Friday Puzzle #7

20 10 2011

A train approaches an observer at the speed of 100 km/h.  The observer tosses a tennis ball at the speed of 100 km/h towards the train.  What will be the speed and direction of the tennis ball after it has bounced off of the collision with the train.  All idealizations apply (bouncing consumes no kinetic energy; front of train is flat and vertical; friction is ignored; train/ball weight ratio is practically infinite; etc.)

 





Friday Puzzle #6

6 10 2011

A chess board has 8*8 squares with alternating black/white colors. The rows are labeled 1-8 and the columns are labeled A-H. The square A1 is black.

However, for the purpose of this week’s friday puzzle, let’s re-label the rows and columns with numbers starting from zero. Then, square (0,0) is black, as well as (1,1), (2,2), etc. Also, the chess board is not restricted to 8*8 squares, it can be arbitrarily large (yet finite), or N*N in size.

Then place pawns on the board according to the following rules:

  • Square (0,0) contains a pawn
  • If and only if square (0,X) contains a pawn, then square (X,0) contains a pawn
  • If and only if both squares (R,0) and (0,S) contain a pawn, then square (R,S) contains a pawn

When the pawns are correctly configured the total number of pawns will be a square number, and they will form a diagonally symmetric pattern. Some pawns will occupy a white square and others will occupy a black square. Call their difference the target.

The puzzle: What is the smallest board where a pattern can be created whose target is a prime number?





Friday Puzzle #5

29 09 2011

A bottle has diameter of 1 unit.  A box having size 2*2 units can contain four bottles at most.  A box having size 2*3 units can contain six bottles at most. A box having size 2*4 units can contain eight bottles at most. Et cetera…

Is there a box having size 2*N units that can contain more than 2*N bottles? Can you suggest an upper limit for N? Can you compute and prove (or at least justify) a minimum value for N?





Friday Puzzle #4

22 09 2011

Set is a card game requiring good perception and quick thinking. It is played with a deck of 81 special cards. Each card has four properties, and every property has three different values. There is exactly one card for every combination of property values, hence the size of the deck is 3^4 = 81. A “set” is three cards for which all four properties have either the same value, or all have a different value. A more elaborate description of the game can be found here.

What is the probability that three randomly chosen Set cards form a “set”?





Friday Puzzle #3

15 09 2011

There is a horizontal plane which has two vertical axles that are 1 unit apart. Both axles have a rotor with two blades, each 1 unit long (hence the assembly of two blades has a diameter of two units). Both rotors can rotate freely without touching each other – but just barely, because the length of a rotor blade equals the distance between the axles. When the rotors are spun, the friction gradually slows them down until eventually they each stop at a random angle.

An attempt to visualize follows:

When both rotors have stopped, what is the probability that their blades overlap?

Again, post your answer and reasoning as a comment to this article. Lengthy and tedious integrals are fine ;-) but let me hint that the puzzle has a very neat intuitive solution.





Friday Puzzle #2

8 09 2011

There’s a departure gate at an airport. Outside the gate there is a plane that has 256 passenger seats. At the gate lobby there is one silly old lady, one grumpy old man, and 254 energetic businessmen waiting to board the plane. Every passenger has a ticket that indicates their assigned seat.

Silly old lady is a bit absent-minded. When it comes her turn to enter the plane, she randomly chooses an empty seat and sits down, regardless of what her ticket indicates.

Energetic businessmen are efficient. They find their assigned seat easily and sit down after they’ve found it. But because they are also polite, they will randomly choose some other seat if they find that their assigned seat is already occupied by someone else.

Grumpy old man is… well… grumpy. He sits tight in the lobby while others stand in the line, and only after everyone else have boarded will he go into the plane as the last passenger. And boy, won’t he raise hell if he sees someone occupy his seat when he comes into the plane. I mean, after all, he’s paid full price for the flight and has the right to demand flawless service.

And so the boarding begins.

What is the probability that the grumpy old man was satisfied with the service after the boarding is over?

Post your answer and reasoning as a comment. I will post the correct answer by some time on Sunday if not already answered.





Puzzle: Guess What I’m Thinking!

7 09 2011

I’m thinking one of numbers 1, 2, or 3.  I’ll truthfully answer with “yes” or “no” to one question, provided that the question can be so answered.  You want to know the number I’m thinking; what will you ask?