A number is called a
palindrome when it is equal to the number you get when all its digits are
reversed. For example, 2772 is a palindrome. We discovered a curious thing. We
took the number 461, reversed the digits, giving the number 164, and calculated
the sum of these two numbers: 461 164 + ------- 625 We repeated the process of
reversing the digits and calculating the sum two more times: 625 526 + -------
1151 1511 + ------- 2662 To our surprise, the result 2662 was a palindrome. We
decided to see if this was a pure coincidence or not. So we took another 3-digit
number, reversed it, which gave a larger number, and added the two. The result
was not a palindrome. We repeated the process, which resulted in another 3-digit
number which was still not a palindrome. We had to repeat the process twice more
to finally arrive at a 4-digit number which was a palindrome. What was the
3-digit number we started with the second time?
Because the reverse of the starting number is greater than the starting number
itself, the first digit of the starting number must be less than the last digit.
Therefore, the starting number must be at least 102. Secondly, we know that
after two summations, the result has still only 3 digits.
abc
cba +
-------
def
fed +
-------
ghi
General Gasslefield, accused
of high treason, is sentenced to death by the court-martial. He is allowed to
make a final statement, after which he will be shot if the statement is false or
will be hung if the statement is true. Gasslefield makes his final statement and
is released. What could he have said?
General Gasslefield said: "I will be shot." If this statement was true, he would
have been hung and thus not be shot. But then his statement would be false,
which implies that he should be shot, making the statement true again, etc... In
other words: the verdict of the court-martial could not be executed and the
general was released.
On a nice summer day two
tourists visit the Dutch city of Gouda. During their tour through the center
they spot a cosy terrace. They decide to have a drink and, as an appetizer, a
portion of hot "bitterballs" (bitterballs are a Dutch delicacy, similar to
croquettes). The waiter tells them that the bitterballs can be served in
portions of 6, 9, or 20. What is the largest number of bitterballs that cannot
be ordered in these portions?
Every natural number is member of one of the following six series:
0, 6, 12, 18, ...
1, 7, 13, 19, ...
2, 8, 14, 20, ...
3, 9, 15, 21, ...
4, 10, 16, 22, ...
5, 11, 17, 23, ...
If for a number in one of these series holds that it can be made using the
numbers 6, 9, and 20, then this also holds for all subsequent numbers in the
series (by adding a multiple of 6). To find out what the largest number is that
cannot be made using the numbers 6, 9, and 20, we therefore only need to know,
for every series, what the smallest number is that can be made in that way. In
the series 0, 6, 12, 18, ... the smallest number that can be made is 0 so there
is no number that cannot be made.In the series 1, 7, 13, 19, ... the smallest
number that can be made is 49 (20+20+9) so 43 is the largest number that cannot
be made.
In the series 2, 8, 14, 20, ... the smallest number that can be made is 20 so 14
is the largest number that cannot be made.In the series 3, 9, 15, 21, ... the
smallest number that can be made is 9 so 3 is the largest number that cannot be
made.In the series 4, 10, 16, 22, ... the smallest number that can be made is 40
(20+20) so 34 is the largest number that cannot be made.In the series 5, 11, 17,
23, ... the smallest number that can be made is 29 (20+9) so 23 is the largest
number that cannot be made.Therefore, 43 is the largest number that cannot be
made using the numbers 6, 9, and 20..
Two friends, Alex and Bob,
go to a bookshop, together with their sons Peter and Tim. All four of them buy
some books; each book costs a whole amount in shillings. When they leave the
bookshop, they notice that both fathers have spent 21 shillings more than their
respective sons. Moreover, each of them paid per book the same amount of
shillings as books that he bought. The difference between the number of books of
Alex and Peter is five. Who is the father of Tim?
For each father-son couple holds: the father bought x books of x shillings, the
son bought y books of y shillings. The difference between their expenses is 21
shillings, thus x2 - y2 = 21. Since x and y are whole numbers (each book costs a
whole amount of shillings), there are two possible solutions: (x=5, y=2) or
(x=11, y=10). Because the difference between Alex and Peter is 5 books, this
means that father Alex bought 5 books and son Peter 10. This means that the
other son, Tim, bought 2 books, and that his father is Alex.
