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Dween
Dween
Joined: Jan 24, 2010
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February 25th, 2011 at 5:42:25 AM permalink
tl;dr version: Skip down to the bold part.

From the 1980's game show Blockbusters, show below are two game boards on which players attempted to make a color-to-color connection. In order to capture a green block, players must give a one-word answer to a question, with each block containing a different letter of the alphabet (not shown in picture), the answer beginning with that letter.

In the earlier version of the show, the board on the left was exclusively used, and two teams competed:
The white team, which consisted of two players, had the advantage of two brains, but a longer path.
The red team, which was a solo player, had the disadvantage of being alone, but a shorter path to win.

On the later version of the show, there were only single players for each color. In game 1, the white player had the side-to-side path, but in game 2 of the match, the red player had the side-to-side path (colors were switched on game board, not shown in picture).

If the match went to game 3, a special board was used, as seen on the right (above). It was stated that neither player had the advantage, as only four spaces were needed to make a connection.

I am not so sure that there was not an advantage. Given the board on the right, which player would you rather be, red or white? And why? Is there a mathematical proof that can be given, or at least a theory, that shows one player would have an easier time making a connection?
-Dween!
JIMMYFOCKER
JIMMYFOCKER
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February 25th, 2011 at 7:00:26 AM permalink
Merry Christmas and happy holidays.
P90
P90
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February 25th, 2011 at 8:20:03 AM permalink
Quote: Dween

I am not so sure that there was not an advantage. Given the board on the right, which player would you rather be, red or white? And why? Is there a mathematical proof that can be given, or at least a theory, that shows one player would have an easier time making a connection?


The white player is at a considerable advantage, it appears. He always has two choices how to continue, while the red player is severely limited.

The white player has the following paths to victory (top to bottom):
Piece 1 - 6 paths
Piece 2 - 8 paths
Piece 3 - 7 paths
Piece 4 - 3 paths

The red player has only the following (left to right):
Piece 1 - 4 paths
Piece 2 - 1 path (!)
Piece 3 - 7 paths
Piece 4 - 1 path

That's a total of 24 versus 13 paths, if I'm not mistaken.

Any starting move by white on column 2 or 4 renders one of red's starting pieces useless. The white player is almost impossible to block as well, any bottleneck has two exits, not so for the red, where bottlenecks have one exit.

I'm not even sure if the board on the left isn't closer to fair than the one on the right (even if biased in the other direction).
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MathExtremist
MathExtremist
Joined: Aug 31, 2010
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February 25th, 2011 at 8:32:07 AM permalink
Quote: Dween

I am not so sure that there was not an advantage. Given the board on the right, which player would you rather be, red or white? And why? Is there a mathematical proof that can be given, or at least a theory, that shows one player would have an easier time making a connection?



White is a clear favorite, and you can use graph theory to analyze. First, create the dual of the graph formed by the green hexagons (i.e. each hex is a node, and there is an edge between each adjacent hex). Then examine the number of paths of length N available to each player. You'll see that the number of shortest paths (length 4) is significantly greater for the white player than the red. Red has only four; white has well over a dozen.
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice." -- Girolamo Cardano, 1563
DJTeddyBear
DJTeddyBear
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February 25th, 2011 at 8:50:26 AM permalink
White. No question about it.

As White moves from one side to the other, it has a choice which does not increase the number of steps to get to the goal.

As Red moves down, if it moves laterally, it cannot move laterally again without increasing the number of steps to the goal.
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