Two really cool dice puzzles that have already been solved to death by bored mathematicians:
Nontransitive dice
Number three blank dice such that one of the dice (say die A) beats another of the dice (die B) more than half of the time, which in turn beats die C more than half of the time. Nothing unusual so far, right? Now make it so that die C beats die A more than half the time as well. Yes, put in simple terms, A > B > C > A. Mathematicians call these nontransitive dice.
Imagine the great bar bets that you can make with these: "choose any of these three dice; then I’ll choose one; we’ll roll 10 times, and whoever rolls a higher number more times wins $20–all you have to do is figure out which die is best."
For extra fun, design a set of dice that enable you to offer to take on two guys at once who each choose a die from a pool of 7 dice, and then you choose a remaining one that beats them both!
Try to come up with at least the 3 dice set on your own, and then check out the Tournament Dice article on Math Games for lots of fun information on both of the above schemes.
"Normal" dice with abnormal numbers
Next, number two 6-sided dice in such a way that if you roll them both and take the sum, it’s the same distribution of sums as that of two normal 6-sided dice (i.e. 1 way to roll a 2 or 12, 2 ways to roll a 3 or 11, and so on). It’s cheating to use non-positive numbers (otherwise, you could just add N to each face on one die and subtract the same N from each face on the other).
I found this one easier to contruct on my own than the nontransitive dice. Give it a shot and then see Wolfram’s Sicherman Dice write-up for more details on this.