Laszlo's puzzles
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1. You have 80 black balls and 70 white balls in a container,
plus a supply of extra black balls. What is the color of the last
ball if you remove balls by the rules:
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Remove them two at a time.
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If the two chosen are both white, then put back a black ball.
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If the two chosen are both black, then put back a black ball
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If the two chosen are one of each color, then put back a white ball?
2. A fleet of boats is assembled at a base on the equator of
the planet Aquaria. This planet is completely covered with water.
The fleet aims to steam around the equator. Fuel is scarce, and each
boat has only enough fuel to go half way around. If no boat is to
be stranded away from the base, what is the minimum number of boats and
amount of fuel such that one boat makes it all the way around (aided by
fuel transfers, of course)?
3. A priest (P) and his friend (F) are walking in the garden.
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P to F: Low attendance at church has me worried.
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F to P: Why? How large was the attendance on Sunday?
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P to F: Besides myself, only three.
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F to P: And what were their ages?
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P to F: Let me give you a puzzle. The product of their ages was 2450
and the sum of their ages was twice as large as your street address.
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F to P: After much thought, I cannot answer.
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P to F: But if I tell you that I was older than all, then you can tell.
Having overheard this conversation, your job is to calculate the age of
the priest.
4. Once there were 3 friends (Anton, Simon, Paul) chatting to each
other in a dining hall. They were talking about numbers:
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A: I've chosen two natural numbers which are both greater than 1. I'm going
to tell Simon the sum and Paul the product of these numbers.
Anton then gave Simon the sum and Paul the product, such that each of our
friends knew "his" number only but not the number of the other friend.
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A: Now, can you guess the numbers which I have in mind?
Simon and Paul liked this sort of game. Their conversation was the following:
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P: I don't know Anton's numbers.
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S: I don't know them either.
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P: Hey, now I know.
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S: Then I know them too.
A math professor had observed this whole situation from the very beginning.
He went to our friends and said "I know Anton's numbers, too!" Do
you?
Warning -- this is a very annoying puzzle. If you want the solution,
click here.
5. Papa Bear, Mama Bear and Baby Bear are having a quiet night at home.
They start to talk about the 10 jars of honey they have in the basement.
They notice that bears always consume a full jar of honey, and whenever
they have honey Papa always eats more than Mama, and Mama always eats more
than Baby.
After all this talk Papa Bear gets hungry, goes to the basement, and
eats a few jars of honey. Mama follows later, and finally Baby also
satisfies his hunger. In the dark basement they have no way of seeing
which jar is full or empty.
A while later Uncle Bear comes over. "I ran out of honey.
Do you have some honey left?" he asks. Papa Bear thinks for a moment
and he responds: "I have no idea". He turns to Mama Bear, asking
"Do you know?" She also thinks for a while and says "Sorry, I do
not know. But perhaps Baby Bear knows". After some thinking
Baby Bear says "I have no idea. Let us ask Papa Bear again".
At this point Papa Bear says "yes, there is some honey left".
Do you know how much?
I was able to solve 1, 2, 4, and 5. Laszlo helped me to see
how to solve #3. Here is another to which Laszlo knows the solution,
but I do not.
6. A wife flees her husband, she by boat on a circular lake, and he
on the footpath surrounding the lake. She rows one quarter as fast
as he runs. At time t=0 she is in the center and he is at the southern-most
point of the shore.
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First question: If she rows directly north, he will obviously catch her
at the northern-most point. Assume that she can turn without changing
speed. What path lands her farthest from him?
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Second question: What path and (constant) speed lets her go as slowly as
possible such that she lands simultaneously with his arrival.
7. Just to lighten up, here is a puzzle which Peter Stephens heard
on "Car Talk." Simplify the expression: (x-a)(x-b)...(x-z).
Valerii Vinokur showed the puzzle below in a seminar
at Stony Brook (4/30/2004) about vortex localization. He said that
it was posed to him on an oral exam in Moscow. When he was unable
to solve it immediately, the professor summoned an audience to exhibit
VV, the student with the singular lack of geometric imagination.
8. Four ships (A,B,C,D) sail on the sea with constant velocities
which are all unequal. Ship A collides at various moments with ships
B, C, and D.
Ship B collides at various other moments with ships C and D.
Prove that ship C collides sometime with ship D.
PBA, 4/30/2004