Cool Board-Gaming "Tools"

I recently saw something cool about choosing which player should go first in a game.  A lot of people roll dice, though ties can happen.  One dice enthusiast came up with a set of four dice that can be used to determine who goes first among two players, with the awesome properties that:

  * The dice only need to be rolled once (no ties), and

  * Each die has an equal chance of being the highest among the four.

To do this, the dice cannot have any ties.  Four twelve-sided dice are used, so the numbers 1 - 48 must be on them.  The more complicated part is how to arrange the numbers so that each player has a 1/4 probability of winning the roll.

Is there a way to generalize this?  If you have n players, what size dice do you need?  (How many sides?)  If you have access to up to 20-sided dice, how many players does this support?  For four players, are 12-sided dice the smallest you need?

It seems like this must all be proven somewhere.

Another awesome thing I saw was this gaming shed someone built, mostly to house all their board games, but also with a place to play them.  That forum post shows all the steps to create the building.  Very cool!  I am extremely jealous!  Someday perhaps I'll have the funds to copy this exact thing. 

There are many other things I desire, but these two are definitely worth sharing.