A man decides to buy a nice
horse. He pays $60 for it, and he is very content with the strong animal. After
a year, the value of the horse has increased to $70 and he decides to sell the
horse. But already a few days later he regrets his decision to sell the
beautiful horse, and he buys it again. Unfortunately he has to pay $80 to get it
back, so he loses $10. After another year of owning the horse, he finally
decides to sell the horse for $90. What is the overall profit the man makes?
Consider the trade-story as if it describes two separate trades, where: In the
first trade, the man buys something for $60 and sells it again for $70, so he
makes a profit of $10.
In the second trade, the man buys something for $80 and sells it again for $90,
so he makes again a profit of $10.
Conclusion: The man makes an overall profit of $10 + $10 = $20.
You can also look at the problem as follows: the total expenses are $60 + $80 =
$140 and the total earnings are $70 + $90 = $160. The overall profit is
therefore $160 - $140 = $20.
Yesterday evening, Helen and
her husband invited their neighbors (two couples) for a dinner at home. The six
of them sat at a round table. Helen tells you the following: "Victor sat on the
left of the woman who sat on the left of the man who sat on the left of Anna.
Esther sat on the left of the man who sat on the left of the woman who sat on
the left of the man who sat on the left of the woman who sat on the left of my
husband. Jim sat on the left of the woman who sat on the left of Roger. I did
not sit beside my husband." What is the name of Helen's husband?
From the second statement, we know that the six people sat at the table in the
following way (clockwise and starting with Helen's husband):
Helen's husband, woman, man, woman, man, Esther Because Helen did not sit beside
her husband, the situation must be as follows: Helen's husband, woman, man,
Helen, man, Esther The remaining woman must be Anna, and combining this with the
first statement, we arrive at the following situation:Helen's husband, Anna,
man, Helen, Victor, Esther Because of the third statement, Jim and Roger can be
placed in only one way, and we now know the complete order:Helen's husband
Roger, Anna, Jim, Helen, Victor, Esther Conclusion: the name of Helen's husband
is Roger. .
In the middle of a round
pool lies a beautiful water-lily. The water-lily doubles in size every day.
After exactly 20 days the complete pool will be covered by the lily. After how
many days will half of the pool be covered by the water-lily?
Because the water-lily doubles its size every day and the complete pool is
covered after 20 days, half of the pool will be covered one day before that,
after 19 days. Conclusion: After 19 days half of the pool will be covered by the
water-lily
Jack and his wife went to a
party where four other married couples were present. Every person shook hands
with everyone he or she was not acquainted with. When the handshaking was over,
Jack asked everyone, including his own wife, how many hands they shook. To his
surprise, Jack got nine different answers. How many hands did Jack's wife shake?
Because, obviously, no person shook hands with his or her partner, nobody shook
hands with more than eight other people. And since nine people shook hands with
different numbers of people, these numbers must be 0, 1, 2, 3, 4, 5, 6, 7, and
8. The person who shook 8 hands only did not shake hands with his or her
partner, and must therefore be married to the person who shook 0 hands. The
person who shook 7 hands, shook hands with all people who also shook hands with
the person who shook 8 hands (so in total at least 2 handshakes per person),
except for his or her partner. So this person must be married to the person who
shook 1 hand. The person who shook 6 hands, shook hands with all people who also
shook hands with the persons who shook 8 and 7 hands (so in total at least 3
handshakes per person), except for his or her partner. So this person must be
married to the person who shook 2 hands. The person who shook 5 hands, shook
hands with all people who also shook hands with the persons who shook 8, 7, and
6 hands (so in total at least 4 handshakes per person), except for his or her
partner. So this person must be married to the person who shook 3 hands. The
only person left, is the one who shook 4 hands, and which must be Jack's wife.
The answer is: Jack's wife shook 4 hands.
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