Puzzle: Give an algorithm for finding a frog that jumps along the number line You are playing a game, your goal in this game is to catch a frog that's leaping between natural numbers. At first, the frog is found at the number $a \in \mathbb N$ which is not known to you. Each turn, you take a guess at where the frog is found. If you are right - you win. If you are wrong - the frog leaps $b \in \mathbb N$ numbers to the right. Meaning, if you got the first guess wrong, the frog is now at $a+b$. If you get the guess wrong again, it's now at $a+2b$. Neither $a$ or $b$ are known to you. All you know is that they are natural numbers. Propose an algorithm that will find the frog in a finite number of steps, regardless of what $a$ and $b$ are. Additional challenge: Same question, but now $a,b \in \mathbb Z$.

For fixed $a$ and $b$, after $m$ steps, the frog will be at $a + mb$ if you haven't guessed right yet. The set of pairs $(a, b) \in \mathbb{N}\times \mathbb{N}$ is countable, so you can enumerate it, say as $(a_0, b_0), (a_1, b_1), \ldots$. Now guess $a_m + mb_m$ at step $m$ and you are bound to guess right eventually.