coan.net: Frog Legs is missing something - it ends up being a very long drawn out game.

IMO, this comes from the fact that it doesn't pay to reveal information.
And that stems from the fact that after you reveal information, your
opponent will be the first one to be able to use it. So, if you make a
move that reveals information from which the location of a frog can be
deduced, your opponent will score, not you.

That is, IMO, the essential flaw in the game. None of the suggestions
that assign different number of points is going to resolve it. Neither
is the auto revealing of squares neighbouring a 0 - that will make the
game shorter, but it still doesn't pay to reveal information.

I can think of one rule change which may fix that: after shooting, you
have the option of guessing as well. So ones turn is one of:

• Shoot
• Shoot, then guess
• Guess

Thus, if your shot reveals information that locates the position of a frog, you're the first one that can use this information.

coan.net: In the current version, I agree it would be nice to see where you have guessed at - but not 100% needed since you can see it in the move list.

By the same argument, showing a chess or checkers board isn't 100% needed, since you can see the move list; and the move list will determine the position.

coan.net: Hide from your opponent where you guess at.

That won't stop people from making "safe" moves. Granted, in some situations when a number other than 0 is shown and there are some unknown neighbours, it's better to guess than to shoot. But since the there's an advantage to guess before your opponent guesses when a number other than 0 is shown, it's better to shoot somewhere knowing where it will reveal a 0 than to shoot somewhere where it may reveal a number other than 0.

But with this proposed variation, it's even more important that squares you've guessed are marked in someway, and we haven't seen guessed squares marked in regular Frog Legs/Finder yet.

joshi tm: So why not award points for shooting on numbers?

What do you mean by "shooting on numbers"?

coan.net: What some had done (and myself at times also) - in your FIRST SHOT - Don't shoot, but guess in a spot. (you will lose 3 points - but that is a lot less then 10 points... and will give you a place to shoot next time with no fear of hitting a frog)

Guessing on your first move is bad. The chance is the first guess/shot hitting a frog is such that on average you lose more points by guessing than shooting. And if you guess wrong, you will NOT have a "safe" shot the next time; assuming your opponent isn't totally stupid, it's your opponent who has a safe shot. And if you are unlucky, your first guess will be next to a frog; not only giving your opponent a three point advantage, your opponent will have a safe place to shoot, you still will not have a safe place (as your opponent shooting on the place you guessed will reveal a number larger than 0).

Baked Alaskan: Eh, no, I am mistaken. I was thinking the question was about frog legs, not frog finder.

A simulation for frog finder suggests that the second player has a slight advantage, winning about 52% of the games, with games lasting about 153 half moves. Again, the assumption is that no player guesses an already guessed square.

Well, for your first shot (assuming you go first), the odd of guessing a frog is about 5.6%, assuming you won't guess one of the middle 9 squares, as they will be empty. The board is 13x13, making 169 squares; the middle 9 are empty, so 9 frogs will be found on 160 squares, or about 5.6% of the squares.

Running a simulation where both players guess randomly (but won't guess a square that was guessed before; and won't guess one of the middle 9 squares) shows the game is pretty even, and lasts about 144 half-moves.

How popular would small versions of Frog finder/frog legs be? For instance, on a 9x9 board; same rules. The number of frogs could be the same, or somewhat less (for instance 2x4 for the small frog finder, and 7 for the small frog legs).

Suggested names: "Spot the tadpole" (mini frog finder) and "Pollywog" (mini frog legs). (Tadpoles are sometimes named pollywogs).

coan.net: Yeah, but once a 1 is shown, there will be often be "safe" shots, that neither reveal where the frog is, and still reveal a 1. Example:

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

If a shot on a2 reveals a 1, the opponent can shoot on a4, also revealing a 1, cancelling any benefits from revealing a non-0.

Currently, the game lasts till all the frogs are found. However, there's usually no point in playing on if one player is more than 5 * F points ahead, with F the number of frogs to be found. (I say usually because theoretically it could be that all unknown squares could potentially contain a frog, forcing the player ahead to guess - but that's a situation that won't occur very often).

