Circle Intersection Game

Here is a game for any plural number of players.

Needed: Blank piece of paper, compass.

First, with the compass, draw two circles of the same radius, with each circle passing through the other's center.

Players take turns. On a player's turn, he/she draws a circle (with the compass, of course) of any positive radius such that the center of the circle is at the intersection of at least two previously-drawn circles. The number of intersections passed through by this new circle is added to the player's score -- where each intersection is of at least two previously drawn circles and of the circle just drawn by the player, for a total of AT LEAST THREE circles per intersection (including the circle just drawn by the player). Note: The number of intersections (each of any number of circles), not the total number of circles in these intersections, is added to the player's score.

No two circles may have both the same radius and same center. (No two circles can coincide completely.)

The first player to reach a predetermined number of points wins.
(I suggest a higher goal score for a larger piece of paper and larger initial circles. I suggest a lower goal score if there a more than just a couple players.)

Note: If 3 or more circles ALMOST intersect at a point -- but their exact point of coinciding is uncertain because the circles were poorly drawn -- use geometric theorems to determine if indeed the three or more circles coincide *officially* at that point.


Thanks,
Leroy Quet

Lines-From-Lines Game

Sorry if this is unoriginal.

Here is a game for two players. It is played on an n-by-n grid drawn (practically perfectly) on paper. I suggest that n be at least 8.

Note: (The lines of the grid do not come into play here. Only the "grid-vertices" {where the lines of THE GRID intersect} are important as far as the grid is concerned.)
By "lines" below, I am referring to straight line-segments drawn by players during play. I suggest these lines be drawn with a straight-edge.

The first player to move draws a line from any grid-vertex to any other grid-vertex.

The players thereafter take turns drawing lines, one line per move, such that:
* each line starts at a grid-vertex intersected by any previously drawn line (drawn by either player).
* each line ends at any grid-vertex not yet touched by a line.
* no line crosses another line or coincides with another line.
* no line starts/ends at or crosses another line's end/start-point (whether or not yet another line passes completely through that vertex).


If you can move, you must. (The other player can help you find allowable moves.)

The LAST player to be able to move LOSES.

And let me clarify things, in case I am a bad writer.
A line may PASS THROUGH a vertex at most once.
A line may START OR END at a vertex at most once.

The same vertex may have one line passing through it and another
single line terminating (ending or starting) there.



There must be an easy strategy to always win. Anyone know of one?


Thanks,
Leroy Quet