## Puzzle : Catch Mickey

A forum for good logic/math puzzles.

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### Puzzle : Catch Mickey

Hi,

Here is a funny puzzle. I hope you will play it for fun and analyze it for fun too.

Carlington
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Location: Sydney, Australia.

### Re: Puzzle : Catch Mickey

The highly symmetrical nature of this one makes it pretty straight forward. My attempt below the spoiler:

Spoiler:
I'm using here notation similar to the standard algebraic notation for chess - moves are given in pairs, red then blue. A move is specified by listing the coordinates of the square upon which the piece moved ends its movement. Where disambiguation is needed, this is done in as minimalistic as possible a fashion - either the starting row or column of the piece moved is prefixed to the move. When absolutely necessary, the entire coordinates of the starting and ending square are given. # at the end of a move means blue has no more moves.

I've chosen certain moves here, and I'm not going to specify that they apply symmetrically each time.

Firstly, the corner case, when blue starts in B4
Red wins by 1. B3 E4 2. E3 A4 3. C4 B4 4. B5# and this is the best blue can do here.

Next, blue starts in C4
1. C3 A4
2. B3 and now we have the corner case, and red wins after 3 more moves.

Finally, blue starts in the centre, C3
1. C2 C4
2. C3 and now we're in the edge case, and red wins after 4 more moves.

Overall, red's strategy in the corner case is to use the first move to confine blue to one row and then force blue to a corner. Once blue is confined to one row, all he can do is move back and forth while red brings in pieces to block squares. The same is true of the edge case - in effect, force the corner case and then win. The centre case requires first waiting for blue to move to an edge, and then it's the corner case.
Kewangji: Posdy zwei tosdy osdy oady. Bork bork bork, hoppity syphilis bork.

Eebster the Great: What specifically is moving faster than light in these examples?
doogly: Hands waving furiously.

Please use he/him/his pronouns when referring to me.

Posts: 431
Joined: Thu Oct 16, 2014 9:28 am UTC

### Re: Puzzle : Catch Mickey

Thank you.
Easy!

More difficult with the same grids but the blue could place 2 (3 or more) pieces instead of one with an added rule : if one of the blue pieces can not move it is captured.
The puzzle is in fact derived from an abstract game invented by myself recently.
I still did not finish designing it. I project larger board with more pieces.

Carlington
Posts: 1588
Joined: Sun Mar 22, 2009 8:46 am UTC
Location: Sydney, Australia.

### Re: Puzzle : Catch Mickey

Spoiler:
I don't think a bigger grid is any more complex. The relevant feature of the puzzle is how long blue can last before being trapped between two red pieces on the edge of the board. Once the blue piece is forced between two red pieces at the edge, it can be confined to one row and then blue only has one possible move each turn, while red is free to build the prison walls around it.
So, the difference between the three cases in the 4x4 version is how many degrees of freedom blue has left after red's first move.
Looking at it this way, we don't need to find how many moves blue lasts in each case - it's enough to notice that the 5x5 board's corner square is really a corner region of size 1x1, which has size 2x2 in the 7x7 board, size 3x3 in the 9x9 board, and so on. Side length (n-3)/2 on an nxn board.
Notice also that the edge case from the first board is really an "adjacent to exactly one red piece" case - because of the symmetrical layout, red can confine blue to one row with the first move.
The only difference is the "centre" case, which now breaks into two cases along the central row and column except the squares immediately adjacent to red pieces. The precise centre is still the best starting point - blue has no first move here that ends up trapped between two reds on an edge. Other squares along the axes give blue only two choices for first move.
Kewangji: Posdy zwei tosdy osdy oady. Bork bork bork, hoppity syphilis bork.

Eebster the Great: What specifically is moving faster than light in these examples?
doogly: Hands waving furiously.

Please use he/him/his pronouns when referring to me.

Posts: 431
Joined: Thu Oct 16, 2014 9:28 am UTC

### Re: Puzzle : Catch Mickey

Thank you.
I said things are harder when the blue places 2 pieces or more.
So the 2 players abstract game I was referring to is based on 8 reds and 4 blues pieces (grid 11x11).
On my abstract game for 2 players the red scores the number of moves required to capture and remove all the 4 blue pieces.
The game is played by rounds (4 rounds). In each round players are taking alternatively the reds.
My only problem remains the set up (which configuration to use?) and the placement protocol before starting the core of the game.
I need to fix the game such as mirroring will be impossible and bounding the maximal number of moves needed to end the game.
I`m thinking to add a diagonal move too. I will need to add some rules in that case.

Soupspoon
You have done something you shouldn't. Or are about to.
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### Re: Puzzle : Catch Mickey

I put this to one side whilst I tried to resolve some things. I could read the solutions to see what others thought, but then I'd be seeing the answers. But I really need to ask...
1) Who starts first? Do we choose (or possibly let the 'defending' player in particular choose according to his/her idea of advantage), as one line says, or is it always red, which the next line somewhat suggests?
2) "Counting the moves"... Is that one for every full (until collision) orthoganal run (i.e. directly related to the number of times control is switched between players), or one for each square travelled upon each orthagonal run (i.e. the full distance travelled)
3) Are we assumng that each player plays the best they can, as well as on the assumption that the other player will also be playing the best they can?
...and finally, to cover all the bases...
4) Have I completely misread some of the instructions so that the above questions are nonsensical or actually explicitly answered?

Posts: 431
Joined: Thu Oct 16, 2014 9:28 am UTC

### Re: Puzzle : Catch Mickey

Easy to understand who starts first after the set up. The hunter (red) starts first.
Counting moves ; as the move if sliding a piece until ...... then each time you move the piece from a starting position to the ending position you count one move.
Yes we assume the best play for the 2 players.
There is a flaw in the puzzle ; if the blue player is trapped in some row or some column he has no means to get out of.
I solved this problem by giving the blue player a new piece ("edging piece) acting like an edge or a virtual occupied square. After each of his turn the blue player can change the position of that piece. The blue player is the only player allowed to use it. For the red player the "edging piece" does not exist at all so he can not be stopped by such piece. This trick will allow the blue player to "survive" making the game more tactical.
No one talked about 2 blue pieces or more.