Introduction to Generating Functions

Transcription

1 Sicherman Dice Problems The purpose of this essay is to introduce the ideas of a generating function to students who have not seen it and to use these ideas to investigate, with exercises and problems, the very nice discovery of a labeling of a pair of dice that yields exactly the same probability distribution as the usual labeling. The subject of Sicherman dice began with a question posed and answered by Colonel George Sicherman which was originally reported in the February 978 issue of Scientific American. Sicherman wanted to determine if there was another way to label a pair of six- sided dice, using only positive integers such that the probabilities of the sums would be the same as a standard dice. Before we get started on problems, let s recall an idea most readers of this essay know already, the area model for multiplication. This model has huge advantages over the commonly taught FOIL method for multiplying pairs of linear polynomials. Suppose we had the problem We can build a 2 by 23 unit rectangle and then compute its area:

2 Exercise. Use the area model to build the product 9. What the reader may not be as familiar with is the possibility of understanding polynomial multiplication using the area model. For example, suppose we want the product (x 2 x + )(x 2 + x + ). 2

3 x 2 x x 2 x x 3 x 2 x x 3 x 2 x x 2 x Now collecting terms of the same type, the product is x + x 2 +. The set of problems offered here will enable the reader to find the Sicherman labeling and also to learn about the very nice connection that enables the counting of certain kinds of objects using polynomials. Thus, this essay can be considered an introduction to generating functions. A generating function is a polynomial f(x) = a n x n + a n x n + + a x + a 0, where a i is the number of objects of type i to be counted. For a regular die, given by the net we have six types of objects each occurring once. So the 6 generating function is the polynomial f(x) = x + x 2 + x 3 + x + x 5 + x 6. If we 3

4 had the die labeled 3 5 5, we would have the generating function 5 g(x) = 2x + x 3 + 3x 5. What s so great about generating functions? Its the way they behave under multiplication. Suppose we want to find the distribution (histogram) of sums of the two dice described above. Notice that the product f(x)g(x) can be computed using the area model we saw above. x x 2 x 3 x x 5 x 6 2x 2x 2 2x 3 2x 2x 5 2x 6 2x 7 x 3 x x 5 x 6 x 7 x 8 x 9 3x 5 3x 6 3x 7 3x 8 3x 9 3x 0 3x Collecting terms of the same type, we get f(x)g(x) = 2x 2 +2x 3 +3x +3x 5 + 6x 6 + 6x 7 + x 8 + x 9 + 3x 0 + 3x. Notice that the sum of the coefficients of the product polynomial is 36, just as we have 36 distinguishable outcomes, assuming we can distinguish the two s and the three 5 s on the second die. In fact, a generating function is an algebraic device in which a sequence a n, n = 0,..., is coded as an infinite series in a variable x. The algebra of the series can then be used to get useful information about the sequence. Formally, the generating function f(x) of a sequence a n is the infinite series f(x) = a k x k = a a x + a 2 x 2 +. k=0 The sequence a n is called the sequence of f(x). Exercise 2. Let a n = n and b n = ( ) n. Let these sequences have generating functions f(x) and g(x), respectively. Compute the product h(x) = f(x)g(x) and find the first six terms of h(x).

5 Standard Dice The matrix shows the number of ways a pair of dice can yield the sums 2 through The matrix has 36 equally likely sums, so the probability distribution of the sum of the dice is given in the table below /36 2/36 3/36 /36 5/36 6/36 5/36 /36 3/36 2/36 /36 So Sicherman s problem can be stated as follows. Find a pair of different labelings of two dice that have the same probability distribution as that above. 2 Exercises. Elizabeth has 3 cubical dice, one numbered,, 2, 2, 3, 3, one numbered 2, 2,,, 6, 6 and one numbered,, 3, 3, 5, 5. She rolls all three simultaneously. What is the probability that the sum of the three faces is odd? What if she rolls two dice of each type? 2. Suppose now that Elizabeth has three spinners like the ones shown below. 5

6 Use polynomial multiplication to find the histogram of values of the sum of the three spinners. IE, complete the table below Build your own pair of dice with wooden cubes. Use any positive integer values for the faces. The dice can be different. Then build the generating functions for each die and finally compute the product of the two generating functions.. Next consider the generating function f(x) of an ordinary die. Its just f(x) = x + x 2 + x 3 + x + x 5 + x 6, so we can pretty quickly arrive at f 2 (x) = x 2 +2x 3 +3x +x 5 +5x 6 +6x 7 +5x 8 +x 9 +3x 0 +2x +x 2. If we can factor this into a pair of polynomials each with sum of coefficients different from those of f, we will know how to build the Sicherman dice. We saw above that x + x 2 + has a rather nice factorization, namely x + x 2 + = (x 2 x + )(x 2 + x + ).. Hence we can write x 6 + x 5 + x + x 3 + x 2 + x = x 2 (x + x 2 + ) + x(x + x 2 + ). 6

7 This leads to (x 6 + x 5 + x + x 3 + x 2 + x) 2 = (x 2 (x + x 2 + ) + x(x + x 2 + )) 2 = (x(x + )) 2 (x + x 2 + ) 2 = x 2 (x + ) 2 (x 2 x + ) 2 (x 2 + x + ) 2 = x(x + )(x 2 + x + ) x(x + )(x 2 + x + )(x 2 x + ) 2 = (x + 2x 3 + 2x 2 + ) (x 8 + x 6 + x 5 + x + x 3 + x) which tells us that we can label the dice as follows: This solves Sicherman s problem and 7