It turns out that the blue men sing between noon and midnight and sleep between midnight and noon while the green men sleep between noon and midnight and sing between midnight and noon. Complaints.
The manager decides to group them. Conveniently, the rooms are in a straight line, numbered from left to right. While there is a green man left to a blue man, he makes them change the rooms.
Eventually, all the blue men leave.
- how many rooms are free?
- how many rooms remain occupied?
- what is the number of the first occupied room?
Prisoners are seated in a circle so they can see all the others. This time the warden flips a fair coin for each prisoner and gives him a yellow or green hat, accordingly. Once all the hats have been placed, and have been seen by the others, prisoners are taken aside singly and given the opportunity to guess the color of his hat. And if instead he chooses not to guess, he is permitted to pass.
Now comes the bad part. Unless at least one prisoner guesses, and all the prisoners who do guess are correct, all the prisoners will be executed. That's right, survival requires perfection from every prisoner who guesses his color.
Prisoners decide on a strategy beforehand, and after the first hat is placed there is no further communication. Clearly, there can be a single "designated guesser" who ... just ... guesses a color. Half the time they all survive. But what kind of a puzzle would that be? Yes, incredibly, the prisoners can do much better. How? Maybe thinking about a three-prisoner case will answer that question.
Once you're convinced they can do better than a coin toss, find their best strategy.
]]>But they were instructed at each event to remove a coin from their respective boxes and discard it. After some thought, Al decided each time to discard his lowest-numbered coin; Bert discarded an even-numbered coin; and Charlie thought what the heck and discarded a coin selected at random. Regardless of strategy, at each event the number of coins in each box grew by unity, so that after N events each box held N coins.
Needless to say when midnight struck their arms were infinitely tired, but it was a small price to pay for infinite riches. But tell us, now, whether their expectations were met.
Describe the contents of each box at midnight.
]]>Here's the puzzle I had intended to post:
You have just lost your 143^{rd} straight game of checkers and have vowed never to play another game. To confirm your vow you decide to saw your wooden checkerboard into pieces that contain no more than a single (red or black) square. With each use of the saw you may pick up a piece of the board and make one straight cut, along boundaries of individual squares, completely through to the other side. You wish to inflict as much damage as possible with each cut, so you first calculate the minimum number of saw cuts needed to finish the job. And that number is ... (spoilers appreciated.)
]]>On average, what fraction of the conveyor belt is not bounded by near-neighbors?
Example:
----- belt segment bounded by near neighbors
===== belt segment not bounded by near neighbors
... --A-----B==========C---D--------E=============F---G-H---I-- ...
]]>
EDIT: for clarification
Why is the total length of the path just 1m?
]]>+ + + + + + + + + + + +
+ + + + + + + + + + + +
+ + + + + + + + O + + +
+ + + O O + ==> + + + O O +
+ + + O O + + + + O + +
+ + + + + + + + + + + +
Is it possible to maneuver the pegs to the corners of a larger square?
+ + + + + +
+ O + O + +
+ + + + + +
+ O + O + +
+ + + + + +
+ + + + + +
]]>
Carroll: Is the total number of coins an even number?
Moriarty: No.
Kurt: Is the total number of coins a prime number?
Moriarty: No.
If Kleene has ﬁve coins in his cell, what question should he ask Moriarty in order to ensure that at least one of the logicians work out the total number of coins in the cells?
Eventually there will be N clusters of cars. What is the expected value of N? (Equivalently, what is the expected cluster size?)
]]>
If one of the small cubes is omitted, four distinct shapes are possible. If two of the small cubes are omitted rather than just one, twenty-two distinct shapes are possible (see previously submitted Cubicle Stack at BrainDen.com).
Now, if three of the cubelets are omitted, how many distinct shapes are possible?
]]>
What is the probability of their meeting (1) if they walk at the same speed, or (2) if Al walks 3 times as fast as Bert?
]]>
He asked, if the thing on the left is a centimeter, what is the thing on the right?
Your turn (spoilers please.)
]]>Example:
The four sides and two diagonals of a square.
]]>
What set of roads minimizes that cost?
A B
C D
]]>Suppose the starting number of jelly beans distributed among three plates is a sufficiently nice multiple of 3, namely b = 3x2^{n}. By making successive doubling moves, as in the first puzzle, is it always possible to end up with an equal number ( 2^{n} ) of jelly beans on the three plates?
]]>For example, consider 123654987. Its first 2 digits (12) are divisible by 2. It's first 5 digits (12365) are divisible by 5.
However this is not a solution, since 1236549 is not a multiple of 7.
]]>Euler, the great mathematician, proposed the task of constructing a similar 6x6 array, but instead it was proven to be impossible.
Does a 5x5 array exist?
]]>Clarifications:
Prove there is a least amount of work you must do (smallest m) that makes Bob complete the job.
]]>