Random Stuff

Another math puzzle.

You've got 1000 jars of milk, and exactly one of them is poisoned. You want to figure out which one is poisoned. What you have at your disposal is an infinite number of rats that you can feed the milk. You have 3 hours to figure out which one is the poisoned one, and the poison takes a nondeterministic amount of time between 2 and 3 hours to kill a rat.

The question is, what's the fewest number of rats you have to use to find out for sure which jar is poisoned?

Here's a trivial hint:
+ Show Spoiler +

This one's not too hard, but pretty cool imo.

There is a room with a cross-shaped table in it. The cross-shaped table can spin around, but it always stops spinning with the arms of the cross facing north, south, east, and west. (i.e. after every spin, the cross will be in one of 4 possible orientations). At the end of each arm of the cross is a cup, either right-side up or upside down, at random.

Now you are in a separate room and cannot see the table, while your friend is in the room with the table. You can command your friend to flip over certain cups by telling him which directions to flip (i.e. if you say flip over the north cup, then your friend will flip the cup on the north arm from right-side up to upside down or vice versa). After your friend flips the cups, he will either say you're done because all the cups are right-side up, or he will spin the table. After he spins it, you can once again give him another command, and the process continues until all the cups are right-side up.

Your goal is to flip all the cups right-side up. Figure out a way to do this in as few spins as possible, and prove that it is optimal.

Street Spirit is a newspaper with the subtitle, "Justice News & Homeless Blues In The Bay Area". In other words, it's a newspaper with news specifically related to the homeless community. They also have many poems interleaved among the news stories.

I thought I'd share a few of the poems from the March issue.


"On The Street"
by Julia Vinograd

I'm not here, don't look at me.
Don't scream when your children look at me,
do you think bad dreams are contagious?
Do you think I'll breathe bad luck on your children?
Suppose you'd spent the afternoon
shining in your lover's bed
and then brought him home to meet your parents
and they screamed
"Don't touch that, you don't know where it's been!"
You can't see me,
I'm just a crack in your contact lens;
I'm just a crack in your mirror.
My hands are dirty,
I've got your shadow under my fingernails,
I can't wash it out.
I've got the shakes,
my own skin isn't speaking to me
and I'm not speaking to you.
I'm not here, I have no past,
no memories, no name, not allowed.
So I eat your memories like garbage,
all your buried broken promises
and the bad dreams you forget
till you see me.
Till you don't see me.
I'm not here.


"Transcendence"
by Sue Ellen Pector

In surges and torrents it is hurled at me
that incessant scolding
"failure failure failure"
until I trust no one ever except
long lost loyal beloveds

who saw in me
a light
an innate glow
that still might
transcend the tirades of conformity.

While others - as if in lockstep -
see round holes rejecting
my square pegs
this body this self this Sue
soars, floats, sings.


"a desert with no oasis"
by Randy Fingland

the new ploy
seems to be
to me
put the poor
on the run
keep them
without possibility
of a real home
of their own
to come back to

nowhere to be
on/in/under
welcome:
yet still within
native geographical
boundaries

new refugees
set to wander
unprotected
in the country
of their nativity,
in this case the U.S.A.

The prisoners are back for more puzzles! And this time there's a light bulb in the mix.

There are 100 prisoners, each in their own isolated rooms. There's another special room with a light bulb and a switch, and nothing else. The light bulb is initially off.

One at a time, a prisoner is chosen at random to go into the special room. The prisoner can choose to either flip the switch or not. The prisoner will then leave the room and a guard will ask him whether he he thinks all 100 prisoners have been in the room at least once. The prisoner can either answer "yes" or "don't know".

If he answers yes and he's correct, all the prisoners go free.
If he answers yes and he's wrong, all the prisoners are killed.
If he answers don't know, then the prisoner goes back to his room and the process continues.

Before the prisoners are put in their rooms, they get to decide beforehand on a strategy. What strategy should the prisoners use to assure the survival of the prisoners?

Key Point: The prisoners can go into the room multiple times, and the order is random. For example, prisoner 1 could potentially go into the room 10 times before prisoner 5 goes in even once.

Do you dare to attempt it?

http://www.superliminal.com/cube/cube.htm

This program simulates a 4-dimensional hypercube. For those of you who think 3 dimensions is too easy, try and give this a shot.

Another puzzle guys. This one is kinda neat, but pretty simple.

You're playing a two-player turn based game. You have a circular table, 1 meter in diameter. You and your opponent take turns putting 1 centimeter pennies on the table. The pennies cannot overlap each other, and they must stay completely on the table.

If you cannot put down a penny on your turn, you lose the game.

