Thursday, February 25, 2010

Arcs And Marks

Needed: blank paper, pencil and compass (the circle-drawing kind), maybe a protractor.

First, draw a circle on a piece of paper, relatively large.

Let m be a multiple of the number of players playing this game. m should be >= 8, at least, I suggest.

Draw m pencil-marks EVENLY SPACED along the circumference of the circle. (This is why you might need a protractor, if you can't just use the compass and straightedge to accomplish this by geometric construction.)

There will be m moves in the game (or in the round).

Players alternately take turns drawing arcs, drawing one arc with the compass on each move. Each arc must be drawn within the circle's interior from the circle's edge back to the circle's edge. On a move, a player uses the appropriate mark as the center of the circular arc that passes through two other marks. The two marks (which the arc intersects the main circle at) must be equidistant from the arc's center mark, and that distant must be nonzero.
The first player starts at any mark to make the center of his arc; and on each move after the first, the moving player uses for her arc's center the mark immediately clockwise from the mark used for the previous player's arc-center.

Count the number of preexisting arcs (not including the "arc" of the main circle) that the moving player's arc intersects or touches. Each player has a running-total of the number of arcs his/her arcs passed through.

The winner has the FEWEST total number of earlier-drawn arcs (drawn by any player) intersected or touched by their arcs.

Notes: An arc may touch/intersect other single arcs more than once each, but each such incident only counts once towards the number of arcs intersected or touched.

Also, multiple arcs touching at a point are counted as touching all other arcs coming together at that point, no matter which arcs are "behind" or "ahead of" which other arcs.

There seems to be an advantage for who moves first, and a disadvantage for who moves last. So, maybe multiple rounds should be played, a different round where each player is the first player to move. (Draw a new circle with the same number of marks each round.) Then the total scores for the rounds are added up for the grand score for each player. Players try to minimize their grand scores, of course.

Leroy Quet

Tuesday, February 16, 2010

An Integer Sequence Game

This is a game for any number of players.

Needed: Pencil/pen, paper, calculator (with long display) perhaps. (Maybe this game could be played via a computer running the appropriate program.)

Start by writing down the integers 1, 2, 3,..., n, where n is at least 8 or more if the number of players is 2, I suggest. n is larger if there are more than 2 players.
This list of integers is called the "r-list".

The variable m starts the game with the value 1. In other words, m(0) = 1.

Players take turns. On the kth move (the kth move among all players together), the moving player lets r(k) = any uncircled integer from the r-list.
The player then circles that number.

m(k) is the value of m after the kth move.
Let m(k) =
r(k)*m(k-1) + (number of composites among m(0),m(1),m(2),...,m(k-1)).

Add to the moving player's score the largest value from m(0),m(1),m(2),...m(k-1) that divides m(k).

The move is complete when the moving player writes down m(k) at the end of the growing list of the values of m.

Players keep taking turns until k = n.


Example game, n = 8: (I may have made a mistake with my math.)
m(0) = 1
r(1) = 2; m(1) = 2*1+0 = 2. (Prime.)
Moving player gets 1 added to score.
r(2) = 8; m(2) = 8*2+0 = 16. (Composite.)
Moving player gets 2 added to score.
r(3) = 3; m(3) = 16*3+1 = 49. (Composite.)
Moving player gets 1 added to score.
r(4) = 5; m(4) = 49*5+2 = 247. (Composite.)
Moving player gets 1 added to score.
r(5) = 1; m(5) = 247*1+3 = 250. (Composite.)
Moving player gets 2 added to score.
r(6) = 4; m(6) = 250*4+4 = 1004. (Composite.)
Moving player gets 2 added to score.
r(7) = 6; m(7) = 1004*6+5 = 6029. (Prime)
Moving player gets 1 added to score.
r(8) = 7; m(8) = 6029*7+5 = 42208. (Composite, but this does not matter.)
Moving player gets 16 added to score.


