A 40 X 40 white sware is divided into 1 X 1 squares by lines parallel to its sides.Some of these 1 X 1 squares are coloured red so that...

I began by drawing a 10x10 white square and divided it
into 1x1 squares.  This was much more manageable than a 40x40
square.

Going from left to
right:

a column of 10 white

a
column of 2 red, 2 white, 2 red, 2 white, 2 red

a column of
10 white

a column of 2 white, 2 red, 2 white, 2 red, 2
white

a column of 10 white

a
column of 2 red, 2 white, 2 red, 2 white, 2 red

a column of
10 white

a column of 2 white, 2 red, 2 white, 2 red, 2
white

a column of 10 white

a
column of 2 red, 2 white, 2 red, 2 white, 2 red

This made
26 red small squares in the 10x10 square.  Multiply that by 4, and you get 104 red small
squares in the 40x40 square.

The largest
possible number of red squares is 104.