So, I suggest to add another winning condition: whenever you are ahead with a number of points more than 5 times the number of unrevealed frogs, you win. This will shorten many games by dozens of moves, where the player ahead is going to play frog finder, not caring who reveals the frog.

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

In a game of Frog Legs, what would you do? Guess on c3 or not? If not, whould you shoot if a1 and a5 are showing 0s as well? Assume the score is equal, and at least 2 frogs are still missing.

joshi tm: Yes? So? What's your point? I'm talking about what happens if a2 reveals a 1.

joshi tm: If shoothing at a2 reveals a 0, there can be everything at d4.

I don't get your point.

troydaniels: Filling in *some* of the squares that are 0 is something that doesn't appeal to me at all. If some of the squares are filled in, but not all of them, you introduce a category of chance in the game where there's no such chance currently. IMO, either all 0-predetermined squares should be revealed, or none. Not all "obviously" zero for some vague notion of obvious.

troydaniels: It's not that simple. There are a lot more cases to consider. For instance:

  +---+---+---+---+---+
6 |   |   | 0 |   |   |   
  +---+---+---+---+---+
5 | 0 |   |   |   |   |
  +---+---+---+---+---+
4 |   | 1 |   |   | 0 |   
  +---+---+---+---+---+
3 |   | 1 |   | 1 |   |   
  +---+---+---+---+---+
2 |   |   |   |   |   |   
  +---+---+---+---+---+
1 |   | 0 |   |   |   |
  +---+---+---+---+---+

If shooting on a2 reveals a 1, it can be determined that d4 must be a zero.

troydaniels: It will be tricky to programmatically determine which squares have to be 0; it's easy to come up with examples, but it'll be hard to find all possibilities.

Here's a table with expected scores, assuming you guess, and subsequent guesses are only done if they give a positive score.

    +-------+-------+-------+-------+-------+-------+-------+-------+
    |   1   |   2   |   3   |   4   |   5   |   6   |   7   |   8   |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 1 | +5.00 | -1.50 | -0.33 | -1.00 | -1.40 | -1.67 | -1.86 | -2.00 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 2 |       |  0.00 | +2.33 | -0.17 | +0.20 | -0.47 | -0.71 | -1.00 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 3 |       |       | +5.00 | -0.00 | +1.80 |  0.00 | +0.43 | -0.27 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 4 |       |       |       |  0.00 | +3.40 |  0.00 | +1.57 |  0.00 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 5 |       |       |       |       | +5.00 | -0.00 | +2.71 | +0.00 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 6 |       |       |       |       |       |  0.00 | +3.86 | +0.00 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 7 |       |       |       |       |       |       | +5.00 | -0.00 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+
| 8 |       |       |       |       |       |       |       |  0.00 |
+---+-------+-------+-------+-------+-------+-------+-------+-------+

'+0.00' means there's a small positive expected score, '-0.00' means there's a small negative expected score.

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.

coan.net: Having the game balanced doesn't necessarely make it fun. Flipping a coin is balanced game as well, but not really fun to play.

I think the "problem" with Frog Legs is that your move reveals information that can immediately be used by your opponent. This, in combination with low density of frogs (only 9 out of 169 squares contain a frog, which means that no matter how the frogs are placed, over half of the squares show a 0) makes for a dull game. The dullness lies in the 60 or 70 moves that are played before the game becomes interesting - no more "waiting moves" can be played.

I don't know if the flaw can be fixed - and I suggest to Fencer he won't make any chances to the game unless it was properly play tested.

Not knowing where one has guessed changes the situation where there's a 50% chance of finding a frog: it will then become advantageous to guess. However, this makes it even better to shoot on squares knowing it will reveal a 0, as revealing a 1 (or an even higher number) becomes worse than it's now.

