Permutations Of +- Integers

This is a game for two players, player 1 and player 2.

There are two lists of numbers being maintained, list A and list B.

Both lists start the game without any integers.

On move number n*, player 1 picks the sign (+ or -) for the integer, which has an absolute value of n, for list B.
And on move number n, player 2 picks the sign (+ or -) for the integer, which has an absolute value of n, for list A.

Then player 1 puts the integer (+ or - n), which had its sign picked by player 2, somewhere among the integers already in list A: either between integers already there, or at the beginning or end of the list of integers.
Then player 2 puts the integer (+ or - n), which had its sign picked by player 1, somewhere among the integers already in list B.

So, the each list, on move n, is a permutation of the integers
(+-1,+-2,+-3,...,+-n). On the nth move, the integers +-1, +-2, +-3,...+-(n-1) each remain in the same order relative to each other, but the +-n is inserted somewhere in each list.

*(A move consists of a series of steps, these being: Each player picking a sign for an integer, then each player placing an integer in a list.)

After a predetermined number of moves, the game is over. (Let the number of total moves be the positive integer m.)

Scoring is as follows:
Say, list A is the permutation (a(1),a(2),a(3),..,a(m)) of (+-1,+-2,+-3,...,+-m).
Let A(n) = sum{k=1 to n}a(k) be a partial sum of the first n terms of the permutation which is list A.
Player 2 (note: player 2) gets a point for every distinct k where |A(k)| = at least one |A(j)|, for 1<= j < k <= m, or where A(k) = 0.

Say, list B is the permutation (b(1),b(2),b(3),..,b(m)) of (+-1,+-2,+-3,...,+-m).
Let B(n) = sum{k=1 to n}b(k) be a partial sum of the first n terms of the permutation which is list B.
Player 1 (note: player 1) gets a point for every distinct k where |B(k)| = at least one |B(j)|, for 1<= j < k <= m, or where B(k) = 0.

Math question: How does the integer sequence {c(k)} start where c(n) = the number of permutations of (+ or - 1, + or - 2, + or - 3,...,+ or - n), where no partial sum of the first k terms of the sequence, 1 <= k <= n, equals any partial sum of the first j terms of the sequence, for all k and j where 0<= j < k <=n, no matter what the signs of the integers are?
(What are the permutations that gain 0 points for the player picking the signs, no matter how they pick the signs?)

Thanks,
Leroy Quet

Making Intersections

This is a game for any plural number of players.

The game consists of a number of rounds, that number being a multiple of the number of players.
Each player takes turns being the offense player.

Before each round, draw an array of n-by-n dots (which correspond to the vertices of a grid of (n-1)-by-(n-1) squares).

On each round, all players take turns drawing straight vertical or horizontal line-segments from any dot to any other in the same row or column such that no line-segments cross any previously drawn segments and such that no line-segment coincides with any previously drawn line-segment. (But any line-segment may connect with a previously-drawn segment at a dot.)

The round ends when the players have drawn m line-segments, where m is a predetermined multiple of the number of players, and is picked by agreement among the players -- and m is the same for every round.
Note: m should be < n^2 - 2n + 4 = (n-1)^2 +3, which is the minimum number of segments needed to connect every dot to each of its orthogonal neighbors. (That this is the minimum number of segments is simply proved; and proving this may be a fun exercise for your amusement, if you don't want too hard a puzzle.)

After m line-segments are drawn, the offense player for the round receives a point for every dot connected to by 3 or 4 line-segments. (For the purpose of counting points, a single line-segment drawn by a player on a move and passing through -- and not terminating at-- a dot counts as 2 "line-segments" meeting at that dot.)

Players add up the points they received on the rounds that each was offense, and the highest total score wins.


Thanks,
Leroy Quet

PS: If this game is played on a rectangular array of j by k dots, the minimum number of straight horizontal and vertical line-segments (each of any length) needed to connect every dot to its orthogonal neighbors, without any line-segments crossing, is:
j*k - k - j + 4, or so I think.

Number Superposition Game

Here is a simple game for any plural number of players.

This game is played on an m-by-m grid drawn on paper.


The first player to move places a 1 in any square of the grid.

Players thereafter take turns, each player placing the integer k in a square adjacent to (and in the direction of above, below, right of, or left of -- with restrictions {see below}) the square where the previous player to move wrote the integer (k-1). (k increases: 1,2,3,4,5,...)

A player may put the integer in an empty square OR may write the integer in a square that already has a single number in it -- but may not write a number in a square that already has two numbers written in it.

A player may also not write a number in the square previously written in by the player who moved before the last player to move.
(In other words: No U-turns. Any integer k cannot be placed in the same square as the integer (k-2).)

A player who writes the second number in a square gets {the absolute difference between the two numbers in the square} added to his/her score.

The game continues until no more moves can be made.

Highest score wins, of course.

Thanks,
Leroy Quet