coan.net: I did some calculating what the best action would be if there's a square showing a 1, and it has N neighbours that may have the frog (frog is still hidden). That is, there are N squares around the 1 that are not showing a number, and from the rest of the field, it cannot be determined whether they have a frog or not.

Obviously, if N == 1, you should guess the square, it will contain the frog with 100% certainty, and you will score 5. If N == 2, guessing one of the squares would be wrong. If you guess right, you score 5, but if you guess wrong, not only do you score -3, your opponent will score 5, so your expected result from guessing is -1.5. For N == 3, guessing is also wrong, but your expected score is less bad as in the N == 2 situation. If N == 3, you have a 1 in 3 chance of guessing right, so the expected score is 5 * (1/3) - 3 * (2/3) == -0.33. Note that after guessing wrong, you leave a situation where there are 2 squares that may contain a frog, and it's in your opponents best interest to leave it like that. In fact, for N >= 3, the expected score from guessing is 5 / N - 3 * (N - 1) / N == (8 - 3N) / N.

This will be a very defensive game.

And what we really need is a marker on the field indicating which squares have been unsuccesfully guessed.