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NIH New user 11 Posts |
Two players take turns choosing one number at a time (without replacement) from the set {-4, -3, -2, -1, 0, 1, 2, 3, 4}. The first to obtain three numbers which sum to 0 wins.
Does either player have a forced win? |
Scott Cram Inner circle 2678 Posts |
Let's arrange a 3x3 grid so that each row totals 0:
..3..|(-4)|..1 ----+----+---- (-2)|..0..|..2 ----+----+---- (-1)|..4..|(-3) When looked at this way, the game merely become a variant of tic-tac-toe. Any three Xs or Os in a row will total zero. I'd say that if only one player realizes they are playing tic-tac-toe, they have an easy victory. If both players realize they are playing tic-tac-toe (and play correctly), then the game will probably end in a draw like it usually does. |
Gregg Tobo New user Denver 64 Posts |
Analysing on Scott Cram's model, we can say that Player A (with the first move) can force a win 50% of the time when Player B makes his/her first response at random.
Player A's first move is to select 0 (the coveted center square in tic-tac-toe). If Player B selects an even number in response (i.e. a side square), Player A can now force a win. For example: Player A selects 0 Player B responds with -4 Player A selects +3 (selecting +1 would work as well) Player B must respond with -3 (or lose on the next turn) Player A selects -2 Player B now faces a dilemma: Player A can win by selecting +2 (0, -2, +2) OR by selecting -1 (+3, -2, -1). Player B cannot prevent both. ----------------- Some further thought reveals that Player A (with the first move) can improve his/her odds (again assuming Player B's response is random) by selecting any odd number on the first move. In this case a win can be forced 75% of the time. Only 2 out of 8 squares can prevent a forced loss on the part of Player B. Player B must respond by selecting either 0 or the inverse of Player A's first selection (i.e. either the square in the center, or the square diagonally opposite of Player A's first move). Failing this, Player A can force a win. Gregg Tobo |
landmark Inner circle within a triangle 5194 Posts |
Great idea with the tic-tac-toe analogy.
Actually, I have a slight amendment to Gregg's analysis. If A chooses +3 and B chooses -3, then A has a forced win by choosing -1. B is forced to respond with -2 and then A can choose +1. B can no longer block A from winning. Jack Shalom
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Gregg Tobo New user Denver 64 Posts |
Jack,
Good catch! I guess we can conclude that, with proper play, the First Mover has an overwhelming advantage -- but not quite a guaranteed win. And the Second Mover who knows that the game is really a tic-tac-toe matrix can always play to a draw (and could even defeat an uneducated First Mover). Sounds like the makings of a successful bar bet (if you can find a mark who's not afraid of a little math...). Gregg |
Scott Cram Inner circle 2678 Posts |
In a past Flim-Flam column by Bob Farmer, he used a similar idea with a 3x3 grid that totaled 21 in every direction.
People thought they were playing blackjack, but they were really playing tic-tac toe. |
Samuel Special user Norway 831 Posts |
Only problem, Jack, is that most players would go for the center fast, in a tic-tac-toe-game - but if we don't consider the tic-tac-toe, then it's all fine
Samuel
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