Permudrome -- Grid Game

Here is a game for 2 players.
The game's name is a combination of the words "permutation" and "palindrome".

Start with an n-by-n grid,
where n is a multiple of 4.
I suggest that n is >= 8.

The players take turns. On a turn a player draws two x's into the grid, each x into an empty square such that no column or row has more than one x.

After there is exactly one x in each row and in each column -- n x's total, n/4 moves for each player -- play is over.

Write down the (n-1) absolute values in order, of the changes in the vertical positions of adjacent x's from column to column, along the bottom of the grid.
Write down the (n-1) absolute values in order, of the changes in the horizontal positions of adjacent x's from row to row, along the left side of the grid.

Player 1 gets as a score the length of the largest palindromic subsequence within the sequence of vertical changes written along the bottom of the grid.

Player 2 gets as a score the length of the largest palindromic subsequence within the sequence of horizontal changes written along the left side of the grid.

Largest score wins. (Ties are possible.)

Example: n=12:

. . x . . . . . . . . .
. . . . x . . . . . . .
. . . . . . x . . . . .
x . . . . . . . . . . .
. . . . . x . . . . . .
. . . . . . . x . . . .
. x . . . . . . . . . .
. . . x . . . . . . . .
. . . . . . . . x . . .
. . . . . . . . . . . x
. . . . . . . . . . x .
. . . . . . . . . x . .

Changes in vertical positions column to column:
3,6,7,6,3,2,3,3,3,1,1
The largest palindromic subsequence is (3,6,7,6,3). Player 1 gets 5 points.

Changes in horizontal positions row to row:
2,2,6,5,2,6,2,5,3,1,1,
The largest palindromic subsequence is (5,2,6,2,5). Player 2 gets 5 points.

It is a tie.

What about strategies for this game?

Thanks,
Leroy Quet

One x Twice

For 2 players.
Start with an n-by-n grid drawn on paper.

A move consists of both players each secretly picking an integer between 1 and n.
Both numbers are then revealed. An x is then drawn in the grid-square that has the column number of player 1's number, and has the row number of player 2's number.
So, in other words, player 1 picks the horizontal position of the number, and player 2 picks the vertical position.

If the x lands in an empty square, then the game continues.

But, however, the first time an x lands in a square that already has an x, then the game is over. Player 1 wins if this final x was written on an oddly numbered move. Player 2 wins if this x was written on an evenly numbered move.
So, in other words, if there are an odd number of x's at game's end -- and an even number of squares with x's -- then player 1 wins. If there are an even number of x's -- and an odd number of squares with x's -- then player 2 wins.

What kind of strategies will help you win at this game (if you cannot read the other player's mind)?

Thanks,
Leroy Quet

Line & Unobscured Dots Game

Here is an unoriginal game for 2 players.

You need a blank piece of paper and a pen/pencil, maybe 2 pens/pencils of different colors.

To start, someone draw a dot in the middle of the piece of paper. Then each player draws a different dot on the paper. (So, you have 3 dots.)


Thereafter, the players take turns forming a line of connected straight line-segments on the paper, plus drawing dots. On their turn, a player draws a straight line segment from THEIR END (of their color, if the players are using differently colored pens/pencils) of the connected string of line-segments (or from the central dot if this is the player's first time drawing a line-segment during the game) to any undrawn-to dot (a dot without a line-segment connected to it), such that the line-segment doesn't pass through any other line-segments or through any other dots along the way.

Then, on the same move, the player draws 2 dots, neither on a line or on another dot. One dot is "visible" by the player's own end of the line of connected line-segments. The other dot is visible by the other player's end of the line of connected line-segments.

By 2 points being "visible" to each other, it is meant that it is possible to draw a straight line-segment between the two points, and that line-segment doesn't pass through any intervening dots or lines.

After a fixed number of moves, the same number of moves for both players, the game is over.

The winner has, at game's end, the most number of undrawn-to dots visible from their end-point of the line of connected line-segments.

Note: If playing with 2 differently colored pens/pencils, it doesn't matter what color the dots are. The only reason for using 2 different colors is to make it easier to see whose end of the line of connected line-segments is whose.

Thanks,
Leroy Quet