Generating Functions or Counting?

Question

Alice has two bags. Each bag has $4$ slips of paper with the numbers $1$ through $4$ on them.

Betty also has two bags, each with $4$ slips of paper with positive integers on them. They decide to play a game whereby each girl pulls a slip from each of her own bags, records the sum of the numbers, then returns each slip to the bag it came from. The numbers in Betty's bags are not $1$ through $4$ in each bag, but the expected distribution of her sums is the same as Alice's. What are all eight numbers in Betty's bags?

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Would a way to solve this problem be using generating functions? Counting?

Answer 1

Neither. Start by computing the frequency distribution of sums for Alice:

$$\begin{array}{rccc} \text{sum}:&2&3&4&5&6&7&8\\ \text{frequency}:&1&2&3&4&3&2&1 \end{array}$$

Betty has the same distribution, and the slips of paper in her bags have positive integers written on them.

• Show that each of Betty’s bags must contain exactly one slip with the number $1$.

• Show that if each of Betty’s bags contained a slip with the number $2$, then her bags would be a copy of Alice’s which is not allowed. Conclude that one of Betty’s bags contains two slips numbered $2$, and the other contains no slips numbered $2$.

At this point we know that one of Betty’s bags contains a $1$ and no $2$s, and the other contains a $1$ and two $2$s, so we have totals of $2,3$, and $3$, and the smallest remaining number in either bag is a $3$.

• Show that in order to get the sum $4$ with frequency $3$, we must have two $3$s in one bag and one $3$ in the other. Which bag is which?

• Now complete the assignment of numbers to Betty’s bags.

Answer 2

Generating functions, I think.

The generating function for Alice’s outcomes is $g(x)=(x+x^2+x^3+x^4)^2$. Can you find polynomials (one for each of Betty’s bags, which have $N$ and $M$ slips in them, respectively) $b_1(x)=\sum\limits_{i=1}^{N}x^{n_i}$ and $b_2(x)=\sum\limits_{i=1}^{M}x^{m_i}$ where $g(x)=b_1(x)\cdot b_2(x)$ and the sets $\{n_i\}_{i=1}^N$ and $\{m_i\}_{i=1}^M$ are sets of positive integers that are different from $\{1,2,3,4\}$?