## Task 30/61

Five housewives want to buy rolls. After counting the available rolls, the baker allows himself a joke: "If each of you buys half of the currently available rolls plus half a roll, none will be left!" How many rolls did the baker have, and how many would each of the customers have received according to this suggestion?
Let $x_{n}$ be the number of Schrippen available before the $n$-th customer buys. The $n$-th customer then receives

$$
y_{n}=\frac{x_{n}}{2}+\frac{1}{2}=\frac{x_{n}+1}{2}
$$

Schrippen. The difference $x_{n}-y_{n}$ is the number of Schrippen remaining after the $n$-th customer has bought, which is before the $(n+1)$-th customer buys:

$$
x_{n+1}=x_{n}-y_{n}=x_{n}-\frac{x_{n}+1}{2}=\frac{x_{n}-1}{2}
$$

Since no Schrippen are left after the fifth customer, we have $x_{6}=0$, thus $\frac{x_{5}-1}{2}=0$, hence $x_{5}=1$ and $y_{5}=1$. Working backwards and substituting the results, we get $x_{4}=3 ; x_{3}=7 ; x_{2}=15 ; x_{1}=31$ and $y_{4}=2 ; y_{3}=4 ; y_{2}=8 ; y_{1}=16$.

Therefore, there were initially 31 Schrippen. The first customer receives 16, the second 8, the third 4, the fourth 2, and the fifth 1 Schrippe. "Half" Schrippen do not appear in the result!