Dice Probability Riddle

A math riddle of sorts:

Here's the riddle my friend told me:

You and I roll a die each.  We can see one another's, but we cannot see our own.  We can communicate before, but not after we roll.  What is the best probability we can have that we will both guess our own die correctly?

It's not 1/36 - it's actually 1/6.

Some days later while talking with my dad, i thought of new extrapolations of the riddle.  I sketched the simple cases on paper...

...but then it got hard to visualize, so I made a sketchup model of my thought process.  It looks a bit obsessive, but I was motivated to make the video because I wanted to show that you can think about riddles like this quickly in 3D space.

Sorry it's kind of hard to read.  If you want just skim through it to get an idea of the process.