not a square Only the combination "number is a multiple of 4" and
"number is a square" results in two numbers, of which one starts with a 3.
William's (lied) answer to Lisa's second question therefore was "Yes", and
William's (true) answer to Lisa's third question was also "Yes". In reality,
William's number is larger than 50, not a multiple of 4, and a square. Of the
squares larger than 50 and at most 100 (these are 64, 81, and 100), this only
holds for 81. Conclusion: William's real house-number is 81.
The poor have it, the rich
want it, but if you eat it you will die. What is this?
Nothing!
The gentlemen Dutch,
English, Painter, and Writer are all teachers at the same secondary school. Each
teacher teaches two different subjects. Furthermore: Three teachers teach Dutch
language There is only one math teacher There are two teachers for chemistry Two
teachers, Simon and mister English, teach history Peter doesn't teach Dutch
language Steven is chemistry teacher Mister Dutch doesn't teach any course that
is tought by Karl or mister Painter. What is the full name of each teacher and
which two subjects does each one teach?
Since Peter as only one doesn't teach Dutch language, and mister Dutch doesn't
teach any course that is tought by Karl or mister Painter, it follows that Peter
and mister Dutch are the same person and that he is at least math teacher. Simon
and mister English both teach history, and are also among the three Dutch
teachers. Peter Dutch therefore has to teach next to math, also chemistry.
Because Steven is also chemistry teacher, he cannot be mister English or mister
Painter, so he must be mister Writer. Since Karl and mister Painter are two
different persons, just like Simon and mister English, the names of the other
two teachers are Karl English and Simon Painter. Summarized:Peter Dutch, math
and chemistrySteven Writer, Dutch and chemistrySimon Painter, Dutch and
historyKarl English, Dutch and history..
You are standing next to a
well, and you have two jugs. One jug has a content of 3 liters and the other one
has a content of 5 liters. How can you get just 4 liters of water using only
these two jugs?
Solution 1: Fill the 5 liter jug. Then fill the 3 liter jug to the top with
water from the 5 liter jug. Now you have 2 liters of water in the 5 liter jug.
Dump out the 3 liter jug and pour what's in the 5 liter jug into the 3 liter
jug. Then refill the 5 liter jug, and fill up the 3 liter jug to the top. Since
there were already 2 liters of water in the 3 liter jug, 1 liter is removed from
the 5 liter jug, leaving 4 liters of water in the 5 liter jug. Solution 2: Fill
the 3 liter jug and pour it into the 5 liter jug. Then refill the 3 liter jug
and fill up the 5 liter jug to the top. Since there were already 3 liters of
water in the 5 liter jug, 2 liters of water are removed from the 3 liter jug,
leaving 1 liter of water in the 3 liter jug. Then dump out the 5 liter jug and
pour what's in the 3 liter jug into the 5 liter jug. Refill the 3 liter jug and
pour it into the 5 liter jug. Now you have 4 liters of water in the 5 liter jug.
On the market of Covent
Garden, mrs. Smith and mrs. Jones sell apples. Mrs. Jones sells her apples for
two per shilling. The apples of Mrs. Smith are a bit smaller; she sells hers for
three per shilling. At a certain moment, when both ladies both have the same
amount of apples left, Mrs. Smith is being called away. She asks her neighbour
to take care of her goods. To make everything not too complicated, Mrs. Jones
simply puts all apples to one big pile, and starts selling them for two shilling
per five apples. When Mrs. Smith returns the next day, all apples have been
sold. But when they start dividing the money, there appears to be a shortage of
seven shilling. Supposing they divide the amount equally, how much does mrs.
Jones lose with this deal?
The big pile of apples contains the same amount of large apples of half a
shilling each (from mrs. Jones), as smaller apples of one third shilling each
(from mrs. Smith). The average price is therefore (1/2 + 1/3)/2 = 5/12 shilling.
But the apples were sold for 2/5 shilling each (5 apples for 2 shilling). Or:
25/60 and 24/60 shilling respectively. This means that per sold apple there is a
shortage of 1/60 shilling. The total shortage is 7 shilling, so the ladies
together started out with 420 apples. These are worth 2/5 × 420 = 168
shilling, or with equal division, 84 shilling for each. If Mrs. Jones would have
sold her apples herself, she would have received 105 shilling. Conclusion: Mrs.
Jones loses 21 shilling in this deal.
A long, long time ago, two
Egyptian camel drivers were fighting for the hand of the daughter of the sheik
of Abbudzjabbu. The sheik, who liked neither of these men to become the future
husband of his daughter, came up with a clever plan: a race would determine who
of the two men would be allowed to marry his daughter. And so the sheik
organized a camel race. Both camel drivers had to travel from Cairo to
Abbudzjabbu, and the one whose camel would arrive last in Abbudzjabbu, would be
allowed to marry the sheik's daughter. The two camel drivers, realizing that
this could become a rather lengthy expedition, finally decided to consult the
Wise Man of their village. Arrived there, they explained him the situation, upon
which the Wise Man raised his cane and spoke four wise words. Relieved, the two
camel drivers left his tent: they were ready for the contest! Which 4 wise words
did the Wise Man speak?
Take each other's camel..
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