3

This is a variation of Edward De Bono’s famous L-game, but played on a 5x5 board (instead of 4x4) with social distancing rules in place. More specifically, two players (Blue and Orange) take turns moving according to the following rules:

  1. (mandatory) Move their L-piece to a new location, not occupied by or orthogonally adjacent to the enemy L-piece or a happy star. Diagonally adjacent is OK.
  2. (optional) Move one of the happy stars to a new location, not occupied by or orthogonally adjacent to the other happy star or either L-piece.

Victory is attained if the opponent has no legal move. If neither side can force victory the game is a draw.

IMPORTANT: if a happy star is moved, steps 1 and 2 must be performed in that order and you can’t have an illegal position “between” steps 1 and 2.

What is the correct result with best play on both sides?

Does the result change if a pie rule is used (i.e. player 2 has the option of swapping sides on his first move)?

enter image description here

happystar
  • 7,534
  • 12
  • 54
  • 2
    Do you believe that you know the answer? Do you have good reason to think there's an elegant solution? How do you feel about computerized answers? – Gareth McCaughan Sep 18 '20 at 11:06
  • The L-pieces can be turned over / reflected, right? – Gareth McCaughan Sep 18 '20 at 11:07
  • Computerized answers are okay, otherwise I would have added a no-computers tag. L pieces can be turned over and reflected (as per the rules explained in the hyperlink). I don't know the answer (and I've already seen several puzzles where AFAIK OP doesn't know the answer) – happystar Sep 18 '20 at 11:16
  • 3
    For the avoidance of doubt, I wasn't saying it's necessarily a problem if you don't know the answer. Just wondering whether you did. (It would make it more likely that there's a nice neat satisfying solution, since if you know the answer you're more likely to post the question here if it's nice than if not.) – Gareth McCaughan Sep 18 '20 at 21:56

2 Answers2

1

Isn't it simply:

Any player can choose:
Odd turns: move the short end of the L to the corner, move the moved happy face back (assuming one was moved)
Even turns: move the short end of the L either 2 horizontally or 2 vertically (only one can be blocked), move the moved happy face back
-> draw

More explicit:
Yellow can enforce that at any time yellow occupies the three yellow spots and one of the orange ones, while blue only can occupy the blue spots.
1 Switch to use of one of the other orange spots.
2 If blue moved a neutral piece: place back to its starting position.
Note that the first move is always possible since blue cannot block at both A and B.
Note that blue can apply the exact same strategy to avoid loosing.
Oeps, I forgot 2 blue spots. But it does not matter for the strategy...
https://i.stack.imgur.com/l8XBq.png

Retudin
  • 8,511
  • 1
  • 13
  • 46
  • 1
    I don't see why that would be optimal play for either player. – Jaap Scherphuis Sep 19 '20 at 18:49
  • 1
    Player A can prevent loosing, so there is no way B can win with optimal play. Player B can prevent loosing, so there is no way A can win with optimal play. Sure there may be moves were the opponent can at least make more mistakes, but the OP asked for the correct result with best play on both sides, which is draw. (note that this does not change with the pie rule in place) – Retudin Sep 20 '20 at 06:15
  • 1
    Can you add a diagram to show what you mean, please? I don't think I fully understand the moves. – hexomino Sep 23 '20 at 16:07
0

By my analysis, discounting rotations and reflections, there are:

2657 possible valid layouts
200 valid losing layouts
240 position where having placed one L and two stars, you can't place a second L. This isn't relevant to the problem, but I found it interesting.

Further testing reveals that 25 of 200 of the losing layouts do not have the loser in the corner.

This image below is shows five losing layouts, magenta to be moved. (Sorry, but I didn't try to match the colors.) (Randomly selected groups until I got a non-corner one.)
enter image description here
The numbers are my programmatic representation of the layout.
Edit: I think it was unintended, but the yellow represents the border around the red. Thus, the green and magenta is were the piece needs to be moved. I now think it works better that way.

If desired, I can post my programs.

Now, what this about strategy or ideal play, I don't know.

Edit: Even further testing shows that for 1 of 200 losing positions, the loser has the middle of the three in the middle of the board. Further, neither player can move from this position, and both can move into it.
enter image description here

David G.
  • 1,606
  • 1
  • 7
  • 12