See https://plus.maths.org/content/mathematical-mysteries-chomp
This is about eating cookies that are placed in a rectangular grid. The cookie in one corner is poisoned, so the person who is left to eat it loses.
There’s a lovely and very simple argument showing that the first player can always win, but the winning strategy is unknown! So getting people to think about this game strategically gets them straight into an unsolved problem.
For certain versions of chomp (“square” and “thin”), however, the winning strategy is known and can be easily described (see http://plus.maths.org/content/os/issue14/xfile/solutions).