# Multiplying English numbers is strange too

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.

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!

• I'm curious about the difference between the english and american solution (ie which is which and what makes one english and one american - any thoughts on that? Great answer though. I was trying to work this out in my head and kept getting lost in circles. :) Mar 16, 2021 at 10:58
• @Chris yes, I'm not sure about this part, I assumed it was more for flavour. One thought that could work - Americans tend to use 'z' in places where British would use 's' in some words so maybe the solution with the '5' (looks like an 's') is British and the one with a '2' in the same place (looks like 'z') is American. Mar 16, 2021 at 11:07