I pick an arbitrary number between $-\infty$ and $+\infty$. In addition, I pick another "jump" number. At each turn, a person needs to guess what my current number is. The only answer for each guess will either be true or false (no more hints are given). For every wrong guess, I update my current number by adding the jump number.
Can you please suggest a winning strategy for this game?
First, we create a surjection $f:\mathbb{Z}\to\mathbb{Z}^2$. An explicit example can be given by “spiraling” around an infinite square grid.
Now, at turn $n$ (indexed at zero), we calculate $f(n)=(s,j)$, and guess the number $s+jn$. We’ll guess correctly if the starting number was $s$ and the jump number was $j$, or if we simply got lucky.
Since we’ll try every single starting number and jump number, we’re guaranteed to eventually guess correctly.