Question

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

Answer 1

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 ?