[Retros] retro Tic-Tac-Toe

[Alain BROBECKER]

Dear retro fans,

I made some extensive searching in the Tic-Tac-Toe & losing
Tic-Tac-Toe game trees, and here are some retro Tic-Tac-Toe
problems. The idea is from a problem by Les Marvin. I assume
that both players were experts (neither one ever afforded the
other an opportunity to force a win). It's not always the same
player that starts a game.

E)
 | |X
-+-+-
X| |O
-+-+-
 | |O
Find the game. (C+)

F)
O| | 
-+-+-
X|X|O
-+-+-
 | |O
Last 2 moves? (C+)

G)
 |X|O
-+-+-
O|O|X
-+-+-
 | |X
Last 3 moves? (C+)

Solutions below,
Happy solving, Alain
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Solutions: (squares numbered like a push button phone)
Rule 1: for a given position (with an even number of symbols), if a
  player can force a win, then it's his opponent to move since no
  player would have allowed the other one to reach a winning position.
Rule 2: if 1st player opens in the center, 2nd player must play in a corner.
Rule 3: if 1st player opens in a corner, 2nd player must play in the center.
Rule 4: if 1st player opens on an edge, 2nd player must play in the center
  or in an adjacent corner or on the opposite edge.

E) X to play could force a win by playing 1, rule 1 implies that 
X played last. So O started and we know from rule 3 that he played 6 then 9.
After 6,3,9 X could force a win by 2,1,5,... so it was not 3 that was played
on 2nd move, and thus the game was 6,4,9,3.

F) If last move was 1, O could have won by playing 3 instead.
If last move was 9, O could have forced a win by playing 3 instead.
So last move was 6. Also we deduce that O started in a corner, and
rule 3 implies that X answered in the centre. So X's last move was 4.
The game was 1,5,9,4,6 or 9,5,1,4,6.

G) O to play could win by playing 7, so rule 1 implies that O
played last. O's last move wasn't 4 or 5 because playing 7 instead
would have won directly or forced a win (respectively). So O's last
move was 3. X's previous move wasn't 2 or 9 since playing in 3 would
have won directly or forced a win (respectively). So X's previous move
was 6. Now, if O's previous move was 5 then whatever was X's 1st move,
2 or 9, the response 4 was incorrect according to rules 3 and 4.
So O's previous move was 4. The game was 2,5,9,4,6,3 or 9,5,2,4,6,3.

Alain Brobecker (abrobecker at yahoo.com) |_  _  _ |_
http://abrobecker.free.fr/             |_)(_|(_|| ) of  Arm's Tech


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