This is an old problem I saw almost 20 years ago: suppose you took two dice and rubbed off the pips (dots) from the faces, and instead put on numbers. How would you number it in such a way that you can roll the two dice and make all the numbers from 1 to 36?
This is a wonderful problem to study combinations, patterns and general problem solving techniques. It is "hard" in that you can't calculate your way through it, and the solution evolves slowly as you work through the problem. But the solution is really pleasing, and many students are surprised that it is possible.
But the fun doesn't stop there, because once your students finish this problem, they can look at other hypothetical situations: suppose we tried the same thing with two dice with 5 faces on each? Or what about a pair of dice with 7 faces? The solution uses the same techniques for 6 faced dice.
But that's not all! What if, instead of re-numbering a pair of 6 faced cubes, we used 3 of them instead? How far could we go? The solution is closely related to the original problems, and uses some interesting mathematics based on "bases" as well as multiples.
Finally, there is an added challenge: instead of three identical dice, suppose they had different numbers of faces? How would that change the outcome of the problem?
This activity includes extensive documentation on how to use this problem in your classroom, how to support diverse learners, and how to link this to other areas of mathematics. It's truly a fun and interesting problem and will engage your students from 5th grade upwards. You could also try it out on younger students who like a good challenge.
Another fine and provocative project from SamizdatMath!