You have nightmares about a pool ball with the number 1 on it, and an empty box. Why?
Imagine, if you will, we are 5000 years into the future. Medicine has evolved and we now are immortal - with one exception, you can be executed and put to death.
Times are much different in 7020. Everyone is required daily to play a game. This game is a matter of life or death.
The game is simple. You choose a pool ball from a box. You then have that ball replaced with other balls with a number lower than the one you chose. If you choose a 1, however, you can't replace it. If the box ends up empty, you die. [Classic Game Dating Back early 1900's]
Each pool ball in the box has a number on it ( a positive integer) and there is an infinite supply. That means there are an infinite number of pool balls with the number 50 on it, as there are with an infinite number of balls with the number 1000 on it.
In a particular game box there are finitely many of these balls, but the box has infinite capacity - it can continue being filled with no limitations.
Each day you are required to choose a ball, discard it, and replace that ball in the box with any number of lower numbered balls. As many as you want. If you choose ball number 44, you can replace it with a million balls with 43 on it. Or a billion. Whatever. Your choice.
If you choose a ball with a 1 on it, you can't replace it with anything. Lost your turn, sort of. It still gets discarded, though.
If the box ever empties, you are executed.
You might as well keep a record of chosen numbers, this will probably be a long game. :)
What is your strategy by which you never get executed, or is death inevitable sooner or later? Please explain your answer.