Math game

[Ravenshire]

The opening is at the bottom of #2, there's a shorter distance to (top to bottom) to #8 than to make it back and up to #3.

Fluid could run equally left and right under #2 but would dump out into #8 before it could ever back up to #3.

The distance to 8 may be shorter (and easier as it's not uphill), but I still don't see how therefore ultimately 13 is the bucket that will be filled first. Unless Bucket 3 never gets touched until Bucket 8 is completely filled? If that's the case, then I have a visualizing problem here, because I'm thinking that some liquid will seep up towards 3 while Bucket 8 is still not completely filled.

Liquid will always flow down, following the path of least resistance. It will flow down from 1 to 2 to 8, until 8 fills to the level of the lowest opening. It will then continue down, backfilling the pipe that's closed off at 9. Once the liquid reaches the pipe into 7 it will fill 7 to the same level as the bottom of the highest section of outflow pipe. After that it fills 6 to the level of the outflow pipe, which in turn fills 5 to the level of it's outflow pipe. After leaving 5, the liquid flows to 13 and the only pipe not blocked is the pipe to twelve who's highest point can't be reached because 13 will overflow before liquid can flow into 12.

4 will never get wet.

This is, of course, assuming zero back-pressure and open topped tanks (as they are shown).

It's the same reason the upstairs bathroom sink will never get drain back flow from a clogged sewer main, even if the basement's flooded.