My Own Math Riddle

My Own Math Riddle

3 answers , last was 15 years ago

There is a game I played once (I now forget the name of the game but that's not important)...

Here are the rules:

You place x dots on a piece of paper, then each player takes turns connecting the dots. When a connection is made a dot is placed in the middle of the line.

Lines need not be straight.
Lines MUST not cross.
The last player able to make a line wins.

HERE ARE THE RIDDLES:
I would rank all these as hard...
In a perfectly played game(up to you to decide what that is exactly) starting with 3 dots, who wins player 1 or player 2? Will this change with the initial number of dots?

What is the minimum number of moves that can happen in a game with 5 dots? What is the maximum number of moves that can happen in a game with 5 dots?

Can you make a general solution for min/max moves with X initial points?

Asked by Ken McWilliams in Mathematics at 2:24pm on September 15th, 2008
Jeremy Devito 1416
Answered at 7:35pm on February 15th, 2009
For any instance of this game it is possible to coneect all the dots with one non straight line. If you had three dots and they can not intersect the maximum possible solution for a game with three dots would be nine lines so the player who went first would be victorious. and for the five dot game all the dots could be connected with one line as well. the minimum for this game could be considered as one or technically speaking if they had to cross the dots each three time would then be three. the maximum allowance of lines in order to connect the five dots with three lines in each non intersecing would be 15.

How did i do?
Jonathan Burley 2375
Answered at 4:16am on September 16th, 2008
I think this is an np-hard problem when extended to "n" points. Which lands it in with most networking problems like the travelling salesman or the minimum path etc... Basically it is solvable but takes too much work to make the solve, the idea of lines crossing is a real mathematical bastard, for any given network you would have to evaluate ALL possible crossings (completely changed by the addition of a single point or the moving an existing point)

Am I right?

The only answer that leaps out is the trivial solution whereby you line up all the points into 2 parallel lines. Then a move linking the extreme points on opposite lines would instantly kill the game in move 1 (ie. link far right of one line with the far left of the other)
Aaron Young 2263
Answered at 11:23pm on September 15th, 2008
Based on your description, the game I play has different results depending on where your initial points are, now I don't know what I'm to do.

Are the initial points in the the game starting with three dots supposed to be in a triangle, or could they all be in a line?
And if the lines need not be straight then could you potentially have an infinite number of lines between the same two points?
Finally, can you make a line from a dot that was made in the middle of a line? (if so there are infinite maximum moves).
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