Re: Multiplying Sub-segments: A Game
From: Leroy Quet (qqquet_at_mindspring.com)
Date: 02/25/05
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Date: 25 Feb 2005 09:17:28 -0800
I wrote:
>Here is another game.
>
>For 2 players.
>
>Start with a blank piece of paper.
>Players take turns drawing a line-segment (of any length) each move
>which connects to the end of the last drawn (by the other player)
segment,
>for a total of n (n is predetermined) connected line-segments, where
the
>line-segments are such that they form a *convex* irregular n-gon.
>(Yes, the last line-segment connects with the first line-segment.)
>
>After the (convex irregular) n-gon is drawn, the players take turns
drawing
>line-segments, each segment connecting the last drawn-to (by the other
>player) vertex of the n-gon to any vertex not yet drawn to.
>The first segment connects any 2 vertexes.
>The last (nth) segment connects the last drawn-to vertex and the first
>drawn-from vertex.
>So, at game's end, we have a closed path of line-segments which visit
>every vertex of the n-gon exactly once.
>
>Scoring: One player gets scored per round.
>The player whose turn it is to score gets points as follows:
>
>When a line-segment intersects another segment, the intersected
>segments are divided into sub-segments; the sub-segments go from a
>segment's intersection with another segment or from the endpoint of
>the segment (on a vertex of the n-gon)
>to another (the nearest) intersection of the segment with another
>segment or to the endpoint of the segment.
>(So, for example, if we have a pentagon with its segments forming
>a 5-pointed star, this particular game has 15 sub-segments.)
>Now the score is *product* of the lengths of all the sub-segments,
>each length rounded up to the nearest muliple of the measuring unit
>(such as millimeters) being used.
>(I have the round-up rule to avoid ambiguity when, say, 3 segments
>cross in such a way as to form a small triangle. Is this one
>intersection of 3 segments or 3 intersections of 2 crossing segments
>each? By rounding up, if the sides of the small triangle are less
>than the measuring unit in length, then each small segment just
>multiplies the score by 1.)
>
>When a round is over, the players switch the order in which they move
>and then draw another n-gon as above, and then draw the paths as
above.
>But in the second round the other player gets the score.
>
>Highest total score from the 2 rounds wins.
>
>What strategies should player of this game use?
>
>thanks,
>Leroy Quet
Actually, this is a pretty dumb game as written, at least when compared
with some of my other games.
It (the obvious) has been pointed out to me that the player who is
scoring will want to draw
his sides of the n-gon as big as possible, while the player on defense
will want to draw her sides of the n-gon a small as possible.
A possible solution, use the same n-gon over again for the second round
by copying the first
n-gon.
Another possible solution, draw both n-gons at the beginning
of the game, *then* randomly assign which is for the first round
and which is for the second round.
thanks,
Leroy Quet
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