Problem of the Week

Suppose you roll two dice, each in the shape of a tetrahedron (i.e. a pyramid).  If the faces of the dice are labeled with the numbers 1, 2 , 3, 4, then the probability of rolling the number n is given in the following table.

Now, if you relabel the faces of one die with the numbers 1, 2, 2, 3 and the other with 1, 3, 3, 5 you get precisely the same table of probabilities!  (Check this out before going on.) These are very bizarre dice.  

(a)  With a pair of standard dice -- in the shape of cubes and labeled  1, 2, 3, 4, 5, 6 -- what is the table of probabilities for rolling a number n?

(b)  If at all possible, how can you create a pair of bizarre dice on a cube by relabeling the faces of the cubes with positive integers so as to get the same table of probabilities? 

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ã2003 Alberto L. Delgado