## Saturday, September 18, 2010

### Lengths Of Lengths Of Lengths Game

This game is for any plural number of players.

There are a predetermined number of rounds. Each player plays the same number of rounds as the offense player.

In a round: A list is made of 0's and 1's, starting at empty-set. The players take turns each appending either a 0 or a 1 to the end (right side) of the list. The total number of digits in the round's list is n, where n is a predetermined number that is a multiple of the number of players.
(n is the same for all rounds.)

After the list of 0's or 1's is made, we determine the score.

We now form a series of lists.

(*) If the new list consists entirely of 1's, then the offense player gets the number of 1's added to his/her score. And the round is over. (If the round is over, then go to **.)

If there is at least one number not equal to 1 in the latest list: Below the last list made, write a new list consisting, in order, of the lengths of the runs of similarly-valued numbers from the previous list.

Go to (*).

(**) Change who is the offense player. Start a new round.

After all rounds have been played, the player with the largest score wins.

Sample round: n = 25:

1010110011101000111001011

1,1,1,1,2,2,3,1,1,3,3,2,1,1,2

4, 2,1,2,2,1,2,1

1,1,1,2,1,1,1

3,1,3

1,1,1

Offense gets 3 points.

Thanks,
Leroy Quet

## Wednesday, September 8, 2010

### Plusses And Minuses Grid Game

This game is for two players.

Start with an n-by-n grid drawn on paper. I suggest that n be an odd integer >= 9.

The first part of the game doesn't involve the grid. In this part of the game, the players take turns each contributing to their own "prediction list" of +'s and -'a. On each move, a player appends to the end of their list either a + or a -. Each player is aware of the other player's list as it is being made. After both players' lists are n-1 symbols long, this part of the game is over. (See below for the significance of the lists.)

I suggest at this point that a dot be put at the lower left corner of the grid so as to keep this corner straight from the others.

In the next part of the game, the players take turns, each move filling in an empty square of the grid. The filled in square must not be in the same row or column as any other filled in square.

(I suggest that the player who moved first in the first part of the game moves second in the second part of the game. Just to be fair.)

After n squares total have been filled in, this part of the game is over.
(There will be exactly one filled in square in each row and in each column.)

Now, we determine the scores.
Take the grid, with the dot in the lower left corner.
Going from the left to the right, the filled in squares represent a permutation P = (p(1),p(2),...p(n)) of (1,2,3,...,n). Player 1 forms a "truth list" of +'s and -'s, where the kth symbol is the sign of p(k+1)-p(k).

Going from bottom to the top, the filled in squares represent a permutation Q = (q(1),q(2),...q(n)) of (1,2,3,...,n), where Q is the inverse permutation of P. Player 2 forms a truth list of +'s and -'s, where the kth symbol is the sign of q(k+1)-q(k).

(The lower-left square represents 1 both in respect to permutation P and permutation Q. And the upper right square represents n for both permutations.)

Write each players truth list below their prediction list so that respective signs are lined up. Each player gets a point for each corresponding pair of signs that match.

Largest score wins.

Here is a sample game:

n = 8.

Grid:
. * . . . . . .
. . . . . . . *
. . . . . . * .
. . . . . * . .
. . * . . . . .
. . . . * . . .
. . . * . . . .
* . . . . . . .

Player 1's prediction list:
+ - + - + - +
Player 1's truth list:
+ - - + + + +
The first pair, second pair, 5th pair, and 7th pair match. So, player 1 gets 4 points.

Player 2's prediction list:
+ + - + + - +
Player 2's truth list:
+ + - + + + -
The first pair, second pair, 3rd pair, 4th pair, and 5th pair match. So, player 2 gets 5 points.

Player 2 wins.

Thanks,
Leroy Quet