2. A kitten bites off a quarter of a sausage from one end, after which a puppy bites off a third of the remaining piece from the opposite end, then the kitten again - a quarter from its end, and the puppy - a third from its end, and so on. It is required to tie a thread around the sausage in advance so that no one eats the thread. In what ratio should it divide the sausage?
Answer: $1: 1$.

Solution. Let $\ell$ be the length of the sausage remaining after the same number of "bites" by the kitten and the puppy. Then on the next step, the kitten will bite off $\frac{\ell}{4}$, and the puppy will bite off $-\frac{1}{3} \frac{3 \ell}{4}=\frac{\ell}{4}$. This means both will eat the same part of the sausage.