Here are some ideas that may work (although only play testing will reveal whether it actually does)

1. Increase the density of the frogs. There will be less no-information revealing moves possible if there are 25 or 49 frogs. It will not lengthen the game, as currently, between good players, most of the board will be shot anyway.


2. Give players two moves per turn. Then, in his/her second move, the player can use the information gained in the first move.


3. No guessing - just shoot the frogs (change them into bunnies). First to shoot 5 frogs wins the game.


4. If you shoot a square revealing a non-zero number, you get an extra turn (or a free guess)

After playing hundreds of moves (but only actually finishing a few games), here's my first evaluation of Frog Legs.

Play wise, it's quite boring. There's no real strategy, except from avoiding playing bad moves. If both players don't play any obvious bad moves, the game boils down to filling up allmost all the squares, delaying having to reveal essential information until there's no other move. And then it becomes a game of luck, with the frog(s) to reveal only having two or three squares. Games will last 70 to 80 moves, which, I think, will rank them along the longest games found here on BK.

As for strategy, I use the following guidelines:

• Never guess unless the chance the square contains a frog exceeds 50%.


• If you can shoot a square that is only surrounded by squares of which it's known whether they contain a frog or not, shoot it.


• Never shoot a square that has only one unknown neighbour (an unknown neighbour is a square of which it's unknown wether it contains a frog or not) - you'll give your opponent 5 points if the square reveals a 1. (See below).


• Avoid shooting squares that have an odd number of unknown neighbours; if the number reveals the same number as the number of unknown neighbours, your opponent can guess one more frog than you do. (This is a generalisation of the previous point).

Of course, the current score can influence things. If there are only N frogs to be found, and your are ahead more then 5 * N, by all means, narrow down where the frogs are as soon as possible - it's ok if your opponent guesses the remaining frogs.

Here's an example of where you shouldn't shoot:

   +---+---+---+
 3 |   |   |   |
   +---+---+---+
 2 | 0 |   |   |
   +---+---+---+
 1 |   | 0 |   |
   +---+---+---+
     a   b   c

Don't shoot at b2. It's already known that a1, a3, c2 and c1 do not contain frogs; c3 is the only unknown neighbour of b2. So, if b2 reveals a 1, there will be a frog at c3 with 100% probability. Unless your opponent is making a very stupid mistake, you will lose 5 points.

coan.net: That has been reported and said to be fixed some hours ago: http://brainking.com/en/ReadBug?bgi=1457.

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.

coan.net: Unfortunally, it also means that it's an advantage to shoot squares where you will already know how it will reveal 0 (like shooting in the corner if your opponent shot (diagonally) one step away from the corner revealing a 0). Such a shot reveals no information at all.

coan.net: Interesting. Two more points: if you go first, and you decide to shoot, you don't have a 'safe' shot, there's always the chance to hit something. Making your first action a guess doesn't give your opponent a field he can shoot knowing there's no frog there.

But here's another thing. Say starting with a guess would be a good thing. Then, wouldn't it be good for the player going second to start with a guess as well? But if both players start with a guess, followed by a shot on their guess, what about their third moves? Shouldn't that be a guess too?

WellyWales: As said, that might give the player going first a bit of an advantage.

Perhaps it's better to have a rule that the second player cannot play his first move adjacent to the first move of the first player.

However, I don't think the second player has that much of a disadvantage - certainly less than white has in chess. Yes, the first player has the disadvantage of having to make a blind shot - but there are only five enemy frogs. He's more likely to make a shot right next to an enemy frog, revealing a non-zero number, leaving the second player to make a blind shot.

A 59 point difference: 44 vs -15

BIG BAD WOLF: In the game I was refering to, I almost did it three times in row as well. Only at the last moment, my mouse hovering above the submit button I notices the move list.

Fencer: Because the field contains information (there's no froggy on that square). And I have made the mistake of guessing the same field twice which just wastes a turn. Sure, one could scan the entire list of turns, but that's awkward.

Would it be possible to mark fields that you have guessed and didn't reveal a frog? Perhaps with a cross, a question mark or a different colour.