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:
• Remove them two at a time.
• If the two chosen are both white, then put back a black ball.
• If the two chosen are both black, then put back a black ball
• 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.
• P to F: Low attendance at church has me worried.
• F to P: Why? How large was the attendance on Sunday?
• P to F: Besides myself, only three.
• F to P: And what were their ages?
• 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.
• F to P: After much thought, I cannot answer.
• 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:
• 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.
• 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:
• P: I don't know Anton's numbers.
• S: I don't know them either.
• P: Hey, now I know.
• 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.

• 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?
• 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