coan.net: in the idea of hiding where your opponent guesses, I'm trying to see how you would think revealing a 1 (or higher) would become worse than it is now. If I shot and reveal a 1, my opponent will still have the same chance of guessing where the frog is as they did before.

Yes, but your opponent goes first, giving him the edge. Take for instance the simplest example, you reveal a 1, with only two possible places for the frog. Assumming both players now guess until the frog has been found, there are three possibilities:

1. Your opponent guesses right. Probability: 0.5. Score: -5.


2. Your opponent guesses wrong, you guess right. Probability: 0.25. Score: 3 + 5 = 8.


3. Your opponent first guesses wrong. You guess wrong. Your opponent finds the frog. Probability: 0.25. Score: 3 - 3 - 5 = -5.

So, your expected score after revealing a 1 with two unknown squares surrounding it: 0.5 * -5 + 0.25 * 8 + 0.25 * -5 = -3.

Things looks less grim if there are 3 unknown squares, but if both players guess until the frog is revealed, the player guessing first (the opponent of the player revealing the 1) has an expected gain in score of 0.44 (if I did my math correctly).

There's no real solution here, even if you play with the rewards/penalties. If, when a 1 (or a different number) is revealed guessing gives an expected positive score, the opponent of the revealer has an edge. Then it doesn't pay to make a move that may reveal a non-zero number. If guessing gives a expected negative score, we have the same situation as we currently have. And if the expected score is 0, it's just a blind luck.