### There are two identical glasses, one glass is half-full, and the other is half-empty?

Half of water from first glass was poured into the second.

Then, half of water from second glass was poured into the first.

This sequence of steps was continued endlessly.

How much water is poured on each step after sufficiently long time?

PoPCorn

on November 30th, -0001

1/3 of a glass.

After a long time, the glasses will have ?2/3 and ?1/3 in each, and then switch in the next step.

After n pours, the fuller glass has:

(1 – 1/2) + 1/4 – 1/8 + 1/16 – … + 1/(-2)^(n+1)

and the emptier glass has:

1/2 – 1/4 + 1/8 – 1/16 + … – 1/(-2)^(n+1)

The limit of these is 2/3, 1/3

Orchid

on November 30th, -0001

1/3 of a glass I think.. that’s when they reach equilibrium

oddperson

on November 30th, -0001

I’m thinking 1/3

Hippie

on November 30th, -0001

1/4 –> 3/8 –> 5/16 –> 11/32 –> 21/64 –> 43/128 –> 85/256

If you look at the even terms, they approach 1/3 from below. If you look at the odd terms, they approach 1/3 from above.

For example:

(2^(2n + 1) + 1) / 3

————————- will approach 1/3 as n –> infinity.

… 2^(2n + 1)

The glasses will approach being 1/3 full and 1/3 empty. 🙂