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I have 3 trasparent Containers A, B and C which can contain water.

It is possible to have an negative amount of water.

Container A start off with a random amount of water .

Container B starts of with a fixed amount of water which never changes.

Container C starts off empty.

Every second I take a bucket and fill it with the exact amount of water which is in Container B,

. . and then pour it into Container C.

Every second I take a bucket and fill it with the exact amount of water which is in Container C,

. . and then pour it into Container A.

. . and so on.

After seconds, how much water is in container A?

Note that the water istransferred from container to container.not

The additional water comes from an outside source.

Here is what I found . . .

The containers start like this:

Code:| | | | | | | | | | | | | | | | | | | | t = 0 | | | | | - - - | | | | | |:::::::| | - - - | | | |:::a:::| |:::b:::| | | * - - - * * - - - * * - - - * A B C

is added to container C.

Code:| | | | | | | | | | | | | | | | | | | | t = 1 | | | | | - - - | | | | | |:::::::| | - - - | | - - - | |:::a:::| |:::b:::| |:::b:::| * - - - * * - - - * * - - - * A B C

Then is added to container A.

Code:| | | | | | | | | | | | | | | | | | | | t = 2 | - - - | | | |:::::::| | | | | |::a+b::| | - - - | | - - - | |:::::::| |:::b:::| |:::b:::| * - - - * * - - - * * - - - * A B C

is added to Container C.

Code:| | | | | | | | | | | | | | | | | | | | t = 3 | - - - | | - - - | |:::::::| | | |:::::::| |::a+b::| | - - - | |::2b:::| |:::::::| |:::b:::| |:::::::| * - - - * * - - - * * - - - * A B C

Then is added to Container A.

Code:| | | | | | | | | | | | | - - - | | | |:::::::| | | t = 4 |:::::::| | - - - | |::a+3b:| | | |:::::::| |:::::::| | - - - | |::2b:::| |:::::::| |:::b:::| |:::::::| * - - - * * - - - * * - - - * A B C

is added to Container C.

Code:| | | | | | | | | | | | | - - - | | | |:::::::| | - - - | t = 5 |:::::::| |:::::::| |::a+3b:| | | |:::::::| |:::::::| | - - - | |:::3b::| |:::::::| |:::b:::| |:::::::| * - - - * * - - - * * - - - * A B C

Then is added to Container A.

Code:| | | | | - - - | | | |:::::::| | | |:::::::| | | |:::::::| | - - - | t = 6 |::a+6b:| |:::::::| |:::::::| | | |:::::::| |:::::::| | - - - | |:::3b::| |:::::::| |:::b:::| |:::::::| * - - - * * - - - * * - - - * A B C

We see a pattern developing.

. .

The coefficients of are triangular numbers: .

If time is annumber of seconds,even

. . then: .