How does the sequence {a(k)} begin, letting a(n) = the largest possible score for a 1-person game where the r-list contains the first n positive integers?

Leroy Quet

Monday, February 8, 2010

Labyrinthine Loop

Here is an (unoriginal) game for any plural number of players.

The game consists of rounds, where every player is the "offense-player" the same predetermined number of rounds.

At the beginning of each round, draw an array of dots (vertices of a grid) on a piece of paper, n rows of dots by n columns. (n is a predetermined integer, which is the same for all rounds. I suggest n be >= 6.)
There is a new array of dots for each round.

At the beginning of the round, the offense-player moves first, connecting any pair of adjacent dots with a straight line-segment. (By "adjacent", one dot must be one of the closest dots to the other dot, and in the direction of E, SE, S, SW, W, NW, N, or NE.)

Players thereafter continue to take turns. On a turn, a player connects (with a straight line-segment) any dot that has AT LEAST ONE line-segment connected to it already, to any ADJACENT dot that has NO line-segments connected to it.
Again: By "adjacent dots", it is meant that one dot is one of the closest to a second dot, where the two dots are in the direction of either E, SE, S, SW, W, NW, N, or NE to each other.

Diagonal line-segments MAY cross each other.

Players continue taking turns until all dots have line-segments connecting to them.
(ie. Players continue taking turns until a total of n^2 -1 line segments are drawn in a round.)

Then, lastly in the round, the offense-player connects any pair of unconnected adjacent dots with a line-segment.

The offense-player's score for that round is the number of dots in the (single) closed loop of line-segments (including the line-segment the offense-player just drew).
(The "loop" is the simple path from dot to dot that connects back to its starting point. The loop does not including dead-ends, of course.)

After each player has played offense the same predetermined number of rounds, then each player adds up his/her scores from those rounds she/he was offense to get her/his grand total. The player with the highest grand total wins.

PS: It should be noted that when determining the shape of the loop created in the final step of a round, each "line-segment" in the final loop goes strictly from a dot to another dot. When two diagonal line-segments cross, they are considered, for our purposes, to not be touching -- one segment goes "over" another. As a result, there is always one and only one loop made each round.

Leroy Quet

Friday, February 5, 2010

Numbers To Number

This is a game for 2 players. Needed: 3 blank pieces of paper, one for each player and one common piece of paper. 2 pens/pencils, one for each player.

Let m be an integer decided on by both players. m should probably be >= 6. Players also decide who is player 1 and who is player 2 at the beginning of the game.

At the beginning of each round, that round's "binary list" is blank. (The binary list is a series of 1's, 0's and _'s written on the common piece of paper, one new binary list for each round.)

On a turn, both players secretly write down on their piece of paper any number from 1 to 2^m -1 that has yet to be written down by that player in the game (in any previous round or in the current round).

The two players' numbers are then revealed to each other.

If player 1's number is > player 2's number, then append a 1 to the right side of that round's binary list.

If player 1's number is < player 2's number, then append a 0 to the right side of that round's binary list.

If player 1's number is = player 2's number, then append a _ (underscore) to the right side of that round's binary list.

After m moves (where one move is both players moving simultaneously) have passed in the round, the round is over.
Treat the binary list as a set of binary numbers, with each _ treated as either a 0 or a 1. Convert each binary integer to decimal.
So, the binary list represents 2^(number of _'s) different integers.
Let the set of decimal integers for that round be D.

So, for instance, if the binary list (m=8) looks like this:


then D for that round contains:
69, 71, 101, and 103.

Each player gets a point for every element of D that was an integer played by the player IN THAT ROUND.

After a round is over, the player crosses off all the numbers played by that player in the round (so that the player can tell the difference between numbers played in that round and numbers player in earlier rounds).

A game consists of floor((2^m -1) /m) rounds.

Add up scores from all rounds to get each player's grand score.
Highest grand score wins.

Leroy Quet