Jack and Jill are playing a game. A natural number $m$ is fixed at the start of the game (say, by rolling a die or using a random number generator). Game play goes as follows:
- Jack picks a natural number $x$ no bigger than $2m$.
- Jill picks $x$ numbers between $1$ and $2m$. Jack wins if Jill picks at least one even number.
Otherwise, Jill wins.
What is the smallest number Jack should pick in order to guarantee a win? Prove your claim.
How do I reason through this?
How many odd numbers are there? Jack needs Jill to pick more than that.