How Many Rocks Will Be in the Hole at 12:00?

[Jim Kata]

Say it's 11:00 and you throw 100 rocks in a hole and at the exact same second a rabbit throws one rock out of the hole. Then at 11:30 you throw a 100 rocks in and the rabbit throws one rock out. At 11:45 you throw 100 rocks in and the rabbit throws one rock out and so on, with the time increments halving each time. How many rocks will be in the hole at 12:00? Well one thought is that you've thrown a countably infinite number of rocks in and the rabbit has thrown countably infinite number out so the answer is 0, but say you threw 99 rocks in the hole each time and there was no rabbit then obviously the answer is infinity. Where is the contradiction?

 

[DaveC426913]

Well one thought is that you've thrown a countably infinite number of rocks in and the rabbit has thrown countably infinite number out so the answer is 0

You assume countably-infinite-X minus countably-infinite-Y is equal to zero, if X=Y.

Is that a valid assumption?

 

[Jim Kata]

I would have to assume that to get an answer of zero.

 

[Office_Shredder]

It depends on the rabbit's algorithm for throwing rocks out of the hole.

Suppose that at each step he throws out the rock most recently thrown into the hole. Then at each step you throw 99 rocks into the hole which are never removed by the rabbit, and you will have infinite rocks in the hole at the end.

On the other hand suppose that at each step he takes the rock which is on the bottom of the pile (the rock which was thrown in before any other rock currently in the hole) and removes that one. Then every rock which is thrown into the hole is removed by the rabbit, and you will end up with no rocks in the hole

 

[blue_raver22]

I would approach this scenario by first clarifying the parameters and variables involved. It is important to note that the time increments are halving each time, and the number of rocks thrown in by the human and the rabbit remains constant. With this information, we can calculate the total number of rocks in the hole at 12:00.

Assuming that the rabbit throws one rock out for every 100 rocks thrown in, the total number of rocks in the hole at 12:00 would be equal to the sum of all the rocks thrown in minus the sum of all the rocks thrown out. This can be represented mathematically as:

(Rocks thrown in at 11:00 + Rocks thrown in at 11:30 + Rocks thrown in at 11:45 + ...) - (Rocks thrown out at 11:00 + Rocks thrown out at 11:30 + Rocks thrown out at 11:45 + ...)

Since we know that the time increments are halving each time, we can express this as a geometric series with a common ratio of 1/2. Using the formula for the sum of a geometric series, we can calculate the total number of rocks in the hole at 12:00:

(Rocks thrown in at 11:00 * (1 - (1/2)^6)) - (Rocks thrown out at 11:00 * (1 - (1/2)^6))

= (100 * (1 - (1/2)^6)) - (1 * (1 - (1/2)^6))

= (100 * (1 - 1/64)) - (1 * (1 - 1/64))

= (100 * (63/64)) - (1 * (63/64))

= 99 rocks in the hole at 12:00.

Therefore, the number of rocks in the hole at 12:00 is not infinite, but rather a finite number of 99 rocks. This is because the time increments halving each time leads to a finite number of rocks being thrown in and out by both the human and the rabbit.

In regards to the perceived contradiction, it is important to note that the concept of infinity is often misunderstood. While it may seem that throwing an infinite number of rocks in and an infinite number of rocks out would cancel each other out, this is not the case. Infinity is not a number, but rather