a) W must be three or higher, in order that the result is 6 digits
b) N must be 0, 1, 5, or 6 in order that N x N ends in N
c) Now we need IN X IN to end in IN.
Obviously, N = 0 doesn't work,
because that would make IN = 00
(10*I + N)^2 = (10*I + N) mod 100
20*I*N + N^2 = (10*I + N) mod 100
If N = 1, then 10*I = 0 mod 100 so I = 0
If N = 5, then 25 = 10*I + 5 mod 100 so I = 2
If N = 6, then 120*I + 36 = 10*I + 6 mod 100
10*I + 30 = 0 mod 100 so I = 7
d) So, the only possibilities are
301 (too low)
401
601
701
801
901
325
425
625
725
825
925
376
476
576
876
976
e) By brute force multiplication, only 425 * 425 = 180625 works
Using a little brute force (spoiler)