Lets consider a symmetric about 0 partition. Let's assign the interval
immediately to the right of zero to B (Black)=and the one immediately to
the left of zero to W (White). Lets assign the other intervals
alternatively to B and W. As a result, for a W segment to the right of
zero the symmetric about zero segment will be assigned to B and vice
versa for a B segment to the right of zero. The integral of "x^2" (which
is symmetric about zero) over this partition will therefore be equal for
B and W. So also for the even constant function "b". For the integral of
"x" over these partitions:
lets choose the special partition [-1,-c,0,c,1]. Lets assign [0,c] to B
and [c,1] to W. If we require integral of "ax" to be the same for both we
need to choose a(c^2-0)=a(1-c^2) which gives c=1/sqrt(2). So a partition
with the required property is [-1,-1/sqrt(2),0,1/sqrt(2),1]
with alternating intervals assigned to black and white.