Math

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a rectangle of integer dimensions is filed with 1x1 tiles. Each tile that touches the perimeter is colored red, while all the "interior" tiles are left white. Let R be the number of red tiles and W the number of whites tiles

find R and W for a 3x10 rectangle

find any and all rectangles for which R and W are equal

Find any and all rectanglef for which R is twice the size of W

Find any and all rectanlges for which E is twice the size of R

  • Math -

    for the 3x10
    total squares = 30
    inner W's = 1x8 = 8
    Reds = 30-8 = 12

    In general
    let width be x, and length be y
    total squares = xy
    W = (x-2)(y-2) = xy -2x -2y + 4
    then R = xy - (xy - 2x - 2y +4)
    = 2x + 2y - 4

    for R = W
    2x+2y-4 = xy - 2x - 2y + 4
    4x+4y-8=xy
    4y - xy = 8-4x
    y(4-x) = 8-4x
    y = (8-4x)/(4-x) = 4(x-2)/(x-4)
    so pick any value of x , x>4 , and y an integer.
    e.g. x = 8, then y = 6
    x=12, y= 5
    x = 60 , y ≠ integer
    since the function approaches 4 as x gets larger, and we are already at y = 5 when x=12
    the only two cases are :
    8 by 6 or 12 by 5


    for R = 2W , try it yourself
    for the last one, I don't know what E is . Did you mean W ?

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