A riddle was posted in this mathoverflow question: https://mathoverflow.net/questions/85439/how-does-intuitionism-handle-this-riddle
A riddle: You and another person are kidnapped and knocked unconscious by a demented villain. When you wake up, you are told that some of you may have an ink dot on your forehead. You can see the other person's forehead but not your own. You must each privately guess as to the status of your forehead. At least one of you must be right, or you will both be killed. No talking or signaling is allowed; this also will result in death.
Now, if you survive this, you and the other person will be transported elsewhere, never to see each other again. You will never know what was on your own forehead.
The solution to this riddle relies on the fact that there are exactly four possibilities: You both have a dot, neither has a dot, you do and she doesn't, she does and you don't. I'll leave it to the reader to figure out the strategy.
Even with the hint, and assuming that the configuration of dots was chosen uniformly at random (which is not stated in the problem), I don't see how to do better than just randomly guessing whether or not I have a dot, with a 25% chance of death.
If I'm allowed to communicate in advance with the other prisoner and devise a strategy, we can guarantee freedom by e.g. one of us guessing what he sees, and the other guessing the opposite of what he sees. But communication is explicitly forbidden in the question.