Multiplying English numbers is strange too

Question

It seems that the English-speaking people multiply the numbers in the very same way that the Portuguese add them: while the latter say that 2+2=8, the former claim that 2×2 is 8:

     T W O
     T W O
× ________
 E I G H T

(Same letters mean the same digits, and the different ones are different.)

P.S. As usual, British and Americans do that in slightly different way (which is true for many things). So, there are 2 (slightly) different solutions.

Answer 1

The two solutions are

$179 \times 179 = 32041$
$189 \times 189 = 35721$

Proof

First of all given that $\sqrt{100000} < 317$, we find that $TWO < 317$ and, in particular, $T$ must be $1,2$ or $3$ (assuming no leading zeroes).
Also, since $T$ is the remainder of a square number divided by $10$, it can only be $0,1,4,5,6,9$ so it must be that $T=1$.
Since $O^2$ must end in $1$, it can only be $1$ or $9$ and since $O\neq T$, it must be that $O=9$.
This actually just leaves us with $8$ options to try ($W=0,2,3,4,5,6,7,8$) so we can test them all by hand and it only requires that the remaining unidentified are distinct and different from $1$ and $9$.
Also, since $E>1$, we only have to try $W=4,5,6,7,8$ as $\sqrt{20000} > 140$.
$149 \times 149 = 22201$ which has double $2$s so this doesn't work.
$159 \times 159 = 25281$ which also has double $2$s.
$169 \times 169 = 28561$ which puts $W=H$ so this doesn't work.
$179 \times 179 = 32041$ and this works!
$189 \times 189 = 35721$ and this also works!