Would you rather go first or second, and what would your strategy be?

Another math problem for you guys. As a side note, is it just me or are prisoners a popular choice for these kind of problems?


There are 100 prisoners in a room. It has been decided by the judge that their fates will be based on a game of luck. The rules of the game are as follows.

There is another room with 100 boxes in a line. Each box contains exactly one piece of paper with the name of one of the prisoners written on it. In some arbitrary order, the prisoners enter the room one at a time and proceed to open 50 of the boxes, in search of their own name. If a prisoner finds his name within 50 box openings, he has succeeded.

If all of the prisoners succeed in finding their own name, then they all get to live. However, if even one person fails to find their name, it's the death penalty for all of them.

The prisoners get to discuss beforehand what strategy they want to use, but there is no communication allowed after the game has begun.

If all the prisoners were to randomly pick 50 boxes, then the total survival chance of the group is (1/2)^100, which is laughably small. The question is, can you devise a strategy for the prisoners that at least grants then some reasonable chance of survival? What is their probability of survival under your strategy?

Points of clarification
-Every prisoner's name appears in exactly one box. There's no case of a prisoner's name being in two boxes.
-There is no swapping of papers among boxes or writing stuff on the paper, or anything that changes the papers or the boxes. The room of boxes is in the exact same state for every prisoner.


As a final note, I do know the solution strategy, but I do not know exactly what the probability of survival is. If you want a ball park figure as a hint:
+ Show Spoiler +

This one I think is pretty nifty, I hope you guys enjoy it. I feel like some of the parts might be confusing to understand, so please ask for clarification if needed.

Wile E. Coyote just got a shipment of an infinite number of bombs, and he's ready to blow up the Road Runner once and for all. However, the rules of the game this time around are a bit different from the usual.

You see, this world is one dimensional, so the Road Runner only runs in a straight line. Also, the world is discrete, and everything has integer values. The Road Runner starts at an integer location and moves at a constant integer velocity. Since time is also discrete, the Road Runner warps from his current location to his next location every second, instead of running continuously on the line.

The problem for Wile E. Coyote is that he doesn't know the Road Runner's velocity or starting location, but he can fire one bomb at any location, every second.

Devise a strategy for Wile E. Coyote that, given an infinite amount of time, assures that he will finally be able to blow up the Road Runner.

Duplicate Entry

The math buffs out there have probably read some variation of this story before. It's a story that somewhat explains the nature of infinity as it is defined by mathematics.

So there exists this place called the Hotel Infinity. It's a hotel that has an infinite number of rooms. However, currently all the rooms are full. A man comes in and asks the desk manager, "Do you have any rooms available for me?" The desk manager replies, “All our rooms are full, but I'll see what I can do."

The manager thinks for a while, and then grabs the intercom and makes the announcement to all the guests, "Attention, if you are currently in room N, please move to room N+1." So the guest in room 1 moved to room 2, the guest in room 2 moved to room 3, etc. After the shift, the manager moved the man into the newly vacant room 1.

The next day, another man comes in and asks the desk manager, "Do you have any room for me and my infinite friends outside?" The desk manager once again replies, "All our rooms are full, but I'll see what I can do."

So the manager again thinks for a while and then makes an announcement to the guests, "Attention, if you are currently in room N, please move to room 2*N." So the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, etc. After the shift, all the odd numbered rooms became vacant, and the manager was able to accommodate all of the man's friends.

On the third day, another man comes in and tells the desk manager, "Outside, I have an infinite number of buses, each with an infinite number of people. Do you have any rooms for us?" The desk manager once again replies, "All our rooms are full, but I'll see what I can do."

So once again after some thought, the manager announces to the hotel, "Attention, if you are currently in room N, please move to room 2^N." So the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, etc. The manager then proceeds to label each of the infinite buses with a different prime number, starting from 3 and going up. Because there are infinite prime numbers, every bus can get a number. The people inside each bus are also counted off starting from 1 and going up. The manager then tells all the people to go to room B^N, where B is their bus's number, and N is their personal number. Once again, the manager was able to accommodate all of the guests.

So the moral of the story is, infinity is pretty awesome.

There was another variation of the third example I heard that, instead of using prime numbers, uses the fact that the 1-dimensional integer plane and the 2-dimensional integer plane are of the same cardinality.

Basically, you can just enumerate all the buses using natural numbers, and enumerate the people in each bus using natural numbers as well. The bus number is the x-coordinate and the person number is the y-coordinate. You then map the corresponding (x,y) pair to some room number and you're done. The interesting part of course is, how you map 2-d to 1-d?