AbigailII: Either a smaller board or more frogs would help reduce the "boring" portion where both players take shots that they know will turn up zeros. Unless there's a rule change, I'm probably not going to start any more Frog Finder games.

AbigailII:  My proposal doesn't introduce any more chance than there already is.  It automatically fills in the ones that are obviously zero (with a precise definition of obvious) but not the ones that a more complication intuitive process would also reveal to be zero.

I'm in several games right now where part of the board looks like below.  The rest of the board consists of squares where, for either player, it gives the opponent an advantage if you guess or shoot in any square.  Consequently, we're going through and filling in this section of the board with solid zeros.  This is rather boring and time consuming and doesn't actually accomplish anything, since we're both making shots that we know will reveal a zero.  I'm trying to create a rule that bypasses this part of the game.  (It will probably be still be there with my rule, but much shorter.) 

  +---+---+---+---+---+
6 | 0 |   | 0 |   | 0 |   
  +---+---+---+---+---+
5 |   | 0 |   | 0 |   |
  +---+---+---+---+---+
4 | 0 |   | 0 |   | 0 |   
  +---+---+---+---+---+
3 |   | 0 |   | 0 |   |   
  +---+---+---+---+---+
2 | 0 |   | 0 |   | 0 |   
  +---+---+---+---+---+
1 |   | 0 |   | 0 |   |
  +---+---+---+---+---+
    a   b   c   d   e

By the way, I'm not neccesarily opposed to long games.  I play anti-backgammon, which can easily run into hundreds of moves.  But most of those moves require actual thought (and auto-pass is there to skip many of the ones that don't).  But under the current rules, Frog Legs usually leads to 50 or so moves where the only thought is to correctly figure out a square that won't matter (which is probably the same calculation you did in the last 10 moves).

AbigailII: I'm advocating for filling in the squares that are "obviously" zero. While obvious hasn't been defined, it doesn't apply here. This is an active part of the board, and most shots will reveal some information about where the frogs are. Figuring out that D4 is 0 is something a good player would do (and a bad player might not).

I'm not trying to automate the game so that there's little actual thinking for the players to do. I'm trying to eliminate the phase where there's a 5x7 block with a checkerboard pattern of zeros, and both players spend almost 20 turns filling it in, because the alternative is to shoot in an area like what you drew and risk giving your opponent an advantage.

AbigailII: It's not that difficult to determine which ones must be zero.  If you shoot in square S1 and get zero, then you need to check

for each unlabled square S2 next to S1

  mustBeZero  = false

  for each labeled square S3 next to S2

     if S3 is labeled with 0

       mustBeZero = true;

     endif

  endfor
  if (mustBeZero)
     label(S2, "0")
  endif

endfor

This would cause the "barren plains" where there are no frogs at all to be quickly filled in.  It also means that if there's a 1 two squares away that's next to a known frog, that won't fill in the square, which might be reasonable, since you might not want to do too much thinking for the players.  If you wanted to also fill in those squares, replace "if S3 is labeled with 0" with "if S3 is labeled with the number of known frogs in neighboring squares".

AbigailII: One thing that would help is to have squares where the result must be a zero automatically be filled in with zeros. For example, if there are zeros at A2, B1, B3 and C2, it's impossible for B2 to be anything but a zero, since all adjoining squares are next to a zero square. This would eliminate (or greatly shorten) the phase of the game where both players shoot squares that they know will return zero, since any other move gives an advantage to your opponent.

Another improvement would be to increase the density of frogs, so that revealing a 2 is more likely. With the current scoring, guessing next to a 1 is always a bad idea unless there's only one possible location. But guessing next to a two has a variety of result. 2/2 and 2/3 are both good. 2/4 is bad, but 2/5 is actually good. (The 40% chance and 5/3 reward mean your guess has a net positive value, but it's likely enough that your opponent would guess wrong and then you would guess right, that at 2/4, it's a bad idea to guess.) If two's were fairly likely, guessing next to a spot with 3 or 5 unknown squares is much more attractive.

It might also work to give points for finding frogs in neighboring squares. For example, it you got the number of points shown in the square for shooting an empty square, that would provide an incentive to shoot in squares that neighbor unknown squares. If the values are balanced correctly, it should be possible to create a situation where it's advantageous for me to shoot a square that might reveal neighboring frogs and then for my opponent to guess where a frog is.

AbigailII:  Interestingly, if you know there are two frogs in four squares, you shouldn't guess.  However, if you know there are two frogs in five squares, you should guess. 

In both cases, you'll guess right often enough that you'll score more points for the right guesses than you'll lose for the wrong guesses.  However, with four squares, if you guess wrong, your opponent can guess with a 2/3 chance of being right, which is good enough that the points he gets from guessing right outweighs the points you might get from guessing right.  With five squares, you still leave your opponent with enough options that guessing is a bad idea, so you just get the payoff from your initial guess.

Another idea would be to randomly rotate and flip the board before starting the game. There are (I think) 8 possible changes, so it's very likely that at least one of them would give you an arrangement where no frogs are on top of each other. There's a little bit of information available from the options that weren't picked, but since it's random and rotations always happen, you won't know which ones were rejected because of your opponents frogs and which ones were because of the random factor. If none of the configurations work, just make both players replace all the frogs.