Balls ! And Locks

[michaellevenson]

One minute before noon , you have an INFINITE number of ping pong balls outside a room, all numbered . You throw in balls 1 +2, immediately ball 1 is thrown back out. Half a minute before noon 3+4 are thrown in and number 2 thrown out. At a quarter of a minute to noon balls 5+6 are thrown in and number 3 is thrown out. At an eighth of a minute to noon 7+8 in and 4 out ad infinitum.... How many balls are in the room at Noon ?

 

[michaellevenson]

Spoiler: the answer

There are three possible answers, all plausible. 1. All the balls are thrown in and out before noon, so noon is actually outside the scope of the puzzle, so how many at noon is indeterminate. 2. However if you really could throw an INFINITE number of balls in, then either they'll be an infinite number in the room at noon or, 3. NONE AT ALL, because an infinite number would have been thrown out. Half the number are thrown out, but half of infinity is a meaningless concept. The set up of the puzzle itself is impossible.

 

[michaellevenson]

Suppose you want to send in the mail a valuable object to a friend. You have a box which is big enough to hold the object. The box has a locking ring which is large enough to have a lock attached and you have several locks with keys. However, your friend does not have the key to any lock that you have. You cannot send the key, your friend cannot send you a key as there is too much risk of these being lost, stolen or copied. How do you send the valuable object, locked, to your friend - so it may be opened by your friend.

 

[michaellevenson]

Spoiler: the answer

You place the object in the box and lock it with a lock that your friend can't open. Send it to your friend, who can't open it, but puts another lock on it, and sends it back. When getting it back you remove your lock and send it back again. It will then have only your friend's lock on it.