# he and she - er und sie

The formula $$HE=\sqrt{SHE}$$ translates in German to $$ER=\sqrt{SIE}$$. Find the solution for both, where each letter represents a digit.

(These are two separate puzzles: digits represented by S and E don't have to be the same between languages.)

If $$HE^2=SHE$$ then
$$HE$$ (call it $$x$$ for convenience) has the property that $$x=x^2$$ mod 100. Hence $$x$$ is either 0 or 1 mod each of 4,25. We can't have either 0,0 or 1,1 because then we actually have $$x^2=x$$ which would mean $$S=0$$ (and also $$H=0$$). 0,1 yields $$HE=76$$ whose square is clearly too big. $$1,0$$ yields $$HE=25$$ which works ($$SHE=625$$).
If $$ER^2=SIE$$ then
clearly $$0 (else the square has too many digits) and $$E$$ is the last digit of a square so $$E=1$$. Then $$R$$ must be 1 (no!) or 9, leading to $$ER=19$$ and $$SIE=361$$.