Consider the game of up-and-down rooks played on a standard chess- board. player i has a set of white rooks initially located in the bottom row, while player ii has a set of black rooks in the top row. in each turn, a player moves one of its rooks up or down a column, without skipping over the other rook or occupying its position. the first player who cannot move loses. this game is not progressively bounded, yet an optimal strategy exists. find such a strategy by relating this game to a nim position with 8 piles.

the thermometer is precise but not accurate

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for the friend request.

when a figure, such as triangle (xyz) is reflected over the line y = x to create x'y'z' the x-coordinates and y-coordinates of the original image (pre-image) will be reversed for the image. (e.g., (x, y) > (y, x)) this means that the pre-image position of the x- and y-coordinates switch places which in turn provides a mirror like image of the original xyz.  

i notice that when i draw a line segment from x to x', that the points are parallel from each other -- they are exactly across from each other and do not intersect with any other points.  

yes, think i would see the same characteristic if you drew the line segment connecting y with the reflecting line and then y' with the reflecting line because of the essence of reflections. because it is a mirror image, each point was effected the same way which leads me to conclude that they same would apply to y and y'.

r = 78539458390583095803458309458039485093485 or just 3

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b

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