Math
posted by Jennifer on .
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

for the 3x10
total squares = 30
inner W's = 1x8 = 8
Reds = 308 = 12
In general
let width be x, and length be y
total squares = xy
W = (x2)(y2) = xy 2x 2y + 4
then R = xy  (xy  2x  2y +4)
= 2x + 2y  4
for R = W
2x+2y4 = xy  2x  2y + 4
4x+4y8=xy
4y  xy = 84x
y(4x) = 84x
y = (84x)/(4x) = 4(x2)/(x4)